Theorem ghmpropd 13356

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 13092	. . . . 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 13092	. . . . 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 13041	. . . . 5 |- ( ph -> ( J MndHom K ) = ( L MndHom M ) )
11	10	eleq2d 2263	. . . 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 13318	. . . . 5 |- ( f e. ( J GrpHom K ) -> J e. Grp )
14		ghmgrp2 13319	. . . . 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 13327	. . . . 5 |- ( ( J e. Grp /\ K e. Grp ) -> ( J GrpHom K ) = ( J MndHom K ) )
17	16	eleq2d 2263	. . . 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 13318	. . . . 5 |- ( f e. ( L GrpHom M ) -> L e. Grp )
20		ghmgrp2 13319	. . . . 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 13327	. . . . 5 |- ( ( L e. Grp /\ M e. Grp ) -> ( L GrpHom M ) = ( L MndHom M ) )
23	22	eleq2d 2263	. . . 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 2191	1 |- ( ph -> ( J GrpHom K ) = ( L GrpHom M ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   MndHom cmhm 13032   Grpcgrp 13075   GrpHom cghm 13313

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-map 6706  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-mhm 13034  df-grp 13078  df-ghm 13314

This theorem is referenced by:  rhmpropd  13753