Theorem resmhm 13119

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 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (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 13096	. . 3 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd)
2		resmhm.u	. . . 4 ⊢ 𝑈 = (𝑆 ↾s 𝑋)
3	2	submmnd 13112	. . 3 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → 𝑈 ∈ Mnd)
4	1, 3	anim12ci 339	. 2 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝑈 ∈ Mnd ∧ 𝑇 ∈ Mnd))
5		eqid 2196	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
6		eqid 2196	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
7	5, 6	mhmf 13097	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
8	5	submss 13108	. . . . 5 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
9		fssres 5433	. . . . 5 ⊢ ((𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑋 ⊆ (Base‘𝑆)) → (𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇))
10	7, 8, 9	syl2an 289	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇))
11	2	a1i 9	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → 𝑈 = (𝑆 ↾s 𝑋))
12		eqidd 2197	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (Base‘𝑆) = (Base‘𝑆))
13		submrcl 13103	. . . . . . 7 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → 𝑆 ∈ Mnd)
14	13	adantl 277	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → 𝑆 ∈ Mnd)
15	8	adantl 277	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → 𝑋 ⊆ (Base‘𝑆))
16	11, 12, 14, 15	ressbas2d 12746	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → 𝑋 = (Base‘𝑈))
17	16	feq2d 5395	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇) ↔ (𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇)))
18	10, 17	mpbid 147	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇))
19		simpll 527	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
20	8	ad2antlr 489	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝑆))
21		simprl 529	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
22	20, 21	sseldd 3184	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ (Base‘𝑆))
23		simprr 531	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
24	20, 23	sseldd 3184	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ (Base‘𝑆))
25		eqid 2196	. . . . . . . 8 ⊢ (+g‘𝑆) = (+g‘𝑆)
26		eqid 2196	. . . . . . . 8 ⊢ (+g‘𝑇) = (+g‘𝑇)
27	5, 25, 26	mhmlin 13099	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
28	19, 22, 24, 27	syl3anc 1249	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
29	25	submcl 13111	. . . . . . . . 9 ⊢ ((𝑋 ∈ (SubMnd‘𝑆) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥(+g‘𝑆)𝑦) ∈ 𝑋)
30	29	3expb 1206	. . . . . . . 8 ⊢ ((𝑋 ∈ (SubMnd‘𝑆) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(+g‘𝑆)𝑦) ∈ 𝑋)
31	30	adantll 476	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(+g‘𝑆)𝑦) ∈ 𝑋)
32	31	fvresd 5583	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑆)𝑦)) = (𝐹‘(𝑥(+g‘𝑆)𝑦)))
33		fvres 5582	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑥) = (𝐹‘𝑥))
34		fvres 5582	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑦) = (𝐹‘𝑦))
35	33, 34	oveqan12d 5941	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
36	35	adantl 277	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
37	28, 32, 36	3eqtr4d 2239	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)))
38	37	ralrimivva 2579	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)))
39	2	a1i 9	. . . . . . . . . 10 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → 𝑈 = (𝑆 ↾s 𝑋))
40		eqidd 2197	. . . . . . . . . 10 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → (+g‘𝑆) = (+g‘𝑆))
41		id 19	. . . . . . . . . 10 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → 𝑋 ∈ (SubMnd‘𝑆))
42	39, 40, 41, 13	ressplusgd 12806	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → (+g‘𝑆) = (+g‘𝑈))
43	42	adantl 277	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (+g‘𝑆) = (+g‘𝑈))
44	43	oveqd 5939	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝑥(+g‘𝑆)𝑦) = (𝑥(+g‘𝑈)𝑦))
45	44	fveqeq2d 5566	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (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 2709	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (∀𝑦 ∈ 𝑋 ((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)) ↔ ∀𝑦 ∈ (Base‘𝑈)((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑈)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦))))
47	16, 46	raleqbidv 2709	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑈)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦))))
48	38, 47	mpbid 147	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑈)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)))
49		eqid 2196	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
50	49	subm0cl 13110	. . . . . 6 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → (0g‘𝑆) ∈ 𝑋)
51	50	adantl 277	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (0g‘𝑆) ∈ 𝑋)
52	51	fvresd 5583	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹 ↾ 𝑋)‘(0g‘𝑆)) = (𝐹‘(0g‘𝑆)))
53	2, 49	subm0 13114	. . . . . 6 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → (0g‘𝑆) = (0g‘𝑈))
54	53	adantl 277	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (0g‘𝑆) = (0g‘𝑈))
55	54	fveq2d 5562	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹 ↾ 𝑋)‘(0g‘𝑆)) = ((𝐹 ↾ 𝑋)‘(0g‘𝑈)))
56		eqid 2196	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
57	49, 56	mhm0 13100	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
58	57	adantr 276	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
59	52, 55, 58	3eqtr3d 2237	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹 ↾ 𝑋)‘(0g‘𝑈)) = (0g‘𝑇))
60	18, 48, 59	3jca 1179	. 2 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑈)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)) ∧ ((𝐹 ↾ 𝑋)‘(0g‘𝑈)) = (0g‘𝑇)))
61		eqid 2196	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
62		eqid 2196	. . 3 ⊢ (+g‘𝑈) = (+g‘𝑈)
63		eqid 2196	. . 3 ⊢ (0g‘𝑈) = (0g‘𝑈)
64	61, 6, 62, 26, 63, 56	ismhm 13093	. 2 ⊢ ((𝐹 ↾ 𝑋) ∈ (𝑈 MndHom 𝑇) ↔ ((𝑈 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ ((𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (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 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 MndHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475   ⊆ wss 3157   ↾ cres 4665  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922  Basecbs 12678   ↾_s cress 12679  +_gcplusg 12755  0_gc0g 12927  Mndcmnd 13057   MndHom cmhm 13089  SubMndcsubmnd 13090

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-map 6709  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-mhm 13091  df-submnd 13092

This theorem is referenced by:  resrhm  13804