Theorem grpoidinvlem1 27358

Description: Lemma for grpoidinv 27362. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
grpfo.1	|- X = ran G

Assertion

Ref	Expression
grpoidinvlem1	|- ( ( ( G e. GrpOp /\ ( Y e. X /\ A e. X ) ) /\ ( ( Y G A ) = U /\ ( A G A ) = A ) ) -> ( U G A ) = U )

Proof of Theorem grpoidinvlem1

Step	Hyp	Ref	Expression
1		id 22	. . . . 5 |- ( ( Y e. X /\ A e. X /\ A e. X ) -> ( Y e. X /\ A e. X /\ A e. X ) )
2	1	3anidm23 1385	. . . 4 |- ( ( Y e. X /\ A e. X ) -> ( Y e. X /\ A e. X /\ A e. X ) )
3		grpfo.1	. . . . 5 |- X = ran G
4	3	grpoass 27357	. . . 4 |- ( ( G e. GrpOp /\ ( Y e. X /\ A e. X /\ A e. X ) ) -> ( ( Y G A ) G A ) = ( Y G ( A G A ) ) )
5	2, 4	sylan2 491	. . 3 |- ( ( G e. GrpOp /\ ( Y e. X /\ A e. X ) ) -> ( ( Y G A ) G A ) = ( Y G ( A G A ) ) )
6	5	adantr 481	. 2 |- ( ( ( G e. GrpOp /\ ( Y e. X /\ A e. X ) ) /\ ( ( Y G A ) = U /\ ( A G A ) = A ) ) -> ( ( Y G A ) G A ) = ( Y G ( A G A ) ) )
7		oveq1 6657	. . 3 |- ( ( Y G A ) = U -> ( ( Y G A ) G A ) = ( U G A ) )
8	7	ad2antrl 764	. 2 |- ( ( ( G e. GrpOp /\ ( Y e. X /\ A e. X ) ) /\ ( ( Y G A ) = U /\ ( A G A ) = A ) ) -> ( ( Y G A ) G A ) = ( U G A ) )
9		oveq2 6658	. . . 4 |- ( ( A G A ) = A -> ( Y G ( A G A ) ) = ( Y G A ) )
10	9	ad2antll 765	. . 3 |- ( ( ( G e. GrpOp /\ ( Y e. X /\ A e. X ) ) /\ ( ( Y G A ) = U /\ ( A G A ) = A ) ) -> ( Y G ( A G A ) ) = ( Y G A ) )
11		simprl 794	. . 3 |- ( ( ( G e. GrpOp /\ ( Y e. X /\ A e. X ) ) /\ ( ( Y G A ) = U /\ ( A G A ) = A ) ) -> ( Y G A ) = U )
12	10, 11	eqtrd 2656	. 2 |- ( ( ( G e. GrpOp /\ ( Y e. X /\ A e. X ) ) /\ ( ( Y G A ) = U /\ ( A G A ) = A ) ) -> ( Y G ( A G A ) ) = U )
13	6, 8, 12	3eqtr3d 2664	1 |- ( ( ( G e. GrpOp /\ ( Y e. X /\ A e. X ) ) /\ ( ( Y G A ) = U /\ ( A G A ) = A ) ) -> ( U G A ) = U )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ran crn 5115  (class class class)co 6650   GrpOpcgr 27343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-ov 6653  df-grpo 27347

This theorem is referenced by:  grpoidinvlem3  27360