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Theorem ghomdiv 28889
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
StepHypRef Expression
1 simpl2 992 . . . 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 2451 . . . . . . 7  |-  ran  H  =  ran  H
42, 3ghomf 28887 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X --> ran  H )
54ffvelrnda 5944 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  A  e.  X )  ->  ( F `  A )  e.  ran  H )
65adantrr 716 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( F `  A
)  e.  ran  H
)
74ffvelrnda 5944 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  B  e.  X )  ->  ( F `  B )  e.  ran  H )
87adantrl 715 . . . 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
)
103, 9grponpcan 23876 . . . 4  |-  ( ( H  e.  GrpOp  /\  ( F `  A )  e.  ran  H  /\  ( F `  B )  e.  ran  H )  -> 
( ( ( F `
 A ) C ( F `  B
) ) H ( F `  B ) )  =  ( F `
 A ) )
111, 6, 8, 10syl3anc 1219 . . 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
)
132, 12grponpcan 23876 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B ) G B )  =  A )
14133expb 1189 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( A D B ) G B )  =  A )
15143ad2antl1 1150 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( ( A D B ) G B )  =  A )
1615fveq2d 5795 . . 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 ) )
172, 12grpodivcl 23871 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  X )
18173expb 1189 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A D B )  e.  X
)
19 simprr 756 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  B  e.  X )
2018, 19jca 532 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( A D B )  e.  X  /\  B  e.  X ) )
21203ad2antl1 1150 . . . 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
) )
222ghomlin 23988 . . . . 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 ) ) )
2322eqcomd 2459 . . . 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 ) ) )
2421, 23syldan 470 . . 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 ) ) )
2511, 16, 243eqtr2rd 2499 . 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
) ) )
26183ad2antl1 1150 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( A D B )  e.  X )
274ffvelrnda 5944 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( A D B )  e.  X
)  ->  ( F `  ( A D B ) )  e.  ran  H )
2826, 27syldan 470 . . 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 )
293, 9grpodivcl 23871 . . . 4  |-  ( ( H  e.  GrpOp  /\  ( F `  A )  e.  ran  H  /\  ( F `  B )  e.  ran  H )  -> 
( ( F `  A ) C ( F `  B ) )  e.  ran  H
)
301, 6, 8, 29syl3anc 1219 . . 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
)
313grporcan 23845 . . 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 ) ) ) )
321, 28, 30, 8, 31syl13anc 1221 . 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
) ) ) )
3325, 32mpbid 210 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
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ran crn 4941   ` cfv 5518  (class class class)co 6192   GrpOpcgr 23810    /g cgs 23813   GrpOpHom cghom 23981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-grpo 23815  df-gid 23816  df-ginv 23817  df-gdiv 23818  df-ghom 23982
This theorem is referenced by:  grpokerinj  28890  rngohomsub  28919
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