Theorem ghomdiv 33691

Description: Group homomorphisms preserve division. (Contributed by Jeff Madsen, 16-Jun-2011.)

Hypotheses

Ref	Expression
ghomdiv.1	|- X = ran G
ghomdiv.2	|- D = ( /g ` G )
ghomdiv.3	|- C = ( /g ` H )

Assertion

Ref	Expression
ghomdiv	|- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( A e. X /\ B e. X ) ) -> ( F ` ( A D B ) ) = ( ( F ` A ) C ( F ` B ) ) )

Proof of Theorem ghomdiv

Step	Hyp	Ref	Expression
1		simpl2 1065	. . . 4 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( A e. X /\ B e. X ) ) -> H e. GrpOp )
2		ghomdiv.1	. . . . . . 7 |- X = ran G
3		eqid 2622	. . . . . . 7 |- ran H = ran H
4	2, 3	ghomf 33689	. . . . . 6 |- ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) -> F : X --> ran H )
5	4	ffvelrnda 6359	. . . . 5 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ A e. X ) -> ( F ` A ) e. ran H )
6	5	adantrr 753	. . . 4 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( A e. X /\ B e. X ) ) -> ( F ` A ) e. ran H )
7	4	ffvelrnda 6359	. . . . 5 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ B e. X ) -> ( F ` B ) e. ran H )
8	7	adantrl 752	. . . 4 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( A e. X /\ B e. X ) ) -> ( F ` B ) e. ran H )
9		ghomdiv.3	. . . . 5 |- C = ( /g ` H )
10	3, 9	grponpcan 27397	. . . 4 |- ( ( H e. GrpOp /\ ( F ` A ) e. ran H /\ ( F ` B ) e. ran H ) -> ( ( ( F ` A ) C ( F ` B ) ) H ( F ` B ) ) = ( F ` A ) )
11	1, 6, 8, 10	syl3anc 1326	. . 3 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( A e. X /\ B e. X ) ) -> ( ( ( F ` A ) C ( F ` B ) ) H ( F ` B ) ) = ( F ` A ) )
12		ghomdiv.2	. . . . . . 7 |- D = ( /g ` G )
13	2, 12	grponpcan 27397	. . . . . 6 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( ( A D B ) G B ) = A )
14	13	3expb 1266	. . . . 5 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X ) ) -> ( ( A D B ) G B ) = A )
15	14	3ad2antl1 1223	. . . 4 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( A e. X /\ B e. X ) ) -> ( ( A D B ) G B ) = A )
16	15	fveq2d 6195	. . 3 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( A e. X /\ B e. X ) ) -> ( F ` ( ( A D B ) G B ) ) = ( F ` A ) )
17	2, 12	grpodivcl 27393	. . . . . . 7 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A D B ) e. X )
18	17	3expb 1266	. . . . . 6 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X ) ) -> ( A D B ) e. X )
19		simprr 796	. . . . . 6 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X ) ) -> B e. X )
20	18, 19	jca 554	. . . . 5 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X ) ) -> ( ( A D B ) e. X /\ B e. X ) )
21	20	3ad2antl1 1223	. . . 4 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( A e. X /\ B e. X ) ) -> ( ( A D B ) e. X /\ B e. X ) )
22	2	ghomlinOLD 33687	. . . . 5 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( ( A D B ) e. X /\ B e. X ) ) -> ( ( F ` ( A D B ) ) H ( F ` B ) ) = ( F ` ( ( A D B ) G B ) ) )
23	22	eqcomd 2628	. . . 4 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( ( A D B ) e. X /\ B e. X ) ) -> ( F ` ( ( A D B ) G B ) ) = ( ( F ` ( A D B ) ) H ( F ` B ) ) )
24	21, 23	syldan 487	. . 3 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( A e. X /\ B e. X ) ) -> ( F ` ( ( A D B ) G B ) ) = ( ( F ` ( A D B ) ) H ( F ` B ) ) )
25	11, 16, 24	3eqtr2rd 2663	. 2 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( A e. X /\ B e. X ) ) -> ( ( F ` ( A D B ) ) H ( F ` B ) ) = ( ( ( F ` A ) C ( F ` B ) ) H ( F ` B ) ) )
26	18	3ad2antl1 1223	. . . 4 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( A e. X /\ B e. X ) ) -> ( A D B ) e. X )
27	4	ffvelrnda 6359	. . . 4 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( A D B ) e. X ) -> ( F ` ( A D B ) ) e. ran H )
28	26, 27	syldan 487	. . 3 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( A e. X /\ B e. X ) ) -> ( F ` ( A D B ) ) e. ran H )
29	3, 9	grpodivcl 27393	. . . 4 |- ( ( H e. GrpOp /\ ( F ` A ) e. ran H /\ ( F ` B ) e. ran H ) -> ( ( F ` A ) C ( F ` B ) ) e. ran H )
30	1, 6, 8, 29	syl3anc 1326	. . 3 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( A e. X /\ B e. X ) ) -> ( ( F ` A ) C ( F ` B ) ) e. ran H )
31	3	grporcan 27372	. . 3 |- ( ( H e. GrpOp /\ ( ( F ` ( A D B ) ) e. ran H /\ ( ( F ` A ) C ( F ` B ) ) e. ran H /\ ( F ` B ) e. ran H ) ) -> ( ( ( F ` ( A D B ) ) H ( F ` B ) ) = ( ( ( F ` A ) C ( F ` B ) ) H ( F ` B ) ) <-> ( F ` ( A D B ) ) = ( ( F ` A ) C ( F ` B ) ) ) )
32	1, 28, 30, 8, 31	syl13anc 1328	. 2 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( A e. X /\ B e. X ) ) -> ( ( ( F ` ( A D B ) ) H ( F ` B ) ) = ( ( ( F ` A ) C ( F ` B ) ) H ( F ` B ) ) <-> ( F ` ( A D B ) ) = ( ( F ` A ) C ( F ` B ) ) ) )
33	25, 32	mpbid 222	1 |- ( ( ( G e. GrpOp /\ H e. GrpOp /\ F e. ( G GrpOpHom H ) ) /\ ( A e. X /\ B e. X ) ) -> ( F ` ( A D B ) ) = ( ( F ` A ) C ( F ` B ) ) )

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   /g cgs 27346   GrpOpHom cghomOLD 33682

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  df-ghomOLD 33683

This theorem is referenced by:  grpokerinj  33692  rngohomsub  33772