Theorem resmhm 13569

Description: Restriction of a monoid homomorphism to a submonoid is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)

Hypothesis

Ref	Expression
resmhm.u	|- U = ( S ↾s X )

Assertion

Ref	Expression
resmhm	|- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( F |` X ) e. ( U MndHom T ) )

Proof of Theorem resmhm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mhmrcl2 13546	. . 3 |- ( F e. ( S MndHom T ) -> T e. Mnd )
2		resmhm.u	. . . 4 |- U = ( S ↾s X )
3	2	submmnd 13562	. . 3 |- ( X e. (SubMnd ` S ) -> U e. Mnd )
4	1, 3	anim12ci 339	. 2 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( U e. Mnd /\ T e. Mnd ) )
5		eqid 2231	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
6		eqid 2231	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
7	5, 6	mhmf 13547	. . . . 5 |- ( F e. ( S MndHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
8	5	submss 13558	. . . . 5 |- ( X e. (SubMnd ` S ) -> X C_ ( Base ` S ) )
9		fssres 5512	. . . . 5 |- ( ( F : ( Base ` S ) --> ( Base ` T ) /\ X C_ ( Base ` S ) ) -> ( F |` X ) : X --> ( Base ` T ) )
10	7, 8, 9	syl2an 289	. . . 4 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( F |` X ) : X --> ( Base ` T ) )
11	2	a1i 9	. . . . . 6 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> U = ( S ↾s X ) )
12		eqidd 2232	. . . . . 6 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( Base ` S ) = ( Base ` S ) )
13		submrcl 13553	. . . . . . 7 |- ( X e. (SubMnd ` S ) -> S e. Mnd )
14	13	adantl 277	. . . . . 6 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> S e. Mnd )
15	8	adantl 277	. . . . . 6 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> X C_ ( Base ` S ) )
16	11, 12, 14, 15	ressbas2d 13150	. . . . 5 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> X = ( Base ` U ) )
17	16	feq2d 5470	. . . 4 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( ( F |` X ) : X --> ( Base ` T ) <-> ( F |` X ) : ( Base ` U ) --> ( Base ` T ) ) )
18	10, 17	mpbid 147	. . 3 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( F |` X ) : ( Base ` U ) --> ( Base ` T ) )
19		simpll 527	. . . . . . 7 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> F e. ( S MndHom T ) )
20	8	ad2antlr 489	. . . . . . . 8 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> X C_ ( Base ` S ) )
21		simprl 531	. . . . . . . 8 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> x e. X )
22	20, 21	sseldd 3228	. . . . . . 7 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> x e. ( Base ` S ) )
23		simprr 533	. . . . . . . 8 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> y e. X )
24	20, 23	sseldd 3228	. . . . . . 7 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> y e. ( Base ` S ) )
25		eqid 2231	. . . . . . . 8 |- ( +g ` S ) = ( +g ` S )
26		eqid 2231	. . . . . . . 8 |- ( +g ` T ) = ( +g ` T )
27	5, 25, 26	mhmlin 13549	. . . . . . 7 |- ( ( F e. ( S MndHom T ) /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
28	19, 22, 24, 27	syl3anc 1273	. . . . . 6 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
29	25	submcl 13561	. . . . . . . . 9 |- ( ( X e. (SubMnd ` S ) /\ x e. X /\ y e. X ) -> ( x ( +g ` S ) y ) e. X )
30	29	3expb 1230	. . . . . . . 8 |- ( ( X e. (SubMnd ` S ) /\ ( x e. X /\ y e. X ) ) -> ( x ( +g ` S ) y ) e. X )
31	30	adantll 476	. . . . . . 7 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> ( x ( +g ` S ) y ) e. X )
32	31	fvresd 5664	. . . . . 6 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> ( ( F |` X ) ` ( x ( +g ` S ) y ) ) = ( F ` ( x ( +g ` S ) y ) ) )
33		fvres 5663	. . . . . . . 8 |- ( x e. X -> ( ( F |` X ) ` x ) = ( F ` x ) )
34		fvres 5663	. . . . . . . 8 |- ( y e. X -> ( ( F |` X ) ` y ) = ( F ` y ) )
35	33, 34	oveqan12d 6036	. . . . . . 7 |- ( ( x e. X /\ y e. X ) -> ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
36	35	adantl 277	. . . . . 6 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
37	28, 32, 36	3eqtr4d 2274	. . . . 5 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> ( ( F |` X ) ` ( x ( +g ` S ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) )
38	37	ralrimivva 2614	. . . 4 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> A. x e. X A. y e. X ( ( F |` X ) ` ( x ( +g ` S ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) )
39	2	a1i 9	. . . . . . . . . 10 |- ( X e. (SubMnd ` S ) -> U = ( S ↾s X ) )
40		eqidd 2232	. . . . . . . . . 10 |- ( X e. (SubMnd ` S ) -> ( +g ` S ) = ( +g ` S ) )
41		id 19	. . . . . . . . . 10 |- ( X e. (SubMnd ` S ) -> X e. (SubMnd ` S ) )
42	39, 40, 41, 13	ressplusgd 13211	. . . . . . . . 9 |- ( X e. (SubMnd ` S ) -> ( +g ` S ) = ( +g ` U ) )
43	42	adantl 277	. . . . . . . 8 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( +g ` S ) = ( +g ` U ) )
44	43	oveqd 6034	. . . . . . 7 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( x ( +g ` S ) y ) = ( x ( +g ` U ) y ) )
45	44	fveqeq2d 5647	. . . . . 6 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( ( ( F |` X ) ` ( x ( +g ` S ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) <-> ( ( F |` X ) ` ( x ( +g ` U ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) ) )
46	16, 45	raleqbidv 2746	. . . . 5 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( A. y e. X ( ( F |` X ) ` ( x ( +g ` S ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) <-> A. y e. ( Base ` U ) ( ( F |` X ) ` ( x ( +g ` U ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) ) )
47	16, 46	raleqbidv 2746	. . . 4 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( A. x e. X A. y e. X ( ( F |` X ) ` ( x ( +g ` S ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) <-> A. x e. ( Base ` U ) A. y e. ( Base ` U ) ( ( F |` X ) ` ( x ( +g ` U ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) ) )
48	38, 47	mpbid 147	. . 3 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> A. x e. ( Base ` U ) A. y e. ( Base ` U ) ( ( F |` X ) ` ( x ( +g ` U ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) )
49		eqid 2231	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
50	49	subm0cl 13560	. . . . . 6 |- ( X e. (SubMnd ` S ) -> ( 0g ` S ) e. X )
51	50	adantl 277	. . . . 5 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( 0g ` S ) e. X )
52	51	fvresd 5664	. . . 4 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( ( F |` X ) ` ( 0g ` S ) ) = ( F ` ( 0g ` S ) ) )
53	2, 49	subm0 13564	. . . . . 6 |- ( X e. (SubMnd ` S ) -> ( 0g ` S ) = ( 0g ` U ) )
54	53	adantl 277	. . . . 5 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( 0g ` S ) = ( 0g ` U ) )
55	54	fveq2d 5643	. . . 4 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( ( F |` X ) ` ( 0g ` S ) ) = ( ( F |` X ) ` ( 0g ` U ) ) )
56		eqid 2231	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
57	49, 56	mhm0 13550	. . . . 5 |- ( F e. ( S MndHom T ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
58	57	adantr 276	. . . 4 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
59	52, 55, 58	3eqtr3d 2272	. . 3 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( ( F |` X ) ` ( 0g ` U ) ) = ( 0g ` T ) )
60	18, 48, 59	3jca 1203	. 2 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( ( F |` X ) : ( Base ` U ) --> ( Base ` T ) /\ A. x e. ( Base ` U ) A. y e. ( Base ` U ) ( ( F |` X ) ` ( x ( +g ` U ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) /\ ( ( F |` X ) ` ( 0g ` U ) ) = ( 0g ` T ) ) )
61		eqid 2231	. . 3 |- ( Base ` U ) = ( Base ` U )
62		eqid 2231	. . 3 |- ( +g ` U ) = ( +g ` U )
63		eqid 2231	. . 3 |- ( 0g ` U ) = ( 0g ` U )
64	61, 6, 62, 26, 63, 56	ismhm 13543	. 2 |- ( ( F |` X ) e. ( U MndHom T ) <-> ( ( U e. Mnd /\ T e. Mnd ) /\ ( ( F |` X ) : ( Base ` U ) --> ( Base ` T ) /\ A. x e. ( Base ` U ) A. y e. ( Base ` U ) ( ( F |` X ) ` ( x ( +g ` U ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) /\ ( ( F |` X ) ` ( 0g ` U ) ) = ( 0g ` T ) ) ) )
65	4, 60, 64	sylanbrc 417	1 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( F |` X ) e. ( U MndHom T ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   A.wral 2510   C_ wss 3200   |` cres 4727   -->wf 5322   ` cfv 5326  (class class class)co 6017   Basecbs 13081   ↾_s cress 13082   +g cplusg 13159   0gc0g 13338   Mndcmnd 13498   MndHom cmhm 13539  SubMndcsubmnd 13540

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-mhm 13541  df-submnd 13542

This theorem is referenced by:  resrhm  14261