Theorem grpidinvlem1 8045

Description: Lemma for grpidinv 8049.

Hypothesis

Ref	Expression
grpfo.1	|- X = ran G

Assertion

Ref	Expression
grpidinvlem1	|- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((YGA) = U /\ (AGA) = A)) -> (UGA) = U)

Proof of Theorem grpidinvlem1

Step	Hyp	Ref	Expression
1		opreq1 3974	. . . 4 |- ((YGA) = U -> ((YGA)GA) = (UGA))
2	1	ad2antrl 408	. . 3 |- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((YGA) = U /\ (AGA) = A)) -> ((YGA)GA) = (UGA))
3		grpfo.1	. . . . . 6 |- X = ran G
4	3	grpass 8044	. . . . 5 |- ((G e. Grp /\ (Y e. X /\ A e. X /\ A e. X)) -> ((YGA)GA) = (YG(AGA)))
5		id 59	. . . . . 6 |- ((Y e. X /\ A e. X /\ A e. X) -> (Y e. X /\ A e. X /\ A e. X))
6	5	3anidm23 886	. . . . 5 |- ((Y e. X /\ A e. X) -> (Y e. X /\ A e. X /\ A e. X))
7	4, 6	sylan2 453	. . . 4 |- ((G e. Grp /\ (Y e. X /\ A e. X)) -> ((YGA)GA) = (YG(AGA)))
8	7	adantr 391	. . 3 |- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((YGA) = U /\ (AGA) = A)) -> ((YGA)GA) = (YG(AGA)))
9	2, 8	eqtr3d 1512	. 2 |- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((YGA) = U /\ (AGA) = A)) -> (UGA) = (YG(AGA)))
10		opreq2 3975	. . 3 |- ((AGA) = A -> (YG(AGA)) = (YGA))
11	10	ad2antll 409	. 2 |- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((YGA) = U /\ (AGA) = A)) -> (YG(AGA)) = (YGA))
12		simprl 416	. 2 |- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((YGA) = U /\ (AGA) = A)) -> (YGA) = U)
13	9, 11, 12	3eqtrd 1514	1 |- (((G e. Grp /\ (Y e. X /\ A e. X)) /\ ((YGA) = U /\ (AGA) = A)) -> (UGA) = U)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  ran crn 3177  (class class class)co 3969  Grpcgr 8030

This theorem is referenced by:  grpidinvlem3 8047

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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