Theorem resmhm 13560

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 13537	. . 3 |- ( F e. ( S MndHom T ) -> T e. Mnd )
2		resmhm.u	. . . 4 |- U = ( S ↾s X )
3	2	submmnd 13553	. . 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 2229	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
6		eqid 2229	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
7	5, 6	mhmf 13538	. . . . 5 |- ( F e. ( S MndHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
8	5	submss 13549	. . . . 5 |- ( X e. (SubMnd ` S ) -> X C_ ( Base ` S ) )
9		fssres 5509	. . . . 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 2230	. . . . . 6 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( Base ` S ) = ( Base ` S ) )
13		submrcl 13544	. . . . . . 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 13141	. . . . 5 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> X = ( Base ` U ) )
17	16	feq2d 5467	. . . 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 529	. . . . . . . 8 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> x e. X )
22	20, 21	sseldd 3226	. . . . . . 7 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> x e. ( Base ` S ) )
23		simprr 531	. . . . . . . 8 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> y e. X )
24	20, 23	sseldd 3226	. . . . . . 7 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> y e. ( Base ` S ) )
25		eqid 2229	. . . . . . . 8 |- ( +g ` S ) = ( +g ` S )
26		eqid 2229	. . . . . . . 8 |- ( +g ` T ) = ( +g ` T )
27	5, 25, 26	mhmlin 13540	. . . . . . 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 1271	. . . . . 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 13552	. . . . . . . . 9 |- ( ( X e. (SubMnd ` S ) /\ x e. X /\ y e. X ) -> ( x ( +g ` S ) y ) e. X )
30	29	3expb 1228	. . . . . . . 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 5660	. . . . . 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 5659	. . . . . . . 8 |- ( x e. X -> ( ( F |` X ) ` x ) = ( F ` x ) )
34		fvres 5659	. . . . . . . 8 |- ( y e. X -> ( ( F |` X ) ` y ) = ( F ` y ) )
35	33, 34	oveqan12d 6032	. . . . . . 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 2272	. . . . 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 2612	. . . 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 2230	. . . . . . . . . 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 13202	. . . . . . . . 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 6030	. . . . . . 7 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( x ( +g ` S ) y ) = ( x ( +g ` U ) y ) )
45	44	fveqeq2d 5643	. . . . . 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 2744	. . . . 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 2744	. . . 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 2229	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
50	49	subm0cl 13551	. . . . . 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 5660	. . . 4 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( ( F |` X ) ` ( 0g ` S ) ) = ( F ` ( 0g ` S ) ) )
53	2, 49	subm0 13555	. . . . . 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 5639	. . . 4 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( ( F |` X ) ` ( 0g ` S ) ) = ( ( F |` X ) ` ( 0g ` U ) ) )
56		eqid 2229	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
57	49, 56	mhm0 13541	. . . . 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 2270	. . 3 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( ( F |` X ) ` ( 0g ` U ) ) = ( 0g ` T ) )
60	18, 48, 59	3jca 1201	. 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 2229	. . 3 |- ( Base ` U ) = ( Base ` U )
62		eqid 2229	. . 3 |- ( +g ` U ) = ( +g ` U )
63		eqid 2229	. . 3 |- ( 0g ` U ) = ( 0g ` U )
64	61, 6, 62, 26, 63, 56	ismhm 13534	. 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 1002   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3198   |` cres 4725   -->wf 5320   ` cfv 5324  (class class class)co 6013   Basecbs 13072   ↾_s cress 13073   +g cplusg 13150   0gc0g 13329   Mndcmnd 13489   MndHom cmhm 13530  SubMndcsubmnd 13531

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-map 6814  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-iress 13080  df-plusg 13163  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-mhm 13532  df-submnd 13533

This theorem is referenced by:  resrhm  14252