Theorem grpidinvlem3NEW 17111

Description: Lemma for grpidinvNEW 17113.

Hypotheses

Ref	Expression
grplem1.1NEW	|- B = (base` G)
grplem1.2NEW	|- P = (+g` G)
grpidinvlem3.2NEW	|- (ph <-> A.x e. B (UPx) = x)
grpidinvlem3.3NEW	|- (ps <-> A.x e. B E.z e. B (zPx) = U)

Assertion

Ref	Expression
grpidinvlem3NEW	|- ((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) -> E.y e. B ((yPA) = U /\ (APy) = U))

Distinct variable groups:   x,y,B   x,P,y   x,A,y   z,B   y,G   z,P,x   x,U   y,z,U   ph,y   ps,y

Proof of Theorem grpidinvlem3NEW

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . . . . 8 |- (x = A -> (yPx) = (yPA))
2	1	eqeq1d 1892	. . . . . . 7 |- (x = A -> ((yPx) = U <-> (yPA) = U))
3	2	rexbidv 2124	. . . . . 6 |- (x = A -> (E.y e. B (yPx) = U <-> E.y e. B (yPA) = U))
4	3	rcla4cva 2379	. . . . 5 |- ((A.x e. B E.y e. B (yPx) = U /\ A e. B) -> E.y e. B (yPA) = U)
5		grpidinvlem3.3NEW	. . . . . 6 |- (ps <-> A.x e. B E.z e. B (zPx) = U)
6		opreq1 4889	. . . . . . . . 9 |- (z = y -> (zPx) = (yPx))
7	6	eqeq1d 1892	. . . . . . . 8 |- (z = y -> ((zPx) = U <-> (yPx) = U))
8	7	cbvrexv 2281	. . . . . . 7 |- (E.z e. B (zPx) = U <-> E.y e. B (yPx) = U)
9	8	ralbii 2127	. . . . . 6 |- (A.x e. B E.z e. B (zPx) = U <-> A.x e. B E.y e. B (yPx) = U)
10	5, 9	bitri 190	. . . . 5 |- (ps <-> A.x e. B E.y e. B (yPx) = U)
11	4, 10	sylanb 498	. . . 4 |- ((ps /\ A e. B) -> E.y e. B (yPA) = U)
12	11	adantll 428	. . 3 |- (((ph /\ ps) /\ A e. B) -> E.y e. B (yPA) = U)
13	12	adantll 428	. 2 |- ((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) -> E.y e. B (yPA) = U)
14		grplem1.1NEW	. . . . . . . . . . . 12 |- B = (base` G)
15		grplem1.2NEW	. . . . . . . . . . . 12 |- P = (+g` G)
16	14, 15	grpclNEW 17106	. . . . . . . . . . 11 |- ((G e. GrpNEW /\ A e. B /\ y e. B) -> (APy) e. B)
17	16	3expa 1067	. . . . . . . . . 10 |- (((G e. GrpNEW /\ A e. B) /\ y e. B) -> (APy) e. B)
18	17	adantllr 433	. . . . . . . . 9 |- ((((G e. GrpNEW /\ U e. B) /\ A e. B) /\ y e. B) -> (APy) e. B)
19	18	adantllr 433	. . . . . . . 8 |- (((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) /\ y e. B) -> (APy) e. B)
20		grpidinvlem3.2NEW	. . . . . . . . . . 11 |- (ph <-> A.x e. B (UPx) = x)
21	20	biimpi 168	. . . . . . . . . 10 |- (ph -> A.x e. B (UPx) = x)
22	21	ad2antrl 442	. . . . . . . . 9 |- (((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) -> A.x e. B (UPx) = x)
23	22	ad2antrr 440	. . . . . . . 8 |- (((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) /\ y e. B) -> A.x e. B (UPx) = x)
24		opreq2 4890	. . . . . . . . . 10 |- (x = (APy) -> (UPx) = (UP(APy)))
25		id 73	. . . . . . . . . 10 |- (x = (APy) -> x = (APy))
26	24, 25	eqeq12d 1899	. . . . . . . . 9 |- (x = (APy) -> ((UPx) = x <-> (UP(APy)) = (APy)))
27	26	rcla4va 2378	. . . . . . . 8 |- (((APy) e. B /\ A.x e. B (UPx) = x) -> (UP(APy)) = (APy))
28	19, 23, 27	syl11anc 524	. . . . . . 7 |- (((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) /\ y e. B) -> (UP(APy)) = (APy))
29	28	adantr 425	. . . . . 6 |- ((((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) /\ y e. B) /\ (yPA) = U) -> (UP(APy)) = (APy))
30		pm3.22 486	. . . . . . . . . . . 12 |- (((y e. B /\ A e. B) /\ G e. GrpNEW) -> (G e. GrpNEW /\ (y e. B /\ A e. B)))
31	30	ancom31s 549	. . . . . . . . . . 11 |- (((G e. GrpNEW /\ A e. B) /\ y e. B) -> (G e. GrpNEW /\ (y e. B /\ A e. B)))
32	31	adantllr 433	. . . . . . . . . 10 |- ((((G e. GrpNEW /\ U e. B) /\ A e. B) /\ y e. B) -> (G e. GrpNEW /\ (y e. B /\ A e. B)))
33	32	adantllr 433	. . . . . . . . 9 |- (((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) /\ y e. B) -> (G e. GrpNEW /\ (y e. B /\ A e. B)))
34	33	adantr 425	. . . . . . . 8 |- ((((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) /\ y e. B) /\ (yPA) = U) -> (G e. GrpNEW /\ (y e. B /\ A e. B)))
35		opreq2 4890	. . . . . . . . . . . . . . 15 |- (x = y -> (UPx) = (UPy))
36		id 73	. . . . . . . . . . . . . . 15 |- (x = y -> x = y)
37	35, 36	eqeq12d 1899	. . . . . . . . . . . . . 14 |- (x = y -> ((UPx) = x <-> (UPy) = y))
38	37	rcla4cva 2379	. . . . . . . . . . . . 13 |- ((A.x e. B (UPx) = x /\ y e. B) -> (UPy) = y)
39	38, 20	sylanb 498	. . . . . . . . . . . 12 |- ((ph /\ y e. B) -> (UPy) = y)
40	39	adantlr 429	. . . . . . . . . . 11 |- (((ph /\ ps) /\ y e. B) -> (UPy) = y)
41	40	adantlr 429	. . . . . . . . . 10 |- ((((ph /\ ps) /\ A e. B) /\ y e. B) -> (UPy) = y)
42	41	adantlll 432	. . . . . . . . 9 |- (((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) /\ y e. B) -> (UPy) = y)
43	42	anim1i 361	. . . . . . . 8 |- ((((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) /\ y e. B) /\ (yPA) = U) -> ((UPy) = y /\ (yPA) = U))
44	14, 15	grpidinvlem2NEW 17110	. . . . . . . 8 |- (((G e. GrpNEW /\ (y e. B /\ A e. B)) /\ ((UPy) = y /\ (yPA) = U)) -> ((APy)P(APy)) = (APy))
45	34, 43, 44	syl11anc 524	. . . . . . 7 |- ((((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) /\ y e. B) /\ (yPA) = U) -> ((APy)P(APy)) = (APy))
46	16	3expb 1068	. . . . . . . . . . . . . . 15 |- ((G e. GrpNEW /\ (A e. B /\ y e. B)) -> (APy) e. B)
47	46	ad2ant2rl 447	. . . . . . . . . . . . . 14 |- (((G e. GrpNEW /\ U e. B) /\ (ph /\ (A e. B /\ y e. B))) -> (APy) e. B)
48	14, 15	grpidinvlem1NEW 17109	. . . . . . . . . . . . . . . . . 18 |- (((G e. GrpNEW /\ (w e. B /\ (APy) e. B)) /\ ((wP(APy)) = U /\ ((APy)P(APy)) = (APy))) -> (UP(APy)) = U)
49		anass 487	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((G e. GrpNEW /\ w e. B) /\ (APy) e. B) <-> (G e. GrpNEW /\ (w e. B /\ (APy) e. B)))
50	49	biimpi 168	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((G e. GrpNEW /\ w e. B) /\ (APy) e. B) -> (G e. GrpNEW /\ (w e. B /\ (APy) e. B)))
51	50	an1rs 547	. . . . . . . . . . . . . . . . . . . . . 22 |- (((G e. GrpNEW /\ (APy) e. B) /\ w e. B) -> (G e. GrpNEW /\ (w e. B /\ (APy) e. B)))
52	51	ex 402	. . . . . . . . . . . . . . . . . . . . 21 |- ((G e. GrpNEW /\ (APy) e. B) -> (w e. B -> (G e. GrpNEW /\ (w e. B /\ (APy) e. B))))
53	46, 52	syldan 516	. . . . . . . . . . . . . . . . . . . 20 |- ((G e. GrpNEW /\ (A e. B /\ y e. B)) -> (w e. B -> (G e. GrpNEW /\ (w e. B /\ (APy) e. B))))
54	53	ad2ant2rl 447	. . . . . . . . . . . . . . . . . . 19 |- (((G e. GrpNEW /\ U e. B) /\ (ph /\ (A e. B /\ y e. B))) -> (w e. B -> (G e. GrpNEW /\ (w e. B /\ (APy) e. B))))
55	54	imp 377	. . . . . . . . . . . . . . . . . 18 |- ((((G e. GrpNEW /\ U e. B) /\ (ph /\ (A e. B /\ y e. B))) /\ w e. B) -> (G e. GrpNEW /\ (w e. B /\ (APy) e. B)))
56	48, 55	sylan 497	. . . . . . . . . . . . . . . . 17 |- (((((G e. GrpNEW /\ U e. B) /\ (ph /\ (A e. B /\ y e. B))) /\ w e. B) /\ ((wP(APy)) = U /\ ((APy)P(APy)) = (APy))) -> (UP(APy)) = U)
57	56	exp43 415	. . . . . . . . . . . . . . . 16 |- (((G e. GrpNEW /\ U e. B) /\ (ph /\ (A e. B /\ y e. B))) -> (w e. B -> ((wP(APy)) = U -> (((APy)P(APy)) = (APy) -> (UP(APy)) = U))))
58	57	r19.23adv 2215	. . . . . . . . . . . . . . 15 |- (((G e. GrpNEW /\ U e. B) /\ (ph /\ (A e. B /\ y e. B))) -> (E.w e. B (wP(APy)) = U -> (((APy)P(APy)) = (APy) -> (UP(APy)) = U)))
59		opreq2 4890	. . . . . . . . . . . . . . . . . . 19 |- (x = (APy) -> (wPx) = (wP(APy)))
60	59	eqeq1d 1892	. . . . . . . . . . . . . . . . . 18 |- (x = (APy) -> ((wPx) = U <-> (wP(APy)) = U))
61	60	rexbidv 2124	. . . . . . . . . . . . . . . . 17 |- (x = (APy) -> (E.w e. B (wPx) = U <-> E.w e. B (wP(APy)) = U))
62	61	rcla4va 2378	. . . . . . . . . . . . . . . 16 |- (((APy) e. B /\ A.x e. B E.w e. B (wPx) = U) -> E.w e. B (wP(APy)) = U)
63		opreq1 4889	. . . . . . . . . . . . . . . . . . . 20 |- (z = w -> (zPx) = (wPx))
64	63	eqeq1d 1892	. . . . . . . . . . . . . . . . . . 19 |- (z = w -> ((zPx) = U <-> (wPx) = U))
65	64	cbvrexv 2281	. . . . . . . . . . . . . . . . . 18 |- (E.z e. B (zPx) = U <-> E.w e. B (wPx) = U)
66	65	ralbii 2127	. . . . . . . . . . . . . . . . 17 |- (A.x e. B E.z e. B (zPx) = U <-> A.x e. B E.w e. B (wPx) = U)
67	5, 66	bitri 190	. . . . . . . . . . . . . . . 16 |- (ps <-> A.x e. B E.w e. B (wPx) = U)
68	62, 67	sylan2b 501	. . . . . . . . . . . . . . 15 |- (((APy) e. B /\ ps) -> E.w e. B (wP(APy)) = U)
69	58, 68	syl5 20	. . . . . . . . . . . . . 14 |- (((G e. GrpNEW /\ U e. B) /\ (ph /\ (A e. B /\ y e. B))) -> (((APy) e. B /\ ps) -> (((APy)P(APy)) = (APy) -> (UP(APy)) = U)))
70	47, 69	mpand 765	. . . . . . . . . . . . 13 |- (((G e. GrpNEW /\ U e. B) /\ (ph /\ (A e. B /\ y e. B))) -> (ps -> (((APy)P(APy)) = (APy) -> (UP(APy)) = U)))
71	70	exp32 408	. . . . . . . . . . . 12 |- ((G e. GrpNEW /\ U e. B) -> (ph -> ((A e. B /\ y e. B) -> (ps -> (((APy)P(APy)) = (APy) -> (UP(APy)) = U)))))
72	71	com34 40	. . . . . . . . . . 11 |- ((G e. GrpNEW /\ U e. B) -> (ph -> (ps -> ((A e. B /\ y e. B) -> (((APy)P(APy)) = (APy) -> (UP(APy)) = U)))))
73	72	imp32 390	. . . . . . . . . 10 |- (((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) -> ((A e. B /\ y e. B) -> (((APy)P(APy)) = (APy) -> (UP(APy)) = U)))
74	73	exp3a 405	. . . . . . . . 9 |- (((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) -> (A e. B -> (y e. B -> (((APy)P(APy)) = (APy) -> (UP(APy)) = U))))
75	74	imp31 389	. . . . . . . 8 |- (((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) /\ y e. B) -> (((APy)P(APy)) = (APy) -> (UP(APy)) = U))
76	75	adantr 425	. . . . . . 7 |- ((((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) /\ y e. B) /\ (yPA) = U) -> (((APy)P(APy)) = (APy) -> (UP(APy)) = U))
77	45, 76	mpd 29	. . . . . 6 |- ((((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) /\ y e. B) /\ (yPA) = U) -> (UP(APy)) = U)
78	29, 77	eqtr3d 1927	. . . . 5 |- ((((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) /\ y e. B) /\ (yPA) = U) -> (APy) = U)
79	78	ex 402	. . . 4 |- (((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) /\ y e. B) -> ((yPA) = U -> (APy) = U))
80	79	ancld 322	. . 3 |- (((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) /\ y e. B) -> ((yPA) = U -> ((yPA) = U /\ (APy) = U)))
81	80	reximdva 2203	. 2 |- ((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) -> (E.y e. B (yPA) = U -> E.y e. B ((yPA) = U /\ (APy) = U)))
82	13, 81	mpd 29	1 |- ((((G e. GrpNEW /\ U e. B) /\ (ph /\ ps)) /\ A e. B) -> E.y e. B ((yPA) = U /\ (APy) = U))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  ` cfv 3998  (class class class)co 4884  basecbs 16758  +_gcplusg 17080  GrpNEWcgrp 17081

This theorem is referenced by:  grpidinvNEW 17113

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-3an 860  df-tru 1262  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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  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-struct 16708  df-grpNEW 17089

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