ordinal arithmetics 2 Let f be a sequence of ordinal numbers and let a be an ordinal number. Observe that f(a) is ordinal-like. Let us consider A, B.

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1 JOURNAL OF FORMALIZED MATHEMATICS Volume 2, Released 1990, Published 2000 Inst. of Computer Science, University of Bia lystok Ordinal Arithmetics Grzegorz Bancerek Warsaw University Bia lystok Summary. At the beginning the article contains some auxiliary theorems concerning the constructors dened in papers [1] and [2]. Next simple properties of addition and multiplication of ordinals are shown, e.g. associativity of addition. Addition and multiplication of a transnite sequence of ordinals and a ordinal are also introduced here. The goal of the article is the proof that the distributivity of multiplication wrt addition and the associativity of multiplication hold. Additionally new binary functors of ordinals are introduced: subtraction, exact division, and remainder and some of their basic properties are presented. MML Identier: ORDINAL3. WWW: The articles [5], [7], [6], [8], [3], [1], [4], and [2] provide the notation and terminology for this paper. We adopt the following convention: f1, p1 denote sequences of ordinal numbers, A, B, C, D denote ordinal numbers, and X, Y denote sets. Next we state a number of propositions: (1) X succ X: (2) If succ X Y; then X Y: (5) 1 A 2 B i succ A 2 succ B: (6) If X A; then S X is an ordinal number. (7) S On X is an ordinal number. (8) If X A; then On X = X: (9) OnfAg = fag: (10) If A 6= ;; then ; 2 A: (11) inf A = ;: (12) inffag = A: (13) T If X A; then X is an ordinal number. 1 The propositions (3) and (4) have been removed. 1 c Association of Mizar Users

2 ordinal arithmetics 2 Let f be a sequence of ordinal numbers and let a be an ordinal number. Observe that f(a) is ordinal-like. Let us consider A, B. One can verify that A [ B is ordinal-like and A \ B is ordinal-like. The following propositions are true: (15) 2 A [ B = A or A [ B = B: (16) A \ B = A or A \ B = B: (17) If A 2 1; then A = ;: (18) 1 = f;g: (19) If A 1; then A = ; or A = 1: (20) If A B or A 2 B and if C 2 D; then A + C 2 B + D: (21) If A B and C D; then A + C B + D: (22) If A 2 B and if C D and D 6= ; or C 2 D; then A C 2 B D: (23) If A B and C D; then A C B D: (24) If B + C = B + D; then C = D: (25) If B + C 2 B + D; then C 2 D: (26) If B + C B + D; then C D: (27) A A + B and B A + B: (28) If A 2 B; then A 2 B + C and A 2 C + B: (29) If A + B = ;; then A = ; and B = ;: (30) If A B; then there exists C such that B = A + C: (31) If A 2 B; then there exists C such that B = A + C and C 6= ;: (32) If A 6= ; and A is a limit ordinal number, then B + A is a limit ordinal number. (33) (A + B) + C = A + (B + C): (34) If A B = ;; then A = ; or B = ;: (35) If A 2 B and C 6= ;; then A 2 B C and A 2 C B: (36) If B A = C A and A 6= ;; then B = C: (37) If B A 2 C A; then B 2 C: (38) If B A C A and A 6= ;; then B C: (39) If B 6= ;; then A A B and A B A: (41) 3 If A B = 1; then A = 1 and B = 1: (42) If A 2 B + C; then A 2 B or there exists D such that D 2 C and A = B + D: Let us consider C, f1. The functor C + f1 yields a sequence of ordinal numbers and is dened as follows: 2 The proposition (14) has been removed. 3 The proposition (40) has been removed.

3 ordinal arithmetics 3 (Def. 2) 4 dom(c + f1) = dom f1 and for every A such that A 2 dom f1 holds (C + f1)(a) = C + f1(a): The functor f1 + C yielding a sequence of ordinal numbers is dened by: (Def. 3) dom(f1 + C) = dom f1 and for every A such that A 2 dom f1 holds (f1 + C)(A) = f1(a) + C: The functor C f1 yielding a sequence of ordinal numbers is dened as follows: (Def. 4) dom(c f1) = dom f1 and for every A such that A 2 dom f1 holds (C f1)(a) = C f1(a): The functor f1 C yielding a sequence of ordinal numbers is dened by: (Def. 5) dom(f1 C) = dom f1 and for every A such that A 2 dom f1 holds (f1 C)(A) = f1(a) C: We now state a number of propositions: (47) 5 If ; 6= dom f1 and dom f1 = dom p1 and for all A, B such that A 2 dom f1 and B = f1(a) holds p1(a) = C + B; then sup p1 = C + sup f1: (48) If A is a limit ordinal number, then A B is a limit ordinal number. (49) If A 2 B C and B is a limit ordinal number, then there exists D such that D 2 B and A 2 D C: (50) Suppose ; 6= dom f1 and dom f1 = dom p1 and C 6= ; and sup f1 is a limit ordinal number and for all A, B such that A 2 dom f1 and B = f1(a) holds p1(a) = B C: Then sup p1 = sup f1 C: (51) If ; 6= dom f1; then sup(c + f1) = C + sup f1: (52) If ; 6= dom f1 and C 6= ; and sup f1 is a limit ordinal number, then sup(f1 C) = sup f1 C: (53) If B 6= ;; then S (A + B) = A + S B: (54) (A + B) C = A C + B C: (55) If A 6= ;; then there exist C, D such that B = C A + D and D 2 A: (56) For all ordinal numbers C1, D1, C2, D2 such that C1 A + D1 = C2 A + D2 and D1 2 A and D2 2 A holds C1 = C2 and D1 = D2: (57) Suppose 1 2 B and A 6= ; and A is a limit ordinal number. Let given f1. If dom f1 = A and for every C such that C 2 A holds f1(c) = C B; then AB = sup f1: (58) (A B) C = A (B C): Let us consider A, B. follows: The functor A? B yielding an ordinal number is dened as (Def. 6)(i) A = B + (A? B) if B A; (ii) A? B = ;; otherwise. The functor A B yields an ordinal number and is dened as follows: (Def. 7)(i) There exists C such that A = (A B) B + C and C 2 B if B 6= ;; (ii) A B = ;; otherwise. 4 The denition (Def. 1) has been removed. 5 The propositions (43){(46) have been removed.

4 ordinal arithmetics 4 Let us consider A, B. The functor A mod B yields an ordinal number and is dened as follows: (Def. 8) A mod B = A? (A B) B: One can prove the following propositions: (59) If A B; then B = A + (B? A): (60) If A 2 B; then B = A + (B? A): (61) If A 6 B; then B? A = ;: (62) If B 6= ;; then there exists C such that A = (A B) B + C and C 2 B: (63) A ; = ;: (64) A mod B = A? (A B) B: (65) (A + B)? A = B: (66) If A 2 B and if C A or C 2 A; then A? C 2 B? C: (67) A? A = ;: (68) If A 2 B; then B? A 6= ; and ; 2 B? A: (69) A? ; = A and ;? A = ;: (70) A? (B + C) = A? B? C: (71) If A B; then C? B C? A: (72) If A B; then A? C B? C: (73) If C 6= ; and A 2 B + C; then A? B 2 C: (74) If A + B 2 C; then B 2 C? A: (75) A B + (A? B): (76) A C? B C = (A? B) C: (77) (A B) B A: (78) A = (A B) B + (A mod B): (79) If A = B C + D and D 2 C; then B = A C and D = A mod C: (80) If A 2 B C; then A C 2 B and A mod C 2 C: (81) If B 6= ;; then A B B = A: (82) A B mod B = ;: (83) ; A = ; and ; mod A = ; and A mod ; = A: (84) A 1 = A and A mod 1 = ;:

5 ordinal arithmetics 5 References [1] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, JFM/Vol1/ordinal1.html. [2] Grzegorz Bancerek. Sequences of ordinal numbers. Journal of Formalized Mathematics, 1, http: //mizar.org/jfm/vol1/ordinal2.html. [3] Czes law Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, http: //mizar.org/jfm/vol1/funct_1.html. [4] Beata Padlewska. Families of sets. Journal of Formalized Mathematics, 1, setfam_1.html. [5] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, [6] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, Vol1/subset_1.html. [7] Zinaida Trybulec and Halina Swieczkowska. Boolean properties of sets. Journal of Formalized Mathematics, 1, [8] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, Received March 1, 1990 Published July 11, 2000