Theorem resmhm 13771

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 13748	. . 3 |- ( F e. ( S MndHom T ) -> T e. Mnd )
2		resmhm.u	. . . 4 |- U = ( S ↾s X )
3	2	submmnd 13764	. . 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 2238	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
6		eqid 2238	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
7	5, 6	mhmf 13749	. . . . 5 |- ( F e. ( S MndHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
8	5	submss 13760	. . . . 5 |- ( X e. (SubMnd ` S ) -> X C_ ( Base ` S ) )
9		fssres 5560	. . . . 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 2239	. . . . . 6 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( Base ` S ) = ( Base ` S ) )
13		submrcl 13755	. . . . . . 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 13399	. . . . 5 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> X = ( Base ` U ) )
17	16	feq2d 5516	. . . 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 531	. . . . . . 7 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> F e. ( S MndHom T ) )
20	8	ad2antlr 493	. . . . . . . 8 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> X C_ ( Base ` S ) )
21		simprl 535	. . . . . . . 8 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> x e. X )
22	20, 21	sseldd 3249	. . . . . . 7 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> x e. ( Base ` S ) )
23		simprr 537	. . . . . . . 8 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> y e. X )
24	20, 23	sseldd 3249	. . . . . . 7 |- ( ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) /\ ( x e. X /\ y e. X ) ) -> y e. ( Base ` S ) )
25		eqid 2238	. . . . . . . 8 |- ( +g ` S ) = ( +g ` S )
26		eqid 2238	. . . . . . . 8 |- ( +g ` T ) = ( +g ` T )
27	5, 25, 26	mhmlin 13751	. . . . . . 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 1278	. . . . . 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 13763	. . . . . . . . 9 |- ( ( X e. (SubMnd ` S ) /\ x e. X /\ y e. X ) -> ( x ( +g ` S ) y ) e. X )
30	29	3expb 1235	. . . . . . . 8 |- ( ( X e. (SubMnd ` S ) /\ ( x e. X /\ y e. X ) ) -> ( x ( +g ` S ) y ) e. X )
31	30	adantll 480	. . . . . . 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 5715	. . . . . 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 5714	. . . . . . . 8 |- ( x e. X -> ( ( F |` X ) ` x ) = ( F ` x ) )
34		fvres 5714	. . . . . . . 8 |- ( y e. X -> ( ( F |` X ) ` y ) = ( F ` y ) )
35	33, 34	oveqan12d 6094	. . . . . . 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 2281	. . . . 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 2632	. . . 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 2239	. . . . . . . . . 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 13460	. . . . . . . . 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 6092	. . . . . . 7 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( x ( +g ` S ) y ) = ( x ( +g ` U ) y ) )
45	44	fveqeq2d 5698	. . . . . 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 2765	. . . . 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 2765	. . . 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 2238	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
50	49	subm0cl 13762	. . . . . 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 5715	. . . 4 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( ( F |` X ) ` ( 0g ` S ) ) = ( F ` ( 0g ` S ) ) )
53	2, 49	subm0 13766	. . . . . 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 5694	. . . 4 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( ( F |` X ) ` ( 0g ` S ) ) = ( ( F |` X ) ` ( 0g ` U ) ) )
56		eqid 2238	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
57	49, 56	mhm0 13752	. . . . 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 2279	. . 3 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( ( F |` X ) ` ( 0g ` U ) ) = ( 0g ` T ) )
60	18, 48, 59	3jca 1208	. 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 2238	. . 3 |- ( Base ` U ) = ( Base ` U )
62		eqid 2238	. . 3 |- ( +g ` U ) = ( +g ` U )
63		eqid 2238	. . 3 |- ( 0g ` U ) = ( 0g ` U )
64	61, 6, 62, 26, 63, 56	ismhm 13745	. 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 421	1 |- ( ( F e. ( S MndHom T ) /\ X e. (SubMnd ` S ) ) -> ( F |` X ) e. ( U MndHom T ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   |` cres 4771   -->wf 5368   ` cfv 5372  (class class class)co 6075   Basecbs 13330   ↾_s cress 13331   +g cplusg 13408   0gc0g 13587   Mndcmnd 13706   MndHom cmhm 13741  SubMndcsubmnd 13742

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-mhm 13743  df-submnd 13744

This theorem is referenced by:  resrhm  14529