Theorem resmhm 13541

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 13518	. . 3 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd)
2		resmhm.u	. . . 4 ⊢ 𝑈 = (𝑆 ↾s 𝑋)
3	2	submmnd 13534	. . 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 2229	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
6		eqid 2229	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
7	5, 6	mhmf 13519	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
8	5	submss 13530	. . . . 5 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
9		fssres 5506	. . . . 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 2230	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (Base‘𝑆) = (Base‘𝑆))
13		submrcl 13525	. . . . . . 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 13122	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → 𝑋 = (Base‘𝑈))
17	16	feq2d 5464	. . . 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 3225	. . . . . . 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 3225	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ (Base‘𝑆))
25		eqid 2229	. . . . . . . 8 ⊢ (+g‘𝑆) = (+g‘𝑆)
26		eqid 2229	. . . . . . . 8 ⊢ (+g‘𝑇) = (+g‘𝑇)
27	5, 25, 26	mhmlin 13521	. . . . . . 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 1271	. . . . . 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 13533	. . . . . . . . 9 ⊢ ((𝑋 ∈ (SubMnd‘𝑆) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥(+g‘𝑆)𝑦) ∈ 𝑋)
30	29	3expb 1228	. . . . . . . 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 5657	. . . . . 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 5656	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑥) = (𝐹‘𝑥))
34		fvres 5656	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑦) = (𝐹‘𝑦))
35	33, 34	oveqan12d 6029	. . . . . . 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 2272	. . . . 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 2612	. . . 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 2230	. . . . . . . . . 10 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → (+g‘𝑆) = (+g‘𝑆))
41		id 19	. . . . . . . . . 10 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → 𝑋 ∈ (SubMnd‘𝑆))
42	39, 40, 41, 13	ressplusgd 13183	. . . . . . . . 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 6027	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝑥(+g‘𝑆)𝑦) = (𝑥(+g‘𝑈)𝑦))
45	44	fveqeq2d 5640	. . . . . 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 2744	. . . . 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 2744	. . . 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 2229	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
50	49	subm0cl 13532	. . . . . 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 5657	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹 ↾ 𝑋)‘(0g‘𝑆)) = (𝐹‘(0g‘𝑆)))
53	2, 49	subm0 13536	. . . . . 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 5636	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹 ↾ 𝑋)‘(0g‘𝑆)) = ((𝐹 ↾ 𝑋)‘(0g‘𝑈)))
56		eqid 2229	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
57	49, 56	mhm0 13522	. . . . 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 2270	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹 ↾ 𝑋)‘(0g‘𝑈)) = (0g‘𝑇))
60	18, 48, 59	3jca 1201	. 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 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 13515	. 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 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508   ⊆ wss 3197   ↾ cres 4722  ⟶wf 5317  ‘cfv 5321  (class class class)co 6010  Basecbs 13053   ↾_s cress 13054  +_gcplusg 13131  0_gc0g 13310  Mndcmnd 13470   MndHom cmhm 13511  SubMndcsubmnd 13512

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 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-pre-ltirr 8127  ax-pre-ltadd 8131

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-map 6810  df-pnf 8199  df-mnf 8200  df-ltxr 8202  df-inn 9127  df-2 9185  df-ndx 13056  df-slot 13057  df-base 13059  df-sets 13060  df-iress 13061  df-plusg 13144  df-0g 13312  df-mgm 13410  df-sgrp 13456  df-mnd 13471  df-mhm 13513  df-submnd 13514

This theorem is referenced by:  resrhm  14233