Theorem resmhm 18910

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 18877	. . 3 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd)
2		resmhm.u	. . . 4 ⊢ 𝑈 = (𝑆 ↾s 𝑋)
3	2	submmnd 18903	. . 3 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → 𝑈 ∈ Mnd)
4	1, 3	anim12ci 626	. 2 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝑈 ∈ Mnd ∧ 𝑇 ∈ Mnd))
5		eqid 2766	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
6		eqid 2766	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
7	5, 6	mhmf 18878	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
8	5	submss 18898	. . . . 5 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
9		fssres 6751	. . . . 5 ⊢ ((𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑋 ⊆ (Base‘𝑆)) → (𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇))
10	7, 8, 9	syl2an 608	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇))
11	8	adantl 487	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → 𝑋 ⊆ (Base‘𝑆))
12	2, 5	ressbas2 17323	. . . . . 6 ⊢ (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑈))
13	11, 12	syl 18	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → 𝑋 = (Base‘𝑈))
14	13	feq2d 6696	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇) ↔ (𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇)))
15	10, 14	mpbid 235	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇))
16		simpll 779	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
17	8	ad2antlr 740	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝑆))
18		simprl 783	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
19	17, 18	sseldd 3941	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ (Base‘𝑆))
20		simprr 785	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
21	17, 20	sseldd 3941	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ (Base‘𝑆))
22		eqid 2766	. . . . . . . 8 ⊢ (+g‘𝑆) = (+g‘𝑆)
23		eqid 2766	. . . . . . . 8 ⊢ (+g‘𝑇) = (+g‘𝑇)
24	5, 22, 23	mhmlin 18882	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
25	16, 19, 21, 24	syl3anc 1398	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
26	22	submcl 18901	. . . . . . . . 9 ⊢ ((𝑋 ∈ (SubMnd‘𝑆) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥(+g‘𝑆)𝑦) ∈ 𝑋)
27	26	3expb 1138	. . . . . . . 8 ⊢ ((𝑋 ∈ (SubMnd‘𝑆) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(+g‘𝑆)𝑦) ∈ 𝑋)
28	27	adantll 727	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(+g‘𝑆)𝑦) ∈ 𝑋)
29	28	fvresd 6908	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑆)𝑦)) = (𝐹‘(𝑥(+g‘𝑆)𝑦)))
30		fvres 6907	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑥) = (𝐹‘𝑥))
31		fvres 6907	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑦) = (𝐹‘𝑦))
32	30, 31	oveqan12d 7442	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
33	32	adantl 487	. . . . . 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)))
34	25, 29, 33	3eqtr4d 2811	. . . . 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)))
35	34	ralrimivva 3211	. . . 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)))
36	2, 22	ressplusg 17369	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → (+g‘𝑆) = (+g‘𝑈))
37	36	adantl 487	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (+g‘𝑆) = (+g‘𝑈))
38	37	oveqd 7440	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝑥(+g‘𝑆)𝑦) = (𝑥(+g‘𝑈)𝑦))
39	38	fveqeq2d 6896	. . . . . 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))))
40	13, 39	raleqbidv 3341	. . . . 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))))
41	13, 40	raleqbidv 3341	. . . 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))))
42	35, 41	mpbid 235	. . 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)))
43		eqid 2766	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
44	43	subm0cl 18900	. . . . . 6 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → (0g‘𝑆) ∈ 𝑋)
45	44	adantl 487	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (0g‘𝑆) ∈ 𝑋)
46	45	fvresd 6908	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹 ↾ 𝑋)‘(0g‘𝑆)) = (𝐹‘(0g‘𝑆)))
47	2, 43	subm0 18905	. . . . . 6 ⊢ (𝑋 ∈ (SubMnd‘𝑆) → (0g‘𝑆) = (0g‘𝑈))
48	47	adantl 487	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (0g‘𝑆) = (0g‘𝑈))
49	48	fveq2d 6892	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹 ↾ 𝑋)‘(0g‘𝑆)) = ((𝐹 ↾ 𝑋)‘(0g‘𝑈)))
50		eqid 2766	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
51	43, 50	mhm0 18883	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
52	51	adantr 486	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
53	46, 49, 52	3eqtr3d 2809	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹 ↾ 𝑋)‘(0g‘𝑈)) = (0g‘𝑇))
54	15, 42, 53	3jca 1146	. 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)))
55		eqid 2766	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
56		eqid 2766	. . 3 ⊢ (+g‘𝑈) = (+g‘𝑈)
57		eqid 2766	. . 3 ⊢ (0g‘𝑈) = (0g‘𝑈)
58	55, 6, 56, 23, 57, 50	ismhm 18874	. 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))))
59	4, 54, 58	sylanbrc 595	1 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 MndHom 𝑇))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082   ⊆ wss 3908   ↾ cres 5668  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423  Basecbs 17294   ↾_s cress 17315  +_gcplusg 17335  0_gc0g 17517  Mndcmnd 18821   MndHom cmhm 18870  SubMndcsubmnd 18871

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-er 8703  df-map 8835  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-ress 17316  df-plusg 17348  df-0g 17519  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-mhm 18872  df-submnd 18873

This theorem is used by:  resrhm  20737  dchrghm  27457