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Theorem ghomdiv 26574
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 959 . . . 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 2283 . . . . . . 7  |-  ran  H  =  ran  H
42, 3ghomf 26572 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X --> ran  H )
5 ffvelrn 5663 . . . . . 6  |-  ( ( F : X --> ran  H  /\  A  e.  X
)  ->  ( F `  A )  e.  ran  H )
64, 5sylan 457 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  A  e.  X )  ->  ( F `  A )  e.  ran  H )
76adantrr 697 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( F `  A
)  e.  ran  H
)
8 ffvelrn 5663 . . . . . 6  |-  ( ( F : X --> ran  H  /\  B  e.  X
)  ->  ( F `  B )  e.  ran  H )
94, 8sylan 457 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  B  e.  X )  ->  ( F `  B )  e.  ran  H )
109adantrl 696 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( F `  B
)  e.  ran  H
)
11 ghomdiv.3 . . . . 5  |-  C  =  (  /g  `  H
)
123, 11grponpcan 20919 . . . 4  |-  ( ( H  e.  GrpOp  /\  ( F `  A )  e.  ran  H  /\  ( F `  B )  e.  ran  H )  -> 
( ( ( F `
 A ) C ( F `  B
) ) H ( F `  B ) )  =  ( F `
 A ) )
131, 7, 10, 12syl3anc 1182 . . 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 ) )
14 ghomdiv.2 . . . . . . 7  |-  D  =  (  /g  `  G
)
152, 14grponpcan 20919 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B ) G B )  =  A )
16153expb 1152 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( A D B ) G B )  =  A )
17163ad2antl1 1117 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( ( A D B ) G B )  =  A )
1817fveq2d 5529 . . 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 ) )
192, 14grpodivcl 20914 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  X )
20193expb 1152 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A D B )  e.  X
)
21 simprr 733 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  B  e.  X )
2220, 21jca 518 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( A D B )  e.  X  /\  B  e.  X ) )
23223ad2antl1 1117 . . . 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
) )
242ghomlin 21031 . . . . 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 ) ) )
2524eqcomd 2288 . . . 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 ) ) )
2623, 25syldan 456 . . 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 ) ) )
2713, 18, 263eqtr2rd 2322 . 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
) ) )
28203ad2antl1 1117 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( A D B )  e.  X )
29 ffvelrn 5663 . . . . 5  |-  ( ( F : X --> ran  H  /\  ( A D B )  e.  X )  ->  ( F `  ( A D B ) )  e.  ran  H
)
304, 29sylan 457 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( A D B )  e.  X
)  ->  ( F `  ( A D B ) )  e.  ran  H )
3128, 30syldan 456 . . 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 )
323, 11grpodivcl 20914 . . . 4  |-  ( ( H  e.  GrpOp  /\  ( F `  A )  e.  ran  H  /\  ( F `  B )  e.  ran  H )  -> 
( ( F `  A ) C ( F `  B ) )  e.  ran  H
)
331, 7, 10, 32syl3anc 1182 . . 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
)
343grporcan 20888 . . 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 ) ) ) )
351, 31, 33, 10, 34syl13anc 1184 . 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
) ) ) )
3627, 35mpbid 201 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853    /g cgs 20856   GrpOpHom cghom 21024
This theorem is referenced by:  grpokerinj  26575  rngohomsub  26604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ghom 21025
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