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Theorem ghmpropd 16919
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 16683 . . . . 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 16683 . . . . 5  |-  ( ph  ->  ( K  e.  Grp  <->  M  e.  Grp ) )
94, 8anbi12d 715 . . . 4  |-  ( ph  ->  ( ( J  e. 
Grp  /\  K  e.  Grp )  <->  ( L  e. 
Grp  /\  M  e.  Grp ) ) )
101, 5, 2, 6, 3, 7mhmpropd 16587 . . . . 5  |-  ( ph  ->  ( J MndHom  K )  =  ( L MndHom  M
) )
1110eleq2d 2492 . . . 4  |-  ( ph  ->  ( f  e.  ( J MndHom  K )  <->  f  e.  ( L MndHom  M ) ) )
129, 11anbi12d 715 . . 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 16884 . . . . 5  |-  ( f  e.  ( J  GrpHom  K )  ->  J  e.  Grp )
14 ghmgrp2 16885 . . . . 5  |-  ( f  e.  ( J  GrpHom  K )  ->  K  e.  Grp )
1513, 14jca 534 . . . 4  |-  ( f  e.  ( J  GrpHom  K )  ->  ( J  e.  Grp  /\  K  e. 
Grp ) )
16 ghmmhmb 16893 . . . . 5  |-  ( ( J  e.  Grp  /\  K  e.  Grp )  ->  ( J  GrpHom  K )  =  ( J MndHom  K
) )
1716eleq2d 2492 . . . 4  |-  ( ( J  e.  Grp  /\  K  e.  Grp )  ->  ( f  e.  ( J  GrpHom  K )  <->  f  e.  ( J MndHom  K ) ) )
1815, 17biadan2 646 . . 3  |-  ( f  e.  ( J  GrpHom  K )  <->  ( ( J  e.  Grp  /\  K  e.  Grp )  /\  f  e.  ( J MndHom  K ) ) )
19 ghmgrp1 16884 . . . . 5  |-  ( f  e.  ( L  GrpHom  M )  ->  L  e.  Grp )
20 ghmgrp2 16885 . . . . 5  |-  ( f  e.  ( L  GrpHom  M )  ->  M  e.  Grp )
2119, 20jca 534 . . . 4  |-  ( f  e.  ( L  GrpHom  M )  ->  ( L  e.  Grp  /\  M  e. 
Grp ) )
22 ghmmhmb 16893 . . . . 5  |-  ( ( L  e.  Grp  /\  M  e.  Grp )  ->  ( L  GrpHom  M )  =  ( L MndHom  M
) )
2322eleq2d 2492 . . . 4  |-  ( ( L  e.  Grp  /\  M  e.  Grp )  ->  ( f  e.  ( L  GrpHom  M )  <->  f  e.  ( L MndHom  M ) ) )
2421, 23biadan2 646 . . 3  |-  ( f  e.  ( L  GrpHom  M )  <->  ( ( L  e.  Grp  /\  M  e.  Grp )  /\  f  e.  ( L MndHom  M ) ) )
2512, 18, 243bitr4g 291 . 2  |-  ( ph  ->  ( f  e.  ( J  GrpHom  K )  <->  f  e.  ( L  GrpHom  M ) ) )
2625eqrdv 2419 1  |-  ( ph  ->  ( J  GrpHom  K )  =  ( L  GrpHom  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872   ` cfv 5601  (class class class)co 6305   Basecbs 15120   +g cplusg 15189   MndHom cmhm 16579   Grpcgrp 16668    GrpHom cghm 16879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7485  df-0g 15339  df-mgm 16487  df-sgrp 16526  df-mnd 16536  df-mhm 16581  df-grp 16672  df-ghm 16880
This theorem is referenced by:  rhmpropd  18042  lmhmpropd  18295
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