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Theorem ghmpropd 14970
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 14750 . . . . 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 14750 . . . . 5  |-  ( ph  ->  ( K  e.  Grp  <->  M  e.  Grp ) )
94, 8anbi12d 692 . . . 4  |-  ( ph  ->  ( ( J  e. 
Grp  /\  K  e.  Grp )  <->  ( L  e. 
Grp  /\  M  e.  Grp ) ) )
101, 5, 2, 6, 3, 7mhmpropd 14671 . . . . 5  |-  ( ph  ->  ( J MndHom  K )  =  ( L MndHom  M
) )
1110eleq2d 2454 . . . 4  |-  ( ph  ->  ( f  e.  ( J MndHom  K )  <->  f  e.  ( L MndHom  M ) ) )
129, 11anbi12d 692 . . 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 14935 . . . . 5  |-  ( f  e.  ( J  GrpHom  K )  ->  J  e.  Grp )
14 ghmgrp2 14936 . . . . 5  |-  ( f  e.  ( J  GrpHom  K )  ->  K  e.  Grp )
1513, 14jca 519 . . . 4  |-  ( f  e.  ( J  GrpHom  K )  ->  ( J  e.  Grp  /\  K  e. 
Grp ) )
16 ghmmhmb 14944 . . . . 5  |-  ( ( J  e.  Grp  /\  K  e.  Grp )  ->  ( J  GrpHom  K )  =  ( J MndHom  K
) )
1716eleq2d 2454 . . . 4  |-  ( ( J  e.  Grp  /\  K  e.  Grp )  ->  ( f  e.  ( J  GrpHom  K )  <->  f  e.  ( J MndHom  K ) ) )
1815, 17biadan2 624 . . 3  |-  ( f  e.  ( J  GrpHom  K )  <->  ( ( J  e.  Grp  /\  K  e.  Grp )  /\  f  e.  ( J MndHom  K ) ) )
19 ghmgrp1 14935 . . . . 5  |-  ( f  e.  ( L  GrpHom  M )  ->  L  e.  Grp )
20 ghmgrp2 14936 . . . . 5  |-  ( f  e.  ( L  GrpHom  M )  ->  M  e.  Grp )
2119, 20jca 519 . . . 4  |-  ( f  e.  ( L  GrpHom  M )  ->  ( L  e.  Grp  /\  M  e. 
Grp ) )
22 ghmmhmb 14944 . . . . 5  |-  ( ( L  e.  Grp  /\  M  e.  Grp )  ->  ( L  GrpHom  M )  =  ( L MndHom  M
) )
2322eleq2d 2454 . . . 4  |-  ( ( L  e.  Grp  /\  M  e.  Grp )  ->  ( f  e.  ( L  GrpHom  M )  <->  f  e.  ( L MndHom  M ) ) )
2421, 23biadan2 624 . . 3  |-  ( f  e.  ( L  GrpHom  M )  <->  ( ( L  e.  Grp  /\  M  e.  Grp )  /\  f  e.  ( L MndHom  M ) ) )
2512, 18, 243bitr4g 280 . 2  |-  ( ph  ->  ( f  e.  ( J  GrpHom  K )  <->  f  e.  ( L  GrpHom  M ) ) )
2625eqrdv 2385 1  |-  ( ph  ->  ( J  GrpHom  K )  =  ( L  GrpHom  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   ` cfv 5394  (class class class)co 6020   Basecbs 13396   +g cplusg 13456   Grpcgrp 14612   MndHom cmhm 14663    GrpHom cghm 14930
This theorem is referenced by:  rhmpropd  15830  lmhmpropd  16072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-map 6956  df-0g 13654  df-mnd 14617  df-mhm 14665  df-grp 14739  df-ghm 14931
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