Theorem oawordri 4174

Description: Weak ordering property of ordinal addition. Proposition 8.7 of [TakeutiZaring] p. 59.

Assertion

Ref	Expression
oawordri	|- ((A e. On /\ B e. On /\ C e. On) -> (A (_ B -> (A +o C) (_ (B +o C)))

Proof of Theorem oawordri

Step	Hyp	Ref	Expression
1		opreq2 3960	. . . . . 6 |- (x = (/) -> (A +o x) = (A +o (/)))
2		opreq2 3960	. . . . . 6 |- (x = (/) -> (B +o x) = (B +o (/)))
3	1, 2	sseq12d 2086	. . . . 5 |- (x = (/) -> ((A +o x) (_ (B +o x) <-> (A +o (/)) (_ (B +o (/))))
4		opreq2 3960	. . . . . 6 |- (x = y -> (A +o x) = (A +o y))
5		opreq2 3960	. . . . . 6 |- (x = y -> (B +o x) = (B +o y))
6	4, 5	sseq12d 2086	. . . . 5 |- (x = y -> ((A +o x) (_ (B +o x) <-> (A +o y) (_ (B +o y)))
7		opreq2 3960	. . . . . 6 |- (x = suc y -> (A +o x) = (A +o suc y))
8		opreq2 3960	. . . . . 6 |- (x = suc y -> (B +o x) = (B +o suc y))
9	7, 8	sseq12d 2086	. . . . 5 |- (x = suc y -> ((A +o x) (_ (B +o x) <-> (A +o suc y) (_ (B +o suc y)))
10		opreq2 3960	. . . . . 6 |- (x = C -> (A +o x) = (A +o C))
11		opreq2 3960	. . . . . 6 |- (x = C -> (B +o x) = (B +o C))
12	10, 11	sseq12d 2086	. . . . 5 |- (x = C -> ((A +o x) (_ (B +o x) <-> (A +o C) (_ (B +o C)))
13		oa0 4145	. . . . . . . 8 |- (A e. On -> (A +o (/)) = A)
14	13	adantr 389	. . . . . . 7 |- ((A e. On /\ B e. On) -> (A +o (/)) = A)
15		oa0 4145	. . . . . . . 8 |- (B e. On -> (B +o (/)) = B)
16	15	adantl 388	. . . . . . 7 |- ((A e. On /\ B e. On) -> (B +o (/)) = B)
17	14, 16	sseq12d 2086	. . . . . 6 |- ((A e. On /\ B e. On) -> ((A +o (/)) (_ (B +o (/)) <-> A (_ B))
18	17	biimpar 417	. . . . 5 |- (((A e. On /\ B e. On) /\ A (_ B) -> (A +o (/)) (_ (B +o (/)))
19		ordsucsssuc 3069	. . . . . . . . . . 11 |- ((Ord (A +o y) /\ Ord (B +o y)) -> ((A +o y) (_ (B +o y) <-> suc (A +o y) (_ suc (B +o y)))
20		oacl 4160	. . . . . . . . . . . 12 |- ((A e. On /\ y e. On) -> (A +o y) e. On)
21		eloni 2953	. . . . . . . . . . . 12 |- ((A +o y) e. On -> Ord (A +o y))
22	20, 21	syl 10	. . . . . . . . . . 11 |- ((A e. On /\ y e. On) -> Ord (A +o y))
23		oacl 4160	. . . . . . . . . . . 12 |- ((B e. On /\ y e. On) -> (B +o y) e. On)
24		eloni 2953	. . . . . . . . . . . 12 |- ((B +o y) e. On -> Ord (B +o y))
25	23, 24	syl 10	. . . . . . . . . . 11 |- ((B e. On /\ y e. On) -> Ord (B +o y))
26	19, 22, 25	syl2an 454	. . . . . . . . . 10 |- (((A e. On /\ y e. On) /\ (B e. On /\ y e. On)) -> ((A +o y) (_ (B +o y) <-> suc (A +o y) (_ suc (B +o y)))
27	26	anandirs 513	. . . . . . . . 9 |- (((A e. On /\ B e. On) /\ y e. On) -> ((A +o y) (_ (B +o y) <-> suc (A +o y) (_ suc (B +o y)))
28		oasuc 4153	. . . . . . . . . . 11 |- ((A e. On /\ y e. On) -> (A +o suc y) = suc (A +o y))
29	28	adantlr 393	. . . . . . . . . 10 |- (((A e. On /\ B e. On) /\ y e. On) -> (A +o suc y) = suc (A +o y))
30		oasuc 4153	. . . . . . . . . . 11 |- ((B e. On /\ y e. On) -> (B +o suc y) = suc (B +o y))
31	30	adantll 392	. . . . . . . . . 10 |- (((A e. On /\ B e. On) /\ y e. On) -> (B +o suc y) = suc (B +o y))
32	29, 31	sseq12d 2086	. . . . . . . . 9 |- (((A e. On /\ B e. On) /\ y e. On) -> ((A +o suc y) (_ (B +o suc y) <-> suc (A +o y) (_ suc (B +o y)))
33	27, 32	bitr4d 530	. . . . . . . 8 |- (((A e. On /\ B e. On) /\ y e. On) -> ((A +o y) (_ (B +o y) <-> (A +o suc y) (_ (B +o suc y)))
34	33	biimpd 153	. . . . . . 7 |- (((A e. On /\ B e. On) /\ y e. On) -> ((A +o y) (_ (B +o y) -> (A +o suc y) (_ (B +o suc y)))
35	34	expcom 374	. . . . . 6 |- (y e. On -> ((A e. On /\ B e. On) -> ((A +o y) (_ (B +o y) -> (A +o suc y) (_ (B +o suc y))))
36	35	adantrd 391	. . . . 5 |- (y e. On -> (((A e. On /\ B e. On) /\ A (_ B) -> ((A +o y) (_ (B +o y) -> (A +o suc y) (_ (B +o suc y))))
37		visset 1809	. . . . . . . 8 |- x e. V
38		oalim 4157	. . . . . . . . . . 11 |- ((A e. On /\ (x e. V /\ Lim x)) -> (A +o x) = U_y e. x (A +o y))
39	38	adantlr 393	. . . . . . . . . 10 |- (((A e. On /\ B e. On) /\ (x e. V /\ Lim x)) -> (A +o x) = U_y e. x (A +o y))
40		oalim 4157	. . . . . . . . . . 11 |- ((B e. On /\ (x e. V /\ Lim x)) -> (B +o x) = U_y e. x (B +o y))
41	40	adantll 392	. . . . . . . . . 10 |- (((A e. On /\ B e. On) /\ (x e. V /\ Lim x)) -> (B +o x) = U_y e. x (B +o y))
42	39, 41	sseq12d 2086	. . . . . . . . 9 |- (((A e. On /\ B e. On) /\ (x e. V /\ Lim x)) -> ((A +o x) (_ (B +o x) <-> U_y e. x (A +o y) (_ U_y e. x (B +o y)))
43		ss2iun 2572	. . . . . . . . 9 |- (A.y e. x (A +o y) (_ (B +o y) -> U_y e. x (A +o y) (_ U_y e. x (B +o y))
44	42, 43	syl5bir 210	. . . . . . . 8 |- (((A e. On /\ B e. On) /\ (x e. V /\ Lim x)) -> (A.y e. x (A +o y) (_ (B +o y) -> (A +o x) (_ (B +o x)))
45	37, 44	mpanr1 708	. . . . . . 7 |- (((A e. On /\ B e. On) /\ Lim x) -> (A.y e. x (A +o y) (_ (B +o y) -> (A +o x) (_ (B +o x)))
46	45	expcom 374	. . . . . 6 |- (Lim x -> ((A e. On /\ B e. On) -> (A.y e. x (A +o y) (_ (B +o y) -> (A +o x) (_ (B +o x))))
47	46	adantrd 391	. . . . 5 |- (Lim x -> (((A e. On /\ B e. On) /\ A (_ B) -> (A.y e. x (A +o y) (_ (B +o y) -> (A +o x) (_ (B +o x))))
48	3, 6, 9, 12, 18, 36, 47	tfinds3 3161	. . . 4 |- (C e. On -> (((A e. On /\ B e. On) /\ A (_ B) -> (A +o C) (_ (B +o C)))
49	48	exp4c 380	. . 3 |- (C e. On -> (A e. On -> (B e. On -> (A (_ B -> (A +o C) (_ (B +o C)))))
50	49	com3l 34	. 2 |- (A e. On -> (B e. On -> (C e. On -> (A (_ B -> (A +o C) (_ (B +o C)))))
51	50	3imp 826	1 |- ((A e. On /\ B e. On /\ C e. On) -> (A (_ B -> (A +o C) (_ (B +o C)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  A.wral 1642  Vcvv 1807   (_ wss 2043  (/)c0 2276  U_ciun 2561  Ord word 2942  Oncon0 2943  Lim wlim 2944  suc csuc 2945  (class class class)co 3954   +o coa 4120

This theorem is referenced by:  oaword2 4177  omwordri 4193

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-oadd 4125

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