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Theorem ghomdiv 31628
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 1001 . . . 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 2402 . . . . . . 7  |-  ran  H  =  ran  H
42, 3ghomf 31626 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X --> ran  H )
54ffvelrnda 6009 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  A  e.  X )  ->  ( F `  A )  e.  ran  H )
65adantrr 715 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( F `  A
)  e.  ran  H
)
74ffvelrnda 6009 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  B  e.  X )  ->  ( F `  B )  e.  ran  H )
87adantrl 714 . . . 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 25668 . . . 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 1230 . . 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 25668 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B ) G B )  =  A )
14133expb 1198 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( A D B ) G B )  =  A )
15143ad2antl1 1159 . . . 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 5853 . . 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 25663 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  X )
18173expb 1198 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A D B )  e.  X
)
19 simprr 758 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  B  e.  X )
2018, 19jca 530 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( A D B )  e.  X  /\  B  e.  X ) )
21203ad2antl1 1159 . . . 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
) )
222ghomlinOLD 25780 . . . . 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 2410 . . . 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 468 . . 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 2450 . 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 1159 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( A D B )  e.  X )
274ffvelrnda 6009 . . . 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 468 . . 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 25663 . . . 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 1230 . . 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 25637 . . 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 1232 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ran crn 4824   ` cfv 5569  (class class class)co 6278   GrpOpcgr 25602    /g cgs 25605   GrpOpHom cghomOLD 25773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-grpo 25607  df-gid 25608  df-ginv 25609  df-gdiv 25610  df-ghomOLD 25774
This theorem is referenced by:  grpokerinj  31629  rngohomsub  31658
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