Theorem ghmpropd 13489

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 13219	. . . . 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 13219	. . . . 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 13168	. . . . 5 |- ( ph -> ( J MndHom K ) = ( L MndHom M ) )
11	10	eleq2d 2266	. . . 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 13451	. . . . 5 |- ( f e. ( J GrpHom K ) -> J e. Grp )
14		ghmgrp2 13452	. . . . 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 13460	. . . . 5 |- ( ( J e. Grp /\ K e. Grp ) -> ( J GrpHom K ) = ( J MndHom K ) )
17	16	eleq2d 2266	. . . 4 |- ( ( J e. Grp /\ K e. Grp ) -> ( f e. ( J GrpHom K ) <-> f e. ( J MndHom K ) ) )
18	15, 17	biadanii 613	. . 3 |- ( f e. ( J GrpHom K ) <-> ( ( J e. Grp /\ K e. Grp ) /\ f e. ( J MndHom K ) ) )
19		ghmgrp1 13451	. . . . 5 |- ( f e. ( L GrpHom M ) -> L e. Grp )
20		ghmgrp2 13452	. . . . 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 13460	. . . . 5 |- ( ( L e. Grp /\ M e. Grp ) -> ( L GrpHom M ) = ( L MndHom M ) )
23	22	eleq2d 2266	. . . 4 |- ( ( L e. Grp /\ M e. Grp ) -> ( f e. ( L GrpHom M ) <-> f e. ( L MndHom M ) ) )
24	21, 23	biadanii 613	. . 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 2194	1 |- ( ph -> ( J GrpHom K ) = ( L GrpHom M ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780   MndHom cmhm 13159   Grpcgrp 13202   GrpHom cghm 13446

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-mhm 13161  df-grp 13205  df-ghm 13447

This theorem is referenced by:  rhmpropd  13886