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Theorem ghmpropd 15035
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 14815 . . . . 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 14815 . . . . 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 14736 . . . . 5  |-  ( ph  ->  ( J MndHom  K )  =  ( L MndHom  M
) )
1110eleq2d 2502 . . . 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 15000 . . . . 5  |-  ( f  e.  ( J  GrpHom  K )  ->  J  e.  Grp )
14 ghmgrp2 15001 . . . . 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 15009 . . . . 5  |-  ( ( J  e.  Grp  /\  K  e.  Grp )  ->  ( J  GrpHom  K )  =  ( J MndHom  K
) )
1716eleq2d 2502 . . . 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 15000 . . . . 5  |-  ( f  e.  ( L  GrpHom  M )  ->  L  e.  Grp )
20 ghmgrp2 15001 . . . . 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 15009 . . . . 5  |-  ( ( L  e.  Grp  /\  M  e.  Grp )  ->  ( L  GrpHom  M )  =  ( L MndHom  M
) )
2322eleq2d 2502 . . . 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 2433 1  |-  ( ph  ->  ( J  GrpHom  K )  =  ( L  GrpHom  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   Grpcgrp 14677   MndHom cmhm 14728    GrpHom cghm 14995
This theorem is referenced by:  rhmpropd  15895  lmhmpropd  16137
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-map 7012  df-0g 13719  df-mnd 14682  df-mhm 14730  df-grp 14804  df-ghm 14996
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