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Theorem ghmpropd 14720
Description: Group homomorphism depends only on the group attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
ghmpropd.a  |-  ( ph  ->  B  =  ( Base `  J ) )
ghmpropd.b  |-  ( ph  ->  C  =  ( Base `  K ) )
ghmpropd.c  |-  ( ph  ->  B  =  ( Base `  L ) )
ghmpropd.d  |-  ( ph  ->  C  =  ( Base `  M ) )
ghmpropd.e  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  J ) y )  =  ( x ( +g  `  L ) y ) )
ghmpropd.f  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  C ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  M ) y ) )
Assertion
Ref Expression
ghmpropd  |-  ( ph  ->  ( J  GrpHom  K )  =  ( L  GrpHom  M ) )
Distinct variable groups:    x, y, J    x, K, y    x, L, y    x, M, y    ph, x, y    x, B, y    x, C, y

Proof of Theorem ghmpropd
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 ghmpropd.a . . . . . 6  |-  ( ph  ->  B  =  ( Base `  J ) )
2 ghmpropd.c . . . . . 6  |-  ( ph  ->  B  =  ( Base `  L ) )
3 ghmpropd.e . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  J ) y )  =  ( x ( +g  `  L ) y ) )
41, 2, 3grppropd 14500 . . . . 5  |-  ( ph  ->  ( J  e.  Grp  <->  L  e.  Grp ) )
5 ghmpropd.b . . . . . 6  |-  ( ph  ->  C  =  ( Base `  K ) )
6 ghmpropd.d . . . . . 6  |-  ( ph  ->  C  =  ( Base `  M ) )
7 ghmpropd.f . . . . . 6  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  C ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  M ) y ) )
85, 6, 7grppropd 14500 . . . . 5  |-  ( ph  ->  ( K  e.  Grp  <->  M  e.  Grp ) )
94, 8anbi12d 691 . . . 4  |-  ( ph  ->  ( ( J  e. 
Grp  /\  K  e.  Grp )  <->  ( L  e. 
Grp  /\  M  e.  Grp ) ) )
101, 5, 2, 6, 3, 7mhmpropd 14421 . . . . 5  |-  ( ph  ->  ( J MndHom  K )  =  ( L MndHom  M
) )
1110eleq2d 2350 . . . 4  |-  ( ph  ->  ( f  e.  ( J MndHom  K )  <->  f  e.  ( L MndHom  M ) ) )
129, 11anbi12d 691 . . 3  |-  ( ph  ->  ( ( ( J  e.  Grp  /\  K  e.  Grp )  /\  f  e.  ( J MndHom  K ) )  <->  ( ( L  e.  Grp  /\  M  e.  Grp )  /\  f  e.  ( L MndHom  M ) ) ) )
13 ghmgrp1 14685 . . . . 5  |-  ( f  e.  ( J  GrpHom  K )  ->  J  e.  Grp )
14 ghmgrp2 14686 . . . . 5  |-  ( f  e.  ( J  GrpHom  K )  ->  K  e.  Grp )
1513, 14jca 518 . . . 4  |-  ( f  e.  ( J  GrpHom  K )  ->  ( J  e.  Grp  /\  K  e. 
Grp ) )
16 ghmmhmb 14694 . . . . 5  |-  ( ( J  e.  Grp  /\  K  e.  Grp )  ->  ( J  GrpHom  K )  =  ( J MndHom  K
) )
1716eleq2d 2350 . . . 4  |-  ( ( J  e.  Grp  /\  K  e.  Grp )  ->  ( f  e.  ( J  GrpHom  K )  <->  f  e.  ( J MndHom  K ) ) )
1815, 17biadan2 623 . . 3  |-  ( f  e.  ( J  GrpHom  K )  <->  ( ( J  e.  Grp  /\  K  e.  Grp )  /\  f  e.  ( J MndHom  K ) ) )
19 ghmgrp1 14685 . . . . 5  |-  ( f  e.  ( L  GrpHom  M )  ->  L  e.  Grp )
20 ghmgrp2 14686 . . . . 5  |-  ( f  e.  ( L  GrpHom  M )  ->  M  e.  Grp )
2119, 20jca 518 . . . 4  |-  ( f  e.  ( L  GrpHom  M )  ->  ( L  e.  Grp  /\  M  e. 
Grp ) )
22 ghmmhmb 14694 . . . . 5  |-  ( ( L  e.  Grp  /\  M  e.  Grp )  ->  ( L  GrpHom  M )  =  ( L MndHom  M
) )
2322eleq2d 2350 . . . 4  |-  ( ( L  e.  Grp  /\  M  e.  Grp )  ->  ( f  e.  ( L  GrpHom  M )  <->  f  e.  ( L MndHom  M ) ) )
2421, 23biadan2 623 . . 3  |-  ( f  e.  ( L  GrpHom  M )  <->  ( ( L  e.  Grp  /\  M  e.  Grp )  /\  f  e.  ( L MndHom  M ) ) )
2512, 18, 243bitr4g 279 . 2  |-  ( ph  ->  ( f  e.  ( J  GrpHom  K )  <->  f  e.  ( L  GrpHom  M ) ) )
2625eqrdv 2281 1  |-  ( ph  ->  ( J  GrpHom  K )  =  ( L  GrpHom  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   MndHom cmhm 14413    GrpHom cghm 14680
This theorem is referenced by:  rhmpropd  15580  lmhmpropd  15826
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-rmo 2551  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-riota 6304  df-map 6774  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-ghm 14681
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