Theorem grpoinvop 27387

Description: The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpasscan1.1	|- X = ran G
grpasscan1.2	|- N = ( inv ` G )

Assertion

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

Proof of Theorem grpoinvop

Step	Hyp	Ref	Expression
1		simp1 1061	. . . 4 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> G e. GrpOp )
2		simp2 1062	. . . 4 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> A e. X )
3		simp3 1063	. . . 4 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> B e. X )
4		grpasscan1.1	. . . . . . 7 |- X = ran G
5		grpasscan1.2	. . . . . . 7 |- N = ( inv ` G )
6	4, 5	grpoinvcl 27378	. . . . . 6 |- ( ( G e. GrpOp /\ B e. X ) -> ( N ` B ) e. X )
7	6	3adant2 1080	. . . . 5 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( N ` B ) e. X )
8	4, 5	grpoinvcl 27378	. . . . . 6 |- ( ( G e. GrpOp /\ A e. X ) -> ( N ` A ) e. X )
9	8	3adant3 1081	. . . . 5 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( N ` A ) e. X )
10	4	grpocl 27354	. . . . 5 |- ( ( G e. GrpOp /\ ( N ` B ) e. X /\ ( N ` A ) e. X ) -> ( ( N ` B ) G ( N ` A ) ) e. X )
11	1, 7, 9, 10	syl3anc 1326	. . . 4 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( N ` B ) G ( N ` A ) ) e. X )
12	4	grpoass 27357	. . . 4 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ ( ( N ` B ) G ( N ` A ) ) e. X ) ) -> ( ( A G B ) G ( ( N ` B ) G ( N ` A ) ) ) = ( A G ( B G ( ( N ` B ) G ( N ` A ) ) ) ) )
13	1, 2, 3, 11, 12	syl13anc 1328	. . 3 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( A G B ) G ( ( N ` B ) G ( N ` A ) ) ) = ( A G ( B G ( ( N ` B ) G ( N ` A ) ) ) ) )
14		eqid 2622	. . . . . . . 8 |- (GId ` G ) = (GId ` G )
15	4, 14, 5	grporinv 27381	. . . . . . 7 |- ( ( G e. GrpOp /\ B e. X ) -> ( B G ( N ` B ) ) = (GId ` G ) )
16	15	3adant2 1080	. . . . . 6 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( B G ( N ` B ) ) = (GId ` G ) )
17	16	oveq1d 6665	. . . . 5 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( B G ( N ` B ) ) G ( N ` A ) ) = ( (GId ` G ) G ( N ` A ) ) )
18	4	grpoass 27357	. . . . . 6 |- ( ( G e. GrpOp /\ ( B e. X /\ ( N ` B ) e. X /\ ( N ` A ) e. X ) ) -> ( ( B G ( N ` B ) ) G ( N ` A ) ) = ( B G ( ( N ` B ) G ( N ` A ) ) ) )
19	1, 3, 7, 9, 18	syl13anc 1328	. . . . 5 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( B G ( N ` B ) ) G ( N ` A ) ) = ( B G ( ( N ` B ) G ( N ` A ) ) ) )
20	4, 14	grpolid 27370	. . . . . . 7 |- ( ( G e. GrpOp /\ ( N ` A ) e. X ) -> ( (GId ` G ) G ( N ` A ) ) = ( N ` A ) )
21	8, 20	syldan 487	. . . . . 6 |- ( ( G e. GrpOp /\ A e. X ) -> ( (GId ` G ) G ( N ` A ) ) = ( N ` A ) )
22	21	3adant3 1081	. . . . 5 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( (GId ` G ) G ( N ` A ) ) = ( N ` A ) )
23	17, 19, 22	3eqtr3d 2664	. . . 4 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( B G ( ( N ` B ) G ( N ` A ) ) ) = ( N ` A ) )
24	23	oveq2d 6666	. . 3 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A G ( B G ( ( N ` B ) G ( N ` A ) ) ) ) = ( A G ( N ` A ) ) )
25	4, 14, 5	grporinv 27381	. . . 4 |- ( ( G e. GrpOp /\ A e. X ) -> ( A G ( N ` A ) ) = (GId ` G ) )
26	25	3adant3 1081	. . 3 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A G ( N ` A ) ) = (GId ` G ) )
27	13, 24, 26	3eqtrd 2660	. 2 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( A G B ) G ( ( N ` B ) G ( N ` A ) ) ) = (GId ` G ) )
28	4	grpocl 27354	. . 3 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A G B ) e. X )
29	4, 14, 5	grpoinvid1 27382	. . 3 |- ( ( G e. GrpOp /\ ( A G B ) e. X /\ ( ( N ` B ) G ( N ` A ) ) e. X ) -> ( ( N ` ( A G B ) ) = ( ( N ` B ) G ( N ` A ) ) <-> ( ( A G B ) G ( ( N ` B ) G ( N ` A ) ) ) = (GId ` G ) ) )
30	1, 28, 11, 29	syl3anc 1326	. 2 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( N ` ( A G B ) ) = ( ( N ` B ) G ( N ` A ) ) <-> ( ( A G B ) G ( ( N ` B ) G ( N ` A ) ) ) = (GId ` G ) ) )
31	27, 30	mpbird 247	1 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( N ` ( A G B ) ) = ( ( N ` B ) G ( N ` A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ 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:  grpoinvdiv  27391