Theorem mhmfmhm 17538

Description: The function fulfilling the conditions of mhmmnd 17537 is a monoid homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.)

Hypotheses

Ref	Expression
ghmgrp.f	|- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
ghmgrp.x	|- X = ( Base ` G )
ghmgrp.y	|- Y = ( Base ` H )
ghmgrp.p	|- .+ = ( +g ` G )
ghmgrp.q	|- .+^ = ( +g ` H )
ghmgrp.1	|- ( ph -> F : X -onto-> Y )
mhmmnd.3	|- ( ph -> G e. Mnd )

Assertion

Ref	Expression
mhmfmhm	|- ( ph -> F e. ( G MndHom H ) )

Distinct variable groups:   x, F, y   x, G, y   x, .+ , y   x, H, y   x, X, y   x, Y, y   x, .+^ , y   ph, x, y

Proof of Theorem mhmfmhm

Step	Hyp	Ref	Expression
1		mhmmnd.3	. . 3 |- ( ph -> G e. Mnd )
2		ghmgrp.f	. . . 4 |- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
3		ghmgrp.x	. . . 4 |- X = ( Base ` G )
4		ghmgrp.y	. . . 4 |- Y = ( Base ` H )
5		ghmgrp.p	. . . 4 |- .+ = ( +g ` G )
6		ghmgrp.q	. . . 4 |- .+^ = ( +g ` H )
7		ghmgrp.1	. . . 4 |- ( ph -> F : X -onto-> Y )
8	2, 3, 4, 5, 6, 7, 1	mhmmnd 17537	. . 3 |- ( ph -> H e. Mnd )
9		fof 6115	. . . . 5 |- ( F : X -onto-> Y -> F : X --> Y )
10	7, 9	syl 17	. . . 4 |- ( ph -> F : X --> Y )
11	2	3expb 1266	. . . . 5 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
12	11	ralrimivva 2971	. . . 4 |- ( ph -> A. x e. X A. y e. X ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
13		eqid 2622	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
14	2, 3, 4, 5, 6, 7, 1, 13	mhmid 17536	. . . 4 |- ( ph -> ( F ` ( 0g ` G ) ) = ( 0g ` H ) )
15	10, 12, 14	3jca 1242	. . 3 |- ( ph -> ( F : X --> Y /\ A. x e. X A. y e. X ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) /\ ( F ` ( 0g ` G ) ) = ( 0g ` H ) ) )
16	1, 8, 15	jca31 557	. 2 |- ( ph -> ( ( G e. Mnd /\ H e. Mnd ) /\ ( F : X --> Y /\ A. x e. X A. y e. X ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) /\ ( F ` ( 0g ` G ) ) = ( 0g ` H ) ) ) )
17		eqid 2622	. . 3 |- ( 0g ` H ) = ( 0g ` H )
18	3, 4, 5, 6, 13, 17	ismhm 17337	. 2 |- ( F e. ( G MndHom H ) <-> ( ( G e. Mnd /\ H e. Mnd ) /\ ( F : X --> Y /\ A. x e. X A. y e. X ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) /\ ( F ` ( 0g ` G ) ) = ( 0g ` H ) ) ) )
19	16, 18	sylibr 224	1 |- ( ph -> F e. ( G MndHom H ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Mndcmnd 17294   MndHom cmhm 17333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335

This theorem is referenced by: (None)