Theorem ghmpropd 14000

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.

Step	Hyp	Ref	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 ) )
4	1, 2, 3	grppropd 13730	. . . . 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 ) )
8	5, 6, 7	grppropd 13730	. . . . 5 |- ( ph -> ( K e. Grp <-> M e. Grp ) )
9	4, 8	anbi12d 473	. . . 4 |- ( ph -> ( ( J e. Grp /\ K e. Grp ) <-> ( L e. Grp /\ M e. Grp ) ) )
10	1, 5, 2, 6, 3, 7	mhmpropd 13679	. . . . 5 |- ( ph -> ( J MndHom K ) = ( L MndHom M ) )
11	10	eleq2d 2302	. . . 4 |- ( ph -> ( f e. ( J MndHom K ) <-> f e. ( L MndHom M ) ) )
12	9, 11	anbi12d 473	. . 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 13962	. . . . 5 |- ( f e. ( J GrpHom K ) -> J e. Grp )
14		ghmgrp2 13963	. . . . 5 |- ( f e. ( J GrpHom K ) -> K e. Grp )
15	13, 14	jca 306	. . . 4 |- ( f e. ( J GrpHom K ) -> ( J e. Grp /\ K e. Grp ) )
16		ghmmhmb 13971	. . . . 5 |- ( ( J e. Grp /\ K e. Grp ) -> ( J GrpHom K ) = ( J MndHom K ) )
17	16	eleq2d 2302	. . . 4 |- ( ( J e. Grp /\ K e. Grp ) -> ( f e. ( J GrpHom K ) <-> f e. ( J MndHom K ) ) )
18	15, 17	biadanii 617	. . 3 |- ( f e. ( J GrpHom K ) <-> ( ( J e. Grp /\ K e. Grp ) /\ f e. ( J MndHom K ) ) )
19		ghmgrp1 13962	. . . . 5 |- ( f e. ( L GrpHom M ) -> L e. Grp )
20		ghmgrp2 13963	. . . . 5 |- ( f e. ( L GrpHom M ) -> M e. Grp )
21	19, 20	jca 306	. . . 4 |- ( f e. ( L GrpHom M ) -> ( L e. Grp /\ M e. Grp ) )
22		ghmmhmb 13971	. . . . 5 |- ( ( L e. Grp /\ M e. Grp ) -> ( L GrpHom M ) = ( L MndHom M ) )
23	22	eleq2d 2302	. . . 4 |- ( ( L e. Grp /\ M e. Grp ) -> ( f e. ( L GrpHom M ) <-> f e. ( L MndHom M ) ) )
24	21, 23	biadanii 617	. . 3 |- ( f e. ( L GrpHom M ) <-> ( ( L e. Grp /\ M e. Grp ) /\ f e. ( L MndHom M ) ) )
25	12, 18, 24	3bitr4g 223	. 2 |- ( ph -> ( f e. ( J GrpHom K ) <-> f e. ( L GrpHom M ) ) )
26	25	eqrdv 2230	1 |- ( ph -> ( J GrpHom K ) = ( L GrpHom M ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   MndHom cmhm 13670   Grpcgrp 13713   GrpHom cghm 13957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-map 6884  df-inn 9238  df-2 9296  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-mhm 13672  df-grp 13716  df-ghm 13958

This theorem is referenced by:  rhmpropd  14399