Theorem ghmpropd 14036

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 13772	. . . . 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 13772	. . . . 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 13721	. . . . 5 |- ( ph -> ( J MndHom K ) = ( L MndHom M ) )
11	10	eleq2d 2304	. . . 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 13998	. . . . 5 |- ( f e. ( J GrpHom K ) -> J e. Grp )
14		ghmgrp2 13999	. . . . 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 14007	. . . . 5 |- ( ( J e. Grp /\ K e. Grp ) -> ( J GrpHom K ) = ( J MndHom K ) )
17	16	eleq2d 2304	. . . 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 13998	. . . . 5 |- ( f e. ( L GrpHom M ) -> L e. Grp )
20		ghmgrp2 13999	. . . . 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 14007	. . . . 5 |- ( ( L e. Grp /\ M e. Grp ) -> ( L GrpHom M ) = ( L MndHom M ) )
23	22	eleq2d 2304	. . . 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 2232	1 |- ( ph -> ( J GrpHom K ) = ( L GrpHom M ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   MndHom cmhm 13712   Grpcgrp 13755   GrpHom cghm 13993

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-mhm 13714  df-grp 13758  df-ghm 13994

This theorem is referenced by:  rhmpropd  14500