Theorem grpidinvlem4 8048

Description: Lemma for grpidinv 8049.

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 8044	. . . . . . . 8 |- ((G e. Grp /\ (A e. X /\ y e. X /\ A e. X)) -> ((AGy)GA) = (AG(yGA)))
3		simpll 414	. . . . . . . 8 |- (((G e. Grp /\ A e. X) /\ y e. X) -> G e. Grp)
4		simplr 415	. . . . . . . . 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 821	. . . . . . . 8 |- (((G e. Grp /\ A e. X) /\ y e. X) -> (A e. X /\ y e. X /\ A e. X))
7	2, 3, 6	sylanc 473	. . . . . . 7 |- (((G e. Grp /\ A e. X) /\ y e. X) -> ((AGy)GA) = (AG(yGA)))
8		opreq2 3975	. . . . . . 7 |- ((yGA) = U -> (AG(yGA)) = (AGU))
9	7, 8	sylan9eq 1530	. . . . . 6 |- ((((G e. Grp /\ A e. X) /\ y e. X) /\ (yGA) = U) -> ((AGy)GA) = (AGU))
10		opreq1 3974	. . . . . 6 |- ((AGy) = U -> ((AGy)GA) = (UGA))
11	9, 10	sylan9req 1531	. . . . 5 |- (((((G e. Grp /\ A e. X) /\ y e. X) /\ (yGA) = U) /\ (AGy) = U) -> (AGU) = (UGA))
12	11	anasss 442	. . . 4 |- ((((G e. Grp /\ A e. X) /\ y e. X) /\ ((yGA) = U /\ (AGy) = U)) -> (AGU) = (UGA))
13	12	exp31 378	. . 3 |- ((G e. Grp /\ A e. X) -> (y e. X -> (((yGA) = U /\ (AGy) = U) -> (AGU) = (UGA))))
14	13	r19.23adv 1749	. 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 777   = wceq 958   e. wcel 960  E.wrex 1649  ran crn 3177  (class class class)co 3969  Grpcgr 8030

This theorem is referenced by:  grpidinv 8049  grpideu 8050

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-grp 8034

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