Theorem grponpcan 27397

Description: Cancellation law for group division. (npcan 10290 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpdivf.1	|- X = ran G
grpdivf.3	|- D = ( /g ` G )

Assertion

Ref	Expression
grponpcan	|- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( A D B ) G B ) = A )

Proof of Theorem grponpcan

Step	Hyp	Ref	Expression
1		grpdivf.1	. . . 4 |- X = ran G
2		eqid 2622	. . . 4 |- ( inv ` G ) = ( inv ` G )
3		grpdivf.3	. . . 4 |- D = ( /g ` G )
4	1, 2, 3	grpodivval 27389	. . 3 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A D B ) = ( A G ( ( inv ` G ) ` B ) ) )
5	4	oveq1d 6665	. 2 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( A D B ) G B ) = ( ( A G ( ( inv ` G ) ` B ) ) G B ) )
6		simp1 1061	. . . 4 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> G e. GrpOp )
7		simp2 1062	. . . 4 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> A e. X )
8	1, 2	grpoinvcl 27378	. . . . 5 |- ( ( G e. GrpOp /\ B e. X ) -> ( ( inv ` G ) ` B ) e. X )
9	8	3adant2 1080	. . . 4 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( inv ` G ) ` B ) e. X )
10		simp3 1063	. . . 4 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> B e. X )
11	1	grpoass 27357	. . . 4 |- ( ( G e. GrpOp /\ ( A e. X /\ ( ( inv ` G ) ` B ) e. X /\ B e. X ) ) -> ( ( A G ( ( inv ` G ) ` B ) ) G B ) = ( A G ( ( ( inv ` G ) ` B ) G B ) ) )
12	6, 7, 9, 10, 11	syl13anc 1328	. . 3 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( A G ( ( inv ` G ) ` B ) ) G B ) = ( A G ( ( ( inv ` G ) ` B ) G B ) ) )
13		eqid 2622	. . . . . . 7 |- (GId ` G ) = (GId ` G )
14	1, 13, 2	grpolinv 27380	. . . . . 6 |- ( ( G e. GrpOp /\ B e. X ) -> ( ( ( inv ` G ) ` B ) G B ) = (GId ` G ) )
15	14	oveq2d 6666	. . . . 5 |- ( ( G e. GrpOp /\ B e. X ) -> ( A G ( ( ( inv ` G ) ` B ) G B ) ) = ( A G (GId ` G ) ) )
16	15	3adant2 1080	. . . 4 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A G ( ( ( inv ` G ) ` B ) G B ) ) = ( A G (GId ` G ) ) )
17	1, 13	grporid 27371	. . . . 5 |- ( ( G e. GrpOp /\ A e. X ) -> ( A G (GId ` G ) ) = A )
18	17	3adant3 1081	. . . 4 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A G (GId ` G ) ) = A )
19	16, 18	eqtrd 2656	. . 3 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A G ( ( ( inv ` G ) ` B ) G B ) ) = A )
20	12, 19	eqtrd 2656	. 2 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( A G ( ( inv ` G ) ` B ) ) G B ) = A )
21	5, 20	eqtrd 2656	1 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( A D B ) G B ) = A )

Syntax hints:   -> wi 4   /\ 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   /g cgs 27346

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-pow 4843  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-pw 4160  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-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350

This theorem is referenced by:  ablonnncan  27410  grpoeqdivid  33680  ghomdiv  33691