Theorem grpidinvlem4 8051

Description: Lemma for grpidinv 8052.

Hypothesis

Ref	Expression
grpfo.1	|- X = ran G

Assertion

Ref	Expression
grpidinvlem4	|- (((G e. Grp /\ A e. X) /\ E.y e. X ((yGA) = U /\ (AGy) = U)) -> (AGU) = (UGA))

Distinct variable groups:   y,A   y,G   y,X   y,U

Proof of Theorem grpidinvlem4

Step	Hyp	Ref	Expression
1		grpfo.1	. . . . . . . . 9 |- X = ran G
2	1	grpass 8047	. . . . . . . 8 |- ((G e. Grp /\ (A e. X /\ y e. X /\ A e. X)) -> ((AGy)GA) = (AG(yGA)))
3		simpll 412	. . . . . . . 8 |- (((G e. Grp /\ A e. X) /\ y e. X) -> G e. Grp)
4		simplr 413	. . . . . . . . 9 |- (((G e. Grp /\ A e. X) /\ y e. X) -> A e. X)
5		pm3.27 323	. . . . . . . . 9 |- (((G e. Grp /\ A e. X) /\ y e. X) -> y e. X)
6	4, 5, 4	3jca 819	. . . . . . . 8 |- (((G e. Grp /\ A e. X) /\ y e. X) -> (A e. X /\ y e. X /\ A e. X))
7	2, 3, 6	sylanc 471	. . . . . . 7 |- (((G e. Grp /\ A e. X) /\ y e. X) -> ((AGy)GA) = (AG(yGA)))
8		opreq2 3969	. . . . . . 7 |- ((yGA) = U -> (AG(yGA)) = (AGU))
9	7, 8	sylan9eq 1527	. . . . . 6 |- ((((G e. Grp /\ A e. X) /\ y e. X) /\ (yGA) = U) -> ((AGy)GA) = (AGU))
10		opreq1 3968	. . . . . 6 |- ((AGy) = U -> ((AGy)GA) = (UGA))
11	9, 10	sylan9req 1528	. . . . 5 |- (((((G e. Grp /\ A e. X) /\ y e. X) /\ (yGA) = U) /\ (AGy) = U) -> (AGU) = (UGA))
12	11	anasss 440	. . . 4 |- ((((G e. Grp /\ A e. X) /\ y e. X) /\ ((yGA) = U /\ (AGy) = U)) -> (AGU) = (UGA))
13	12	exp31 376	. . 3 |- ((G e. Grp /\ A e. X) -> (y e. X -> (((yGA) = U /\ (AGy) = U) -> (AGU) = (UGA))))
14	13	r19.23adv 1746	. 2 |- ((G e. Grp /\ A e. X) -> (E.y e. X ((yGA) = U /\ (AGy) = U) -> (AGU) = (UGA)))
15	14	imp 350	1 |- (((G e. Grp /\ A e. X) /\ E.y e. X ((yGA) = U /\ (AGy) = U)) -> (AGU) = (UGA))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  E.wrex 1646  ran crn 3171  (class class class)co 3963  Grpcgr 8033

This theorem is referenced by:  grpidinv 8052  grpideu 8053

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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