Theorem grpidinvlem2NEW 17110

Description: Lemma for grpidinvNEW 17113.

Hypotheses

Ref	Expression
grplem1.1NEW	|- B = (base` G)
grplem1.2NEW	|- P = (+g` G)

Assertion

Ref	Expression
grpidinvlem2NEW	|- (((G e. GrpNEW /\ (Y e. B /\ A e. B)) /\ ((UPY) = Y /\ (YPA) = U)) -> ((APY)P(APY)) = (APY))

Proof of Theorem grpidinvlem2NEW

Step	Hyp	Ref	Expression
1		simprr 451	. . . . 5 |- ((G e. GrpNEW /\ (Y e. B /\ A e. B)) -> A e. B)
2		simprl 450	. . . . 5 |- ((G e. GrpNEW /\ (Y e. B /\ A e. B)) -> Y e. B)
3		grplem1.1NEW	. . . . . . . 8 |- B = (base` G)
4		grplem1.2NEW	. . . . . . . 8 |- P = (+g` G)
5	3, 4	grpclNEW 17106	. . . . . . 7 |- ((G e. GrpNEW /\ A e. B /\ Y e. B) -> (APY) e. B)
6	5	3com23 1074	. . . . . 6 |- ((G e. GrpNEW /\ Y e. B /\ A e. B) -> (APY) e. B)
7	6	3expb 1068	. . . . 5 |- ((G e. GrpNEW /\ (Y e. B /\ A e. B)) -> (APY) e. B)
8	1, 2, 7	3jca 1050	. . . 4 |- ((G e. GrpNEW /\ (Y e. B /\ A e. B)) -> (A e. B /\ Y e. B /\ (APY) e. B))
9	3, 4	grpassNEW 17107	. . . 4 |- ((G e. GrpNEW /\ (A e. B /\ Y e. B /\ (APY) e. B)) -> ((APY)P(APY)) = (AP(YP(APY))))
10	8, 9	syldan 516	. . 3 |- ((G e. GrpNEW /\ (Y e. B /\ A e. B)) -> ((APY)P(APY)) = (AP(YP(APY))))
11	10	adantr 425	. 2 |- (((G e. GrpNEW /\ (Y e. B /\ A e. B)) /\ ((UPY) = Y /\ (YPA) = U)) -> ((APY)P(APY)) = (AP(YP(APY))))
12		opreq1 4889	. . . . . . 7 |- ((YPA) = U -> ((YPA)PY) = (UPY))
13	12	adantl 424	. . . . . 6 |- (((UPY) = Y /\ (YPA) = U) -> ((YPA)PY) = (UPY))
14		simpl 346	. . . . . 6 |- (((UPY) = Y /\ (YPA) = U) -> (UPY) = Y)
15	13, 14	eqtr2d 1926	. . . . 5 |- (((UPY) = Y /\ (YPA) = U) -> Y = ((YPA)PY))
16	3, 4	grpassNEW 17107	. . . . . 6 |- ((G e. GrpNEW /\ (Y e. B /\ A e. B /\ Y e. B)) -> ((YPA)PY) = (YP(APY)))
17		id 73	. . . . . . 7 |- ((Y e. B /\ A e. B /\ Y e. B) -> (Y e. B /\ A e. B /\ Y e. B))
18	17	3anidm13 1155	. . . . . 6 |- ((Y e. B /\ A e. B) -> (Y e. B /\ A e. B /\ Y e. B))
19	16, 18	sylan2 500	. . . . 5 |- ((G e. GrpNEW /\ (Y e. B /\ A e. B)) -> ((YPA)PY) = (YP(APY)))
20	15, 19	sylan9eqr 1951	. . . 4 |- (((G e. GrpNEW /\ (Y e. B /\ A e. B)) /\ ((UPY) = Y /\ (YPA) = U)) -> Y = (YP(APY)))
21	20	eqcomd 1889	. . 3 |- (((G e. GrpNEW /\ (Y e. B /\ A e. B)) /\ ((UPY) = Y /\ (YPA) = U)) -> (YP(APY)) = Y)
22	21	opreq2d 4898	. 2 |- (((G e. GrpNEW /\ (Y e. B /\ A e. B)) /\ ((UPY) = Y /\ (YPA) = U)) -> (AP(YP(APY))) = (APY))
23	11, 22	eqtrd 1925	1 |- (((G e. GrpNEW /\ (Y e. B /\ A e. B)) /\ ((UPY) = Y /\ (YPA) = U)) -> ((APY)P(APY)) = (APY))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ` cfv 3998  (class class class)co 4884  basecbs 16758  +_gcplusg 17080  GrpNEWcgrp 17081

This theorem is referenced by:  grpidinvlem3NEW 17111

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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