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Theorem ghomdiv 29800
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 995 . . . 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 2460 . . . . . . 7  |-  ran  H  =  ran  H
42, 3ghomf 29798 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X --> ran  H )
54ffvelrnda 6012 . . . . 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 6012 . . . . 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 24780 . . . 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 1223 . . 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 24780 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B ) G B )  =  A )
14133expb 1192 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( A D B ) G B )  =  A )
15143ad2antl1 1153 . . . 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 5861 . . 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 24775 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  X )
18173expb 1192 . . . . . 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 1153 . . . 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 24892 . . . . 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 2468 . . . 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 2508 . 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 1153 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( A D B )  e.  X )
274ffvelrnda 6012 . . . 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 24775 . . . 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 1223 . . 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 24749 . . 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 1225 . 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 968    = wceq 1374    e. wcel 1762   ran crn 4993   ` cfv 5579  (class class class)co 6275   GrpOpcgr 24714    /g cgs 24717   GrpOpHom cghom 24885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-grpo 24719  df-gid 24720  df-ginv 24721  df-gdiv 24722  df-ghom 24886
This theorem is referenced by:  grpokerinj  29801  rngohomsub  29830
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