Theorem ghmpropd 13619

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 13349	. . . . 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 13349	. . . . 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 13298	. . . . 5 |- ( ph -> ( J MndHom K ) = ( L MndHom M ) )
11	10	eleq2d 2275	. . . 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 13581	. . . . 5 |- ( f e. ( J GrpHom K ) -> J e. Grp )
14		ghmgrp2 13582	. . . . 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 13590	. . . . 5 |- ( ( J e. Grp /\ K e. Grp ) -> ( J GrpHom K ) = ( J MndHom K ) )
17	16	eleq2d 2275	. . . 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 13581	. . . . 5 |- ( f e. ( L GrpHom M ) -> L e. Grp )
20		ghmgrp2 13582	. . . . 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 13590	. . . . 5 |- ( ( L e. Grp /\ M e. Grp ) -> ( L GrpHom M ) = ( L MndHom M ) )
23	22	eleq2d 2275	. . . 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 2203	1 |- ( ph -> ( J GrpHom K ) = ( L GrpHom M ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   MndHom cmhm 13289   Grpcgrp 13332   GrpHom cghm 13576

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-map 6737  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-mhm 13291  df-grp 13335  df-ghm 13577

This theorem is referenced by:  rhmpropd  14016