Theorem oawordri 5232

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 C_ B -> (A +o C) C_ (B +o C)))

Proof of Theorem oawordri

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . . 6 |- (x = (/) -> (A +o x) = (A +o (/)))
2		opreq2 4890	. . . . . 6 |- (x = (/) -> (B +o x) = (B +o (/)))
3	1, 2	sseq12d 2646	. . . . 5 |- (x = (/) -> ((A +o x) C_ (B +o x) <-> (A +o (/)) C_ (B +o (/))))
4		opreq2 4890	. . . . . 6 |- (x = y -> (A +o x) = (A +o y))
5		opreq2 4890	. . . . . 6 |- (x = y -> (B +o x) = (B +o y))
6	4, 5	sseq12d 2646	. . . . 5 |- (x = y -> ((A +o x) C_ (B +o x) <-> (A +o y) C_ (B +o y)))
7		opreq2 4890	. . . . . 6 |- (x = suc y -> (A +o x) = (A +o suc y))
8		opreq2 4890	. . . . . 6 |- (x = suc y -> (B +o x) = (B +o suc y))
9	7, 8	sseq12d 2646	. . . . 5 |- (x = suc y -> ((A +o x) C_ (B +o x) <-> (A +o suc y) C_ (B +o suc y)))
10		opreq2 4890	. . . . . 6 |- (x = C -> (A +o x) = (A +o C))
11		opreq2 4890	. . . . . 6 |- (x = C -> (B +o x) = (B +o C))
12	10, 11	sseq12d 2646	. . . . 5 |- (x = C -> ((A +o x) C_ (B +o x) <-> (A +o C) C_ (B +o C)))
13		oa0 5200	. . . . . . . 8 |- (A e. On -> (A +o (/)) = A)
14	13	adantr 425	. . . . . . 7 |- ((A e. On /\ B e. On) -> (A +o (/)) = A)
15		oa0 5200	. . . . . . . 8 |- (B e. On -> (B +o (/)) = B)
16	15	adantl 424	. . . . . . 7 |- ((A e. On /\ B e. On) -> (B +o (/)) = B)
17	14, 16	sseq12d 2646	. . . . . 6 |- ((A e. On /\ B e. On) -> ((A +o (/)) C_ (B +o (/)) <-> A C_ B))
18	17	biimpar 461	. . . . 5 |- (((A e. On /\ B e. On) /\ A C_ B) -> (A +o (/)) C_ (B +o (/)))
19		ordsucsssuc 3904	. . . . . . . . . . 11 |- ((Ord (A +o y) /\ Ord (B +o y)) -> ((A +o y) C_ (B +o y) <-> suc (A +o y) C_ suc (B +o y)))
20		oacl 5215	. . . . . . . . . . . 12 |- ((A e. On /\ y e. On) -> (A +o y) e. On)
21		eloni 3667	. . . . . . . . . . . 12 |- ((A +o y) e. On -> Ord (A +o y))
22	20, 21	syl 12	. . . . . . . . . . 11 |- ((A e. On /\ y e. On) -> Ord (A +o y))
23		oacl 5215	. . . . . . . . . . . 12 |- ((B e. On /\ y e. On) -> (B +o y) e. On)
24		eloni 3667	. . . . . . . . . . . 12 |- ((B +o y) e. On -> Ord (B +o y))
25	23, 24	syl 12	. . . . . . . . . . 11 |- ((B e. On /\ y e. On) -> Ord (B +o y))
26	19, 22, 25	syl2an 503	. . . . . . . . . 10 |- (((A e. On /\ y e. On) /\ (B e. On /\ y e. On)) -> ((A +o y) C_ (B +o y) <-> suc (A +o y) C_ suc (B +o y)))
27	26	anandirs 571	. . . . . . . . 9 |- (((A e. On /\ B e. On) /\ y e. On) -> ((A +o y) C_ (B +o y) <-> suc (A +o y) C_ suc (B +o y)))
28		oasuc 5208	. . . . . . . . . . 11 |- ((A e. On /\ y e. On) -> (A +o suc y) = suc (A +o y))
29	28	adantlr 429	. . . . . . . . . 10 |- (((A e. On /\ B e. On) /\ y e. On) -> (A +o suc y) = suc (A +o y))
30		oasuc 5208	. . . . . . . . . . 11 |- ((B e. On /\ y e. On) -> (B +o suc y) = suc (B +o y))
31	30	adantll 428	. . . . . . . . . 10 |- (((A e. On /\ B e. On) /\ y e. On) -> (B +o suc y) = suc (B +o y))
32	29, 31	sseq12d 2646	. . . . . . . . 9 |- (((A e. On /\ B e. On) /\ y e. On) -> ((A +o suc y) C_ (B +o suc y) <-> suc (A +o y) C_ suc (B +o y)))
33	27, 32	bitr4d 590	. . . . . . . 8 |- (((A e. On /\ B e. On) /\ y e. On) -> ((A +o y) C_ (B +o y) <-> (A +o suc y) C_ (B +o suc y)))
34	33	biimpd 170	. . . . . . 7 |- (((A e. On /\ B e. On) /\ y e. On) -> ((A +o y) C_ (B +o y) -> (A +o suc y) C_ (B +o suc y)))
35	34	expcom 403	. . . . . 6 |- (y e. On -> ((A e. On /\ B e. On) -> ((A +o y) C_ (B +o y) -> (A +o suc y) C_ (B +o suc y))))
36	35	adantrd 427	. . . . 5 |- (y e. On -> (((A e. On /\ B e. On) /\ A C_ B) -> ((A +o y) C_ (B +o y) -> (A +o suc y) C_ (B +o suc y))))
37		visset 2295	. . . . . . . 8 |- x e. _V
38		oalim 5212	. . . . . . . . . . 11 |- ((A e. On /\ (x e. _V /\ Lim x)) -> (A +o x) = U_y e. x (A +o y))
39	38	adantlr 429	. . . . . . . . . 10 |- (((A e. On /\ B e. On) /\ (x e. _V /\ Lim x)) -> (A +o x) = U_y e. x (A +o y))
40		oalim 5212	. . . . . . . . . . 11 |- ((B e. On /\ (x e. _V /\ Lim x)) -> (B +o x) = U_y e. x (B +o y))
41	40	adantll 428	. . . . . . . . . 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 2646	. . . . . . . . 9 |- (((A e. On /\ B e. On) /\ (x e. _V /\ Lim x)) -> ((A +o x) C_ (B +o x) <-> U_y e. x (A +o y) C_ U_y e. x (B +o y)))
43		ss2iun 3271	. . . . . . . . 9 |- (A.y e. x (A +o y) C_ (B +o y) -> U_y e. x (A +o y) C_ U_y e. x (B +o y))
44	42, 43	syl5bir 227	. . . . . . . 8 |- (((A e. On /\ B e. On) /\ (x e. _V /\ Lim x)) -> (A.y e. x (A +o y) C_ (B +o y) -> (A +o x) C_ (B +o x)))
45	37, 44	mpanr1 774	. . . . . . 7 |- (((A e. On /\ B e. On) /\ Lim x) -> (A.y e. x (A +o y) C_ (B +o y) -> (A +o x) C_ (B +o x)))
46	45	expcom 403	. . . . . 6 |- (Lim x -> ((A e. On /\ B e. On) -> (A.y e. x (A +o y) C_ (B +o y) -> (A +o x) C_ (B +o x))))
47	46	adantrd 427	. . . . 5 |- (Lim x -> (((A e. On /\ B e. On) /\ A C_ B) -> (A.y e. x (A +o y) C_ (B +o y) -> (A +o x) C_ (B +o x))))
48	3, 6, 9, 12, 18, 36, 47	tfinds3 3948	. . . 4 |- (C e. On -> (((A e. On /\ B e. On) /\ A C_ B) -> (A +o C) C_ (B +o C)))
49	48	exp4c 411	. . 3 |- (C e. On -> (A e. On -> (B e. On -> (A C_ B -> (A +o C) C_ (B +o C)))))
50	49	com3l 38	. 2 |- (A e. On -> (B e. On -> (C e. On -> (A C_ B -> (A +o C) C_ (B +o C)))))
51	50	3imp 1061	1 |- ((A e. On /\ B e. On /\ C e. On) -> (A C_ B -> (A +o C) C_ (B +o C)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  (/)c0 2875  U_ciun 3255  Ord word 3656  Oncon0 3657  Lim wlim 3658  suc csuc 3659  (class class class)co 4884   +o coa 5174

This theorem is referenced by:  oaword2 5235  omwordri 5251

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-oadd 5179

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