Theorem resmhm 13633

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 13610	. . 3 |- ( F e. ( S MndHom T ) -> T e. Mnd )
2		resmhm.u	. . . 4 |- U = ( S ↾s X )
3	2	submmnd 13626	. . 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 13611	. . . . 5 |- ( F e. ( S MndHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
8	5	submss 13622	. . . . 5 |- ( X e. (SubMnd ` S ) -> X C_ ( Base ` S ) )
9		fssres 5520	. . . . 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 13617	. . . . . . 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 13214	. . . . 5 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> X = ( Base ` U ) )
17	16	feq2d 5477	. . . 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 3229	. . . . . . 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 3229	. . . . . . 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 13613	. . . . . . 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 1274	. . . . . 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 13625	. . . . . . . . 9 |- ( ( X e. (SubMnd ` S ) /\ x e. X /\ y e. X ) -> ( x ( +g ` S ) y ) e. X )
30	29	3expb 1231	. . . . . . . 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 5673	. . . . . 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 5672	. . . . . . . 8 |- ( x e. X -> ( ( F |` X ) ` x ) = ( F ` x ) )
34		fvres 5672	. . . . . . . 8 |- ( y e. X -> ( ( F |` X ) ` y ) = ( F ` y ) )
35	33, 34	oveqan12d 6047	. . . . . . 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 2615	. . . 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 13275	. . . . . . . . 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 6045	. . . . . . 7 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( x ( +g ` S ) y ) = ( x ( +g ` U ) y ) )
45	44	fveqeq2d 5656	. . . . . 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 2747	. . . . 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 2747	. . . 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 13624	. . . . . 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 5673	. . . 4 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( ( F |` X ) ` ( 0g ` S ) ) = ( F ` ( 0g ` S ) ) )
53	2, 49	subm0 13628	. . . . . 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 5652	. . . 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 13614	. . . . 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 1204	. 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 13607	. 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 1005   = wceq 1398   e. wcel 2202   A.wral 2511   C_ wss 3201   |` cres 4733   -->wf 5329   ` cfv 5333  (class class class)co 6028   Basecbs 13145   ↾_s cress 13146   +g cplusg 13223   0gc0g 13402   Mndcmnd 13562   MndHom cmhm 13603  SubMndcsubmnd 13604

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-iress 13153  df-plusg 13236  df-0g 13404  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-mhm 13605  df-submnd 13606

This theorem is referenced by:  resrhm  14326