Theorem oaabs 5309

Description: Ordinal addition absorbs a natural number added to the left of a transfinite number. Proposition 8.10 of [TakeutiZaring] p. 59.

Assertion

Ref	Expression
oaabs	|- (((A e. om /\ B e. On) /\ om C_ B) -> (A +o B) = B)

Proof of Theorem oaabs

Step	Hyp	Ref	Expression
1		ssexg 3457	. . . . . . . 8 |- ((om C_ B /\ B e. On) -> om e. _V)
2	1	ex 402	. . . . . . 7 |- (om C_ B -> (B e. On -> om e. _V))
3		ordom 3960	. . . . . . . 8 |- Ord om
4		elong 3665	. . . . . . . 8 |- (om e. _V -> (om e. On <-> Ord om))
5	3, 4	mpbiri 211	. . . . . . 7 |- (om e. _V -> om e. On)
6	2, 5	syl6com 64	. . . . . 6 |- (B e. On -> (om C_ B -> om e. On))
7	6	imp 377	. . . . 5 |- ((B e. On /\ om C_ B) -> om e. On)
8		opreq2 4890	. . . . . . . . 9 |- (x = om -> (A +o x) = (A +o om))
9		id 73	. . . . . . . . 9 |- (x = om -> x = om)
10	8, 9	eqeq12d 1899	. . . . . . . 8 |- (x = om -> ((A +o x) = x <-> (A +o om) = om))
11	10	imbi2d 674	. . . . . . 7 |- (x = om -> ((A e. om -> (A +o x) = x) <-> (A e. om -> (A +o om) = om)))
12		opreq2 4890	. . . . . . . . 9 |- (x = y -> (A +o x) = (A +o y))
13		id 73	. . . . . . . . 9 |- (x = y -> x = y)
14	12, 13	eqeq12d 1899	. . . . . . . 8 |- (x = y -> ((A +o x) = x <-> (A +o y) = y))
15	14	imbi2d 674	. . . . . . 7 |- (x = y -> ((A e. om -> (A +o x) = x) <-> (A e. om -> (A +o y) = y)))
16		opreq2 4890	. . . . . . . . 9 |- (x = suc y -> (A +o x) = (A +o suc y))
17		id 73	. . . . . . . . 9 |- (x = suc y -> x = suc y)
18	16, 17	eqeq12d 1899	. . . . . . . 8 |- (x = suc y -> ((A +o x) = x <-> (A +o suc y) = suc y))
19	18	imbi2d 674	. . . . . . 7 |- (x = suc y -> ((A e. om -> (A +o x) = x) <-> (A e. om -> (A +o suc y) = suc y)))
20		opreq2 4890	. . . . . . . . 9 |- (x = B -> (A +o x) = (A +o B))
21		id 73	. . . . . . . . 9 |- (x = B -> x = B)
22	20, 21	eqeq12d 1899	. . . . . . . 8 |- (x = B -> ((A +o x) = x <-> (A +o B) = B))
23	22	imbi2d 674	. . . . . . 7 |- (x = B -> ((A e. om -> (A +o x) = x) <-> (A e. om -> (A +o B) = B)))
24		oaabslem 5308	. . . . . . . 8 |- ((om e. On /\ A e. om) -> (A +o om) = om)
25	24	ex 402	. . . . . . 7 |- (om e. On -> (A e. om -> (A +o om) = om))
26		oasuc 5208	. . . . . . . . . . . . 13 |- ((A e. On /\ y e. On) -> (A +o suc y) = suc (A +o y))
27		nnon 3957	. . . . . . . . . . . . 13 |- (A e. om -> A e. On)
28	26, 27	sylan 497	. . . . . . . . . . . 12 |- ((A e. om /\ y e. On) -> (A +o suc y) = suc (A +o y))
29		suceq 3729	. . . . . . . . . . . 12 |- ((A +o y) = y -> suc (A +o y) = suc y)
30	28, 29	sylan9eq 1948	. . . . . . . . . . 11 |- (((A e. om /\ y e. On) /\ (A +o y) = y) -> (A +o suc y) = suc y)
31	30	exp31 407	. . . . . . . . . 10 |- (A e. om -> (y e. On -> ((A +o y) = y -> (A +o suc y) = suc y)))
32	31	com12 14	. . . . . . . . 9 |- (y e. On -> (A e. om -> ((A +o y) = y -> (A +o suc y) = suc y)))
33	32	ad2antrr 440	. . . . . . . 8 |- (((y e. On /\ om e. On) /\ om C_ y) -> (A e. om -> ((A +o y) = y -> (A +o suc y) = suc y)))
34	33	a2d 16	. . . . . . 7 |- (((y e. On /\ om e. On) /\ om C_ y) -> ((A e. om -> (A +o y) = y) -> (A e. om -> (A +o suc y) = suc y)))
35		oalim 5212	. . . . . . . . . . . . . . . . . . 19 |- ((A e. On /\ (om e. On /\ Lim om)) -> (A +o om) = U_y e. om (A +o y))
36		limom 3967	. . . . . . . . . . . . . . . . . . . 20 |- Lim om
37	36	jctr 315	. . . . . . . . . . . . . . . . . . 19 |- (om e. On -> (om e. On /\ Lim om))
38	35, 27, 37	syl2an 503	. . . . . . . . . . . . . . . . . 18 |- ((A e. om /\ om e. On) -> (A +o om) = U_y e. om (A +o y))
39	24	ancoms 484	. . . . . . . . . . . . . . . . . . 19 |- ((A e. om /\ om e. On) -> (A +o om) = om)
40		limuni 3724	. . . . . . . . . . . . . . . . . . . 20 |- (Lim om -> om = U.om)
41	36, 40	ax-mp 7	. . . . . . . . . . . . . . . . . . 19 |- om = U.om
42	39, 41	syl6eq 1944	. . . . . . . . . . . . . . . . . 18 |- ((A e. om /\ om e. On) -> (A +o om) = U.om)
43	38, 42	eqtr3d 1927	. . . . . . . . . . . . . . . . 17 |- ((A e. om /\ om e. On) -> U_y e. om (A +o y) = U.om)
44	43	adantl 424	. . . . . . . . . . . . . . . 16 |- ((((Lim x /\ om C_ x) /\ A.y e. x (om C_ y -> (A +o y) = y)) /\ (A e. om /\ om e. On)) -> U_y e. om (A +o y) = U.om)
45		ordelon 3682	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((Ord x /\ y e. x) -> y e. On)
46		limord 3723	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (Lim x -> Ord x)
47	45, 46	sylan 497	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((Lim x /\ y e. x) -> y e. On)
48		eloni 3667	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (y e. On -> Ord y)
49		ordtri1 3693	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((Ord om /\ Ord y) -> (om C_ y <-> -. y e. om))
50	3, 49	mpan 759	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (Ord y -> (om C_ y <-> -. y e. om))
51	47, 48, 50	3syl 24	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((Lim x /\ y e. x) -> (om C_ y <-> -. y e. om))
52	51	pm5.32da 711	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (Lim x -> ((y e. x /\ om C_ y) <-> (y e. x /\ -. y e. om)))
53		eldif 2609	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y e. (x \ om) <-> (y e. x /\ -. y e. om))
54	52, 53	syl6bbr 597	. . . . . . . . . . . . . . . . . . . . . . 23 |- (Lim x -> ((y e. x /\ om C_ y) <-> y e. (x \ om)))
55	54	imbi1d 675	. . . . . . . . . . . . . . . . . . . . . 22 |- (Lim x -> (((y e. x /\ om C_ y) -> (A +o y) = y) <-> (y e. (x \ om) -> (A +o y) = y)))
56		impexp 374	. . . . . . . . . . . . . . . . . . . . . 22 |- (((y e. x /\ om C_ y) -> (A +o y) = y) <-> (y e. x -> (om C_ y -> (A +o y) = y)))
57	55, 56	syl5bbr 593	. . . . . . . . . . . . . . . . . . . . 21 |- (Lim x -> ((y e. x -> (om C_ y -> (A +o y) = y)) <-> (y e. (x \ om) -> (A +o y) = y)))
58	57	ralbidv2 2125	. . . . . . . . . . . . . . . . . . . 20 |- (Lim x -> (A.y e. x (om C_ y -> (A +o y) = y) <-> A.y e. (x \ om)(A +o y) = y))
59		iuneq2 3273	. . . . . . . . . . . . . . . . . . . . 21 |- (A.y e. (x \ om)(A +o y) = y -> U_y e. (x \ om)(A +o y) = U_y e. (x \ om)y)
60		uniiun 3306	. . . . . . . . . . . . . . . . . . . . 21 |- U.(x \ om) = U_y e. (x \ om)y
61	59, 60	syl6eqr 1946	. . . . . . . . . . . . . . . . . . . 20 |- (A.y e. (x \ om)(A +o y) = y -> U_y e. (x \ om)(A +o y) = U.(x \ om))
62	58, 61	syl6bi 231	. . . . . . . . . . . . . . . . . . 19 |- (Lim x -> (A.y e. x (om C_ y -> (A +o y) = y) -> U_y e. (x \ om)(A +o y) = U.(x \ om)))
63	62	imp 377	. . . . . . . . . . . . . . . . . 18 |- ((Lim x /\ A.y e. x (om C_ y -> (A +o y) = y)) -> U_y e. (x \ om)(A +o y) = U.(x \ om))
64	63	adantlr 429	. . . . . . . . . . . . . . . . 17 |- (((Lim x /\ om C_ x) /\ A.y e. x (om C_ y -> (A +o y) = y)) -> U_y e. (x \ om)(A +o y) = U.(x \ om))
65	64	adantr 425	. . . . . . . . . . . . . . . 16 |- ((((Lim x /\ om C_ x) /\ A.y e. x (om C_ y -> (A +o y) = y)) /\ (A e. om /\ om e. On)) -> U_y e. (x \ om)(A +o y) = U.(x \ om))
66	44, 65	uneq12d 2756	. . . . . . . . . . . . . . 15 |- ((((Lim x /\ om C_ x) /\ A.y e. x (om C_ y -> (A +o y) = y)) /\ (A e. om /\ om e. On)) -> (U_y e. om (A +o y) u. U_y e. (x \ om)(A +o y)) = (U.om u. U.(x \ om)))
67		oalim 5212	. . . . . . . . . . . . . . . . . . . 20 |- ((A e. On /\ (x e. _V /\ Lim x)) -> (A +o x) = U_y e. x (A +o y))
68		visset 2295	. . . . . . . . . . . . . . . . . . . . 21 |- x e. _V
69	68	jctl 314	. . . . . . . . . . . . . . . . . . . 20 |- (Lim x -> (x e. _V /\ Lim x))
70	67, 27, 69	syl2an 503	. . . . . . . . . . . . . . . . . . 19 |- ((A e. om /\ Lim x) -> (A +o x) = U_y e. x (A +o y))
71	70	ancoms 484	. . . . . . . . . . . . . . . . . 18 |- ((Lim x /\ A e. om) -> (A +o x) = U_y e. x (A +o y))
72		ssequn1 2775	. . . . . . . . . . . . . . . . . . . . . 22 |- (om C_ x <-> (om u. x) = x)
73	72	biimpi 168	. . . . . . . . . . . . . . . . . . . . 21 |- (om C_ x -> (om u. x) = x)
74		undif2 2950	. . . . . . . . . . . . . . . . . . . . 21 |- (om u. (x \ om)) = (om u. x)
75	73, 74	syl5eq 1940	. . . . . . . . . . . . . . . . . . . 20 |- (om C_ x -> (om u. (x \ om)) = x)
76		iuneq1 3269	. . . . . . . . . . . . . . . . . . . 20 |- ((om u. (x \ om)) = x -> U_y e. (om u. (x \ om))(A +o y) = U_y e. x (A +o y))
77	75, 76	syl 12	. . . . . . . . . . . . . . . . . . 19 |- (om C_ x -> U_y e. (om u. (x \ om))(A +o y) = U_y e. x (A +o y))
78		iunxun 3329	. . . . . . . . . . . . . . . . . . 19 |- U_y e. (om u. (x \ om))(A +o y) = (U_y e. om (A +o y) u. U_y e. (x \ om)(A +o y))
79	77, 78	syl5reqr 1943	. . . . . . . . . . . . . . . . . 18 |- (om C_ x -> U_y e. x (A +o y) = (U_y e. om (A +o y) u. U_y e. (x \ om)(A +o y)))
80	71, 79	sylan9eq 1948	. . . . . . . . . . . . . . . . 17 |- (((Lim x /\ A e. om) /\ om C_ x) -> (A +o x) = (U_y e. om (A +o y) u. U_y e. (x \ om)(A +o y)))
81	80	an1rs 547	. . . . . . . . . . . . . . . 16 |- (((Lim x /\ om C_ x) /\ A e. om) -> (A +o x) = (U_y e. om (A +o y) u. U_y e. (x \ om)(A +o y)))
82	81	ad2ant2r 445	. . . . . . . . . . . . . . 15 |- ((((Lim x /\ om C_ x) /\ A.y e. x (om C_ y -> (A +o y) = y)) /\ (A e. om /\ om e. On)) -> (A +o x) = (U_y e. om (A +o y) u. U_y e. (x \ om)(A +o y)))
83		limuni 3724	. . . . . . . . . . . . . . . . 17 |- (Lim x -> x = U.x)
84	75	unieqd 3188	. . . . . . . . . . . . . . . . . 18 |- (om C_ x -> U.(om u. (x \ om)) = U.x)
85		uniun 3196	. . . . . . . . . . . . . . . . . 18 |- U.(om u. (x \ om)) = (U.om u. U.(x \ om))
86	84, 85	syl5reqr 1943	. . . . . . . . . . . . . . . . 17 |- (om C_ x -> U.x = (U.om u. U.(x \ om)))
87	83, 86	sylan9eq 1948	. . . . . . . . . . . . . . . 16 |- ((Lim x /\ om C_ x) -> x = (U.om u. U.(x \ om)))
88	87	ad2antrr 440	. . . . . . . . . . . . . . 15 |- ((((Lim x /\ om C_ x) /\ A.y e. x (om C_ y -> (A +o y) = y)) /\ (A e. om /\ om e. On)) -> x = (U.om u. U.(x \ om)))
89	66, 82, 88	3eqtr4d 1937	. . . . . . . . . . . . . 14 |- ((((Lim x /\ om C_ x) /\ A.y e. x (om C_ y -> (A +o y) = y)) /\ (A e. om /\ om e. On)) -> (A +o x) = x)
90	89	exp43 415	. . . . . . . . . . . . 13 |- ((Lim x /\ om C_ x) -> (A.y e. x (om C_ y -> (A +o y) = y) -> (A e. om -> (om e. On -> (A +o x) = x))))
91	90	com23 36	. . . . . . . . . . . 12 |- ((Lim x /\ om C_ x) -> (A e. om -> (A.y e. x (om C_ y -> (A +o y) = y) -> (om e. On -> (A +o x) = x))))
92	91	a2d 16	. . . . . . . . . . 11 |- ((Lim x /\ om C_ x) -> ((A e. om -> A.y e. x (om C_ y -> (A +o y) = y)) -> (A e. om -> (om e. On -> (A +o x) = x))))
93		bi2.04 177	. . . . . . . . . . . . 13 |- ((om C_ y -> (A e. om -> (A +o y) = y)) <-> (A e. om -> (om C_ y -> (A +o y) = y)))
94	93	ralbii 2127	. . . . . . . . . . . 12 |- (A.y e. x (om C_ y -> (A e. om -> (A +o y) = y)) <-> A.y e. x (A e. om -> (om C_ y -> (A +o y) = y)))
95		r19.21v 2178	. . . . . . . . . . . 12 |- (A.y e. x (A e. om -> (om C_ y -> (A +o y) = y)) <-> (A e. om -> A.y e. x (om C_ y -> (A +o y) = y)))
96	94, 95	bitri 190	. . . . . . . . . . 11 |- (A.y e. x (om C_ y -> (A e. om -> (A +o y) = y)) <-> (A e. om -> A.y e. x (om C_ y -> (A +o y) = y)))
97	92, 96	syl5ib 223	. . . . . . . . . 10 |- ((Lim x /\ om C_ x) -> (A.y e. x (om C_ y -> (A e. om -> (A +o y) = y)) -> (A e. om -> (om e. On -> (A +o x) = x))))
98	97	com4r 45	. . . . . . . . 9 |- (om e. On -> ((Lim x /\ om C_ x) -> (A.y e. x (om C_ y -> (A e. om -> (A +o y) = y)) -> (A e. om -> (A +o x) = x))))
99	98	impcom 378	. . . . . . . 8 |- (((Lim x /\ om C_ x) /\ om e. On) -> (A.y e. x (om C_ y -> (A e. om -> (A +o y) = y)) -> (A e. om -> (A +o x) = x)))
100	99	an1rs 547	. . . . . . 7 |- (((Lim x /\ om e. On) /\ om C_ x) -> (A.y e. x (om C_ y -> (A e. om -> (A +o y) = y)) -> (A e. om -> (A +o x) = x)))
101	11, 15, 19, 23, 25, 34, 100	tfindsg 3944	. . . . . 6 |- (((B e. On /\ om e. On) /\ om C_ B) -> (A e. om -> (A +o B) = B))
102	101	an1rs 547	. . . . 5 |- (((B e. On /\ om C_ B) /\ om e. On) -> (A e. om -> (A +o B) = B))
103	7, 102	mpdan 768	. . . 4 |- ((B e. On /\ om C_ B) -> (A e. om -> (A +o B) = B))
104	103	ex 402	. . 3 |- (B e. On -> (om C_ B -> (A e. om -> (A +o B) = B)))
105	104	com3r 39	. 2 |- (A e. om -> (B e. On -> (om C_ B -> (A +o B) = B)))
106	105	imp31 389	1 |- (((A e. om /\ B e. On) /\ om C_ B) -> (A +o B) = B)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   \ cdif 2590   u. cun 2591   C_ wss 2593  U.cuni 3177  U_ciun 3255  Ord word 3656  Oncon0 3657  Lim wlim 3658  suc csuc 3659  omcom 3949  (class class class)co 4884   +o coa 5174

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-om 3950  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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