Theorem grpidinvlem2 8049

Description: Lemma for grpidinv 8052.

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 415	. . . . 5 |- ((G e. Grp /\ (Y e. X /\ A e. X)) -> A e. X)
2		simprl 414	. . . . 5 |- ((G e. Grp /\ (Y e. X /\ A e. X)) -> Y e. X)
3		grpfo.1	. . . . . . . 8 |- X = ran G
4	3	grpcl 8044	. . . . . . 7 |- ((G e. Grp /\ A e. X /\ Y e. X) -> (AGY) e. X)
5	4	3com23 839	. . . . . 6 |- ((G e. Grp /\ Y e. X /\ A e. X) -> (AGY) e. X)
6	5	3expb 834	. . . . 5 |- ((G e. Grp /\ (Y e. X /\ A e. X)) -> (AGY) e. X)
7	1, 2, 6	3jca 819	. . . 4 |- ((G e. Grp /\ (Y e. X /\ A e. X)) -> (A e. X /\ Y e. X /\ (AGY) e. X))
8	3	grpass 8047	. . . 4 |- ((G e. Grp /\ (A e. X /\ Y e. X /\ (AGY) e. X)) -> ((AGY)G(AGY)) = (AG(YG(AGY))))
9	7, 8	syldan 467	. . 3 |- ((G e. Grp /\ (Y e. X /\ A e. X)) -> ((AGY)G(AGY)) = (AG(YG(AGY))))
10	9	adantr 389	. 2 |- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((UGY) = Y /\ (YGA) = U)) -> ((AGY)G(AGY)) = (AG(YG(AGY))))
11		opreq1 3968	. . . . . . 7 |- ((YGA) = U -> ((YGA)GY) = (UGY))
12	11	adantl 388	. . . . . 6 |- (((UGY) = Y /\ (YGA) = U) -> ((YGA)GY) = (UGY))
13		pm3.26 319	. . . . . 6 |- (((UGY) = Y /\ (YGA) = U) -> (UGY) = Y)
14	12, 13	eqtr2d 1508	. . . . 5 |- (((UGY) = Y /\ (YGA) = U) -> Y = ((YGA)GY))
15	3	grpass 8047	. . . . . 6 |- ((G e. Grp /\ (Y e. X /\ A e. X /\ Y e. X)) -> ((YGA)GY) = (YG(AGY)))
16		id 59	. . . . . . 7 |- ((Y e. X /\ A e. X /\ Y e. X) -> (Y e. X /\ A e. X /\ Y e. X))
17	16	3anidm13 883	. . . . . 6 |- ((Y e. X /\ A e. X) -> (Y e. X /\ A e. X /\ Y e. X))
18	15, 17	sylan2 451	. . . . 5 |- ((G e. Grp /\ (Y e. X /\ A e. X)) -> ((YGA)GY) = (YG(AGY)))
19	14, 18	sylan9eqr 1529	. . . 4 |- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((UGY) = Y /\ (YGA) = U)) -> Y = (YG(AGY)))
20	19	eqcomd 1480	. . 3 |- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((UGY) = Y /\ (YGA) = U)) -> (YG(AGY)) = Y)
21	20	opreq2d 3976	. 2 |- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((UGY) = Y /\ (YGA) = U)) -> (AG(YG(AGY))) = (AGY))
22	10, 21	eqtrd 1507	1 |- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((UGY) = Y /\ (YGA) = U)) -> ((AGY)G(AGY)) = (AGY))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  ran crn 3171  (class class class)co 3963  Grpcgr 8033

This theorem is referenced by:  grpidinvlem3 8050

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-grp 8037

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