Theorem oawordri 4182

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 3967	. . . . . 6 |- (x = (/) -> (A +o x) = (A +o (/)))
2		opreq2 3967	. . . . . 6 |- (x = (/) -> (B +o x) = (B +o (/)))
3	1, 2	sseq12d 2088	. . . . 5 |- (x = (/) -> ((A +o x) (_ (B +o x) <-> (A +o (/)) (_ (B +o (/))))
4		opreq2 3967	. . . . . 6 |- (x = y -> (A +o x) = (A +o y))
5		opreq2 3967	. . . . . 6 |- (x = y -> (B +o x) = (B +o y))
6	4, 5	sseq12d 2088	. . . . 5 |- (x = y -> ((A +o x) (_ (B +o x) <-> (A +o y) (_ (B +o y)))
7		opreq2 3967	. . . . . 6 |- (x = suc y -> (A +o x) = (A +o suc y))
8		opreq2 3967	. . . . . 6 |- (x = suc y -> (B +o x) = (B +o suc y))
9	7, 8	sseq12d 2088	. . . . 5 |- (x = suc y -> ((A +o x) (_ (B +o x) <-> (A +o suc y) (_ (B +o suc y)))
10		opreq2 3967	. . . . . 6 |- (x = C -> (A +o x) = (A +o C))
11		opreq2 3967	. . . . . 6 |- (x = C -> (B +o x) = (B +o C))
12	10, 11	sseq12d 2088	. . . . 5 |- (x = C -> ((A +o x) (_ (B +o x) <-> (A +o C) (_ (B +o C)))
13		oa0 4153	. . . . . . . 8 |- (A e. On -> (A +o (/)) = A)
14	13	adantr 389	. . . . . . 7 |- ((A e. On /\ B e. On) -> (A +o (/)) = A)
15		oa0 4153	. . . . . . . 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 2088	. . . . . 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 3072	. . . . . . . . . . 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 4168	. . . . . . . . . . . 12 |- ((A e. On /\ y e. On) -> (A +o y) e. On)
21		eloni 2956	. . . . . . . . . . . 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 4168	. . . . . . . . . . . 12 |- ((B e. On /\ y e. On) -> (B +o y) e. On)
24		eloni 2956	. . . . . . . . . . . 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 4161	. . . . . . . . . . 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 4161	. . . . . . . . . . 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 2088	. . . . . . . . 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 531	. . . . . . . 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 1811	. . . . . . . 8 |- x e. V
38		oalim 4165	. . . . . . . . . . 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 4165	. . . . . . . . . . 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 2088	. . . . . . . . 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 2575	. . . . . . . . 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 709	. . . . . . 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 3164	. . . 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 827	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 775   = wceq 956   e. wcel 958  A.wral 1644  Vcvv 1809   (_ wss 2045  (/)c0 2278  U_ciun 2564  Ord word 2945  Oncon0 2946  Lim wlim 2947  suc csuc 2948  (class class class)co 3961   +o coa 4128

This theorem is referenced by:  oaword2 4185  omwordri 4201

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2691  ax-sep 2701  ax-nul 2708  ax-pow 2740  ax-pr 2777  ax-un 2864

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2502  df-iun 2566  df-br 2618  df-opab 2665  df-tr 2679  df-eprel 2830  df-id 2833  df-po 2838  df-so 2848  df-fr 2915  df-we 2932  df-ord 2949  df-on 2950  df-lim 2951  df-suc 2952  df-xp 3182  df-rel 3183  df-cnv 3184  df-co 3185  df-dm 3186  df-rn 3187  df-res 3188  df-ima 3189  df-fun 3190  df-fn 3191  df-fv 3196  df-rdg 3930  df-opr 3963  df-oprab 3964  df-oadd 4133

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