Theorem grpidinvlem2 9329

Description: Lemma for grpidinv 9332.

Hypothesis

Ref	Expression
grpfo.1	|- X = ran G

Assertion

Ref	Expression
grpidinvlem2	|- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((UGY) = Y /\ (YGA) = U)) -> ((AGY)G(AGY)) = (AGY))

Proof of Theorem grpidinvlem2

Step	Hyp	Ref	Expression
1		simprr 451	. . . . 5 |- ((G e. Grp /\ (Y e. X /\ A e. X)) -> A e. X)
2		simprl 450	. . . . 5 |- ((G e. Grp /\ (Y e. X /\ A e. X)) -> Y e. X)
3		grpfo.1	. . . . . . . 8 |- X = ran G
4	3	grpcl 9324	. . . . . . 7 |- ((G e. Grp /\ A e. X /\ Y e. X) -> (AGY) e. X)
5	4	3com23 1074	. . . . . 6 |- ((G e. Grp /\ Y e. X /\ A e. X) -> (AGY) e. X)
6	5	3expb 1068	. . . . 5 |- ((G e. Grp /\ (Y e. X /\ A e. X)) -> (AGY) e. X)
7	1, 2, 6	3jca 1050	. . . 4 |- ((G e. Grp /\ (Y e. X /\ A e. X)) -> (A e. X /\ Y e. X /\ (AGY) e. X))
8	3	grpass 9327	. . . 4 |- ((G e. Grp /\ (A e. X /\ Y e. X /\ (AGY) e. X)) -> ((AGY)G(AGY)) = (AG(YG(AGY))))
9	7, 8	syldan 516	. . 3 |- ((G e. Grp /\ (Y e. X /\ A e. X)) -> ((AGY)G(AGY)) = (AG(YG(AGY))))
10	9	adantr 425	. 2 |- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((UGY) = Y /\ (YGA) = U)) -> ((AGY)G(AGY)) = (AG(YG(AGY))))
11		opreq1 4889	. . . . . . 7 |- ((YGA) = U -> ((YGA)GY) = (UGY))
12	11	adantl 424	. . . . . 6 |- (((UGY) = Y /\ (YGA) = U) -> ((YGA)GY) = (UGY))
13		simpl 346	. . . . . 6 |- (((UGY) = Y /\ (YGA) = U) -> (UGY) = Y)
14	12, 13	eqtr2d 1926	. . . . 5 |- (((UGY) = Y /\ (YGA) = U) -> Y = ((YGA)GY))
15	3	grpass 9327	. . . . . 6 |- ((G e. Grp /\ (Y e. X /\ A e. X /\ Y e. X)) -> ((YGA)GY) = (YG(AGY)))
16		id 73	. . . . . . 7 |- ((Y e. X /\ A e. X /\ Y e. X) -> (Y e. X /\ A e. X /\ Y e. X))
17	16	3anidm13 1155	. . . . . 6 |- ((Y e. X /\ A e. X) -> (Y e. X /\ A e. X /\ Y e. X))
18	15, 17	sylan2 500	. . . . 5 |- ((G e. Grp /\ (Y e. X /\ A e. X)) -> ((YGA)GY) = (YG(AGY)))
19	14, 18	sylan9eqr 1951	. . . 4 |- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((UGY) = Y /\ (YGA) = U)) -> Y = (YG(AGY)))
20	19	eqcomd 1889	. . 3 |- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((UGY) = Y /\ (YGA) = U)) -> (YG(AGY)) = Y)
21	20	opreq2d 4898	. 2 |- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((UGY) = Y /\ (YGA) = U)) -> (AG(YG(AGY))) = (AGY))
22	10, 21	eqtrd 1925	1 |- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((UGY) = Y /\ (YGA) = U)) -> ((AGY)G(AGY)) = (AGY))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ran crn 3987  (class class class)co 4884  Grpcgr 9311

This theorem is referenced by:  grpidinvlem3 9330

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-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-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-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-grp 9316

Copyright terms: Public domain