Theorem resmhm 13319

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 13296	. . 3 |- ( F e. ( S MndHom T ) -> T e. Mnd )
2		resmhm.u	. . . 4 |- U = ( S ↾s X )
3	2	submmnd 13312	. . 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 2205	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
6		eqid 2205	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
7	5, 6	mhmf 13297	. . . . 5 |- ( F e. ( S MndHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
8	5	submss 13308	. . . . 5 |- ( X e. (SubMnd ` S ) -> X C_ ( Base ` S ) )
9		fssres 5451	. . . . 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 2206	. . . . . 6 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( Base ` S ) = ( Base ` S ) )
13		submrcl 13303	. . . . . . 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 12900	. . . . 5 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> X = ( Base ` U ) )
17	16	feq2d 5413	. . . 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 3194	. . . . . . 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 3194	. . . . . . 7 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> y e. ( Base ` S ) )
25		eqid 2205	. . . . . . . 8 |- ( +g ` S ) = ( +g ` S )
26		eqid 2205	. . . . . . . 8 |- ( +g ` T ) = ( +g ` T )
27	5, 25, 26	mhmlin 13299	. . . . . . 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 1250	. . . . . 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 13311	. . . . . . . . 9 |- ( ( X e. (SubMnd ` S ) /\ x e. X /\ y e. X ) -> ( x ( +g ` S ) y ) e. X )
30	29	3expb 1207	. . . . . . . 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 5601	. . . . . 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 5600	. . . . . . . 8 |- ( x e. X -> ( ( F |` X ) ` x ) = ( F ` x ) )
34		fvres 5600	. . . . . . . 8 |- ( y e. X -> ( ( F |` X ) ` y ) = ( F ` y ) )
35	33, 34	oveqan12d 5963	. . . . . . 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 2248	. . . . 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 2588	. . . 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 2206	. . . . . . . . . 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 12961	. . . . . . . . 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 5961	. . . . . . 7 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( x ( +g ` S ) y ) = ( x ( +g ` U ) y ) )
45	44	fveqeq2d 5584	. . . . . 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 2718	. . . . 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 2718	. . . 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 2205	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
50	49	subm0cl 13310	. . . . . 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 5601	. . . 4 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( ( F |` X ) ` ( 0g ` S ) ) = ( F ` ( 0g ` S ) ) )
53	2, 49	subm0 13314	. . . . . 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 5580	. . . 4 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( ( F |` X ) ` ( 0g ` S ) ) = ( ( F |` X ) ` ( 0g ` U ) ) )
56		eqid 2205	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
57	49, 56	mhm0 13300	. . . . 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 2246	. . 3 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( ( F |` X ) ` ( 0g ` U ) ) = ( 0g ` T ) )
60	18, 48, 59	3jca 1180	. 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 2205	. . 3 |- ( Base ` U ) = ( Base ` U )
62		eqid 2205	. . 3 |- ( +g ` U ) = ( +g ` U )
63		eqid 2205	. . 3 |- ( 0g ` U ) = ( 0g ` U )
64	61, 6, 62, 26, 63, 56	ismhm 13293	. 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 981   = wceq 1373   e. wcel 2176   A.wral 2484   C_ wss 3166   |` cres 4677   -->wf 5267   ` cfv 5271  (class class class)co 5944   Basecbs 12832   ↾_s cress 12833   +g cplusg 12909   0gc0g 13088   Mndcmnd 13248   MndHom cmhm 13289  SubMndcsubmnd 13290

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-map 6737  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840  df-plusg 12922  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-mhm 13291  df-submnd 13292

This theorem is referenced by:  resrhm  14010