Theorem grpoinvid1 27382

Description: The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpinv.1	|- X = ran G
grpinv.2	|- U = (GId ` G )
grpinv.3	|- N = ( inv ` G )

Assertion

Ref	Expression
grpoinvid1	|- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( N ` A ) = B <-> ( A G B ) = U ) )

Proof of Theorem grpoinvid1

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . 4 |- ( ( N ` A ) = B -> ( A G ( N ` A ) ) = ( A G B ) )
2	1	adantl 482	. . 3 |- ( ( ( G e. GrpOp /\ A e. X /\ B e. X ) /\ ( N ` A ) = B ) -> ( A G ( N ` A ) ) = ( A G B ) )
3		grpinv.1	. . . . . 6 |- X = ran G
4		grpinv.2	. . . . . 6 |- U = (GId ` G )
5		grpinv.3	. . . . . 6 |- N = ( inv ` G )
6	3, 4, 5	grporinv 27381	. . . . 5 |- ( ( G e. GrpOp /\ A e. X ) -> ( A G ( N ` A ) ) = U )
7	6	3adant3 1081	. . . 4 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A G ( N ` A ) ) = U )
8	7	adantr 481	. . 3 |- ( ( ( G e. GrpOp /\ A e. X /\ B e. X ) /\ ( N ` A ) = B ) -> ( A G ( N ` A ) ) = U )
9	2, 8	eqtr3d 2658	. 2 |- ( ( ( G e. GrpOp /\ A e. X /\ B e. X ) /\ ( N ` A ) = B ) -> ( A G B ) = U )
10		oveq2 6658	. . . 4 |- ( ( A G B ) = U -> ( ( N ` A ) G ( A G B ) ) = ( ( N ` A ) G U ) )
11	10	adantl 482	. . 3 |- ( ( ( G e. GrpOp /\ A e. X /\ B e. X ) /\ ( A G B ) = U ) -> ( ( N ` A ) G ( A G B ) ) = ( ( N ` A ) G U ) )
12	3, 4, 5	grpolinv 27380	. . . . . . . 8 |- ( ( G e. GrpOp /\ A e. X ) -> ( ( N ` A ) G A ) = U )
13	12	oveq1d 6665	. . . . . . 7 |- ( ( G e. GrpOp /\ A e. X ) -> ( ( ( N ` A ) G A ) G B ) = ( U G B ) )
14	13	3adant3 1081	. . . . . 6 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( ( N ` A ) G A ) G B ) = ( U G B ) )
15	3, 5	grpoinvcl 27378	. . . . . . . . . 10 |- ( ( G e. GrpOp /\ A e. X ) -> ( N ` A ) e. X )
16	15	adantrr 753	. . . . . . . . 9 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X ) ) -> ( N ` A ) e. X )
17		simprl 794	. . . . . . . . 9 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X ) ) -> A e. X )
18		simprr 796	. . . . . . . . 9 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X ) ) -> B e. X )
19	16, 17, 18	3jca 1242	. . . . . . . 8 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X ) ) -> ( ( N ` A ) e. X /\ A e. X /\ B e. X ) )
20	3	grpoass 27357	. . . . . . . 8 |- ( ( G e. GrpOp /\ ( ( N ` A ) e. X /\ A e. X /\ B e. X ) ) -> ( ( ( N ` A ) G A ) G B ) = ( ( N ` A ) G ( A G B ) ) )
21	19, 20	syldan 487	. . . . . . 7 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X ) ) -> ( ( ( N ` A ) G A ) G B ) = ( ( N ` A ) G ( A G B ) ) )
22	21	3impb 1260	. . . . . 6 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( ( N ` A ) G A ) G B ) = ( ( N ` A ) G ( A G B ) ) )
23	14, 22	eqtr3d 2658	. . . . 5 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( U G B ) = ( ( N ` A ) G ( A G B ) ) )
24	3, 4	grpolid 27370	. . . . . 6 |- ( ( G e. GrpOp /\ B e. X ) -> ( U G B ) = B )
25	24	3adant2 1080	. . . . 5 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( U G B ) = B )
26	23, 25	eqtr3d 2658	. . . 4 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( N ` A ) G ( A G B ) ) = B )
27	26	adantr 481	. . 3 |- ( ( ( G e. GrpOp /\ A e. X /\ B e. X ) /\ ( A G B ) = U ) -> ( ( N ` A ) G ( A G B ) ) = B )
28	3, 4	grporid 27371	. . . . . 6 |- ( ( G e. GrpOp /\ ( N ` A ) e. X ) -> ( ( N ` A ) G U ) = ( N ` A ) )
29	15, 28	syldan 487	. . . . 5 |- ( ( G e. GrpOp /\ A e. X ) -> ( ( N ` A ) G U ) = ( N ` A ) )
30	29	3adant3 1081	. . . 4 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( N ` A ) G U ) = ( N ` A ) )
31	30	adantr 481	. . 3 |- ( ( ( G e. GrpOp /\ A e. X /\ B e. X ) /\ ( A G B ) = U ) -> ( ( N ` A ) G U ) = ( N ` A ) )
32	11, 27, 31	3eqtr3rd 2665	. 2 |- ( ( ( G e. GrpOp /\ A e. X /\ B e. X ) /\ ( A G B ) = U ) -> ( N ` A ) = B )
33	9, 32	impbida 877	1 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( N ` A ) = B <-> ( A G B ) = U ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ran crn 5115   ` cfv 5888  (class class class)co 6650   GrpOpcgr 27343  GIdcgi 27344   invcgn 27345

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-rep 4771  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-grpo 27347  df-gid 27348  df-ginv 27349

This theorem is referenced by:  grpoinvop  27387  rngonegmn1l  33740