Theorem resmhm2 13632

Description: One direction of resmhm2b 13633. (Contributed by Mario Carneiro, 18-Jun-2015.)

Hypothesis

Ref	Expression
resmhm2.u	⊢ 𝑈 = (𝑇 ↾s 𝑋)

Assertion

Ref	Expression
resmhm2	⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑇))

Proof of Theorem resmhm2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mhmrcl1 13607	. . 3 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑈) → 𝑆 ∈ Mnd)
2		submrcl 13615	. . 3 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → 𝑇 ∈ Mnd)
3	1, 2	anim12i 338	. 2 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → (𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd))
4		eqid 2231	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
5		eqid 2231	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
6	4, 5	mhmf 13609	. . . 4 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑈) → 𝐹:(Base‘𝑆)⟶(Base‘𝑈))
7		resmhm2.u	. . . . . 6 ⊢ 𝑈 = (𝑇 ↾s 𝑋)
8	7	submbas 13625	. . . . 5 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → 𝑋 = (Base‘𝑈))
9		eqid 2231	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
10	9	submss 13620	. . . . 5 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → 𝑋 ⊆ (Base‘𝑇))
11	8, 10	eqsstrrd 3265	. . . 4 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → (Base‘𝑈) ⊆ (Base‘𝑇))
12		fss 5501	. . . 4 ⊢ ((𝐹:(Base‘𝑆)⟶(Base‘𝑈) ∧ (Base‘𝑈) ⊆ (Base‘𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
13	6, 11, 12	syl2an 289	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
14		eqid 2231	. . . . . . . 8 ⊢ (+g‘𝑆) = (+g‘𝑆)
15		eqid 2231	. . . . . . . 8 ⊢ (+g‘𝑈) = (+g‘𝑈)
16	4, 14, 15	mhmlin 13611	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)))
17	16	3expb 1231	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)))
18	17	adantlr 477	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)))
19	7	a1i 9	. . . . . . . 8 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → 𝑈 = (𝑇 ↾s 𝑋))
20		eqidd 2232	. . . . . . . 8 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → (+g‘𝑇) = (+g‘𝑇))
21		id 19	. . . . . . . 8 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → 𝑋 ∈ (SubMnd‘𝑇))
22	19, 20, 21, 2	ressplusgd 13273	. . . . . . 7 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → (+g‘𝑇) = (+g‘𝑈))
23	22	ad2antlr 489	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (+g‘𝑇) = (+g‘𝑈))
24	23	oveqd 6045	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)))
25	18, 24	eqtr4d 2267	. . . 4 ⊢ (((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
26	25	ralrimivva 2615	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
27		eqid 2231	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
28		eqid 2231	. . . . . 6 ⊢ (0g‘𝑈) = (0g‘𝑈)
29	27, 28	mhm0 13612	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑈) → (𝐹‘(0g‘𝑆)) = (0g‘𝑈))
30	29	adantr 276	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑈))
31		eqid 2231	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
32	7, 31	subm0 13626	. . . . 5 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → (0g‘𝑇) = (0g‘𝑈))
33	32	adantl 277	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → (0g‘𝑇) = (0g‘𝑈))
34	30, 33	eqtr4d 2267	. . 3 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
35	13, 26, 34	3jca 1204	. 2 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇)))
36		eqid 2231	. . 3 ⊢ (+g‘𝑇) = (+g‘𝑇)
37	4, 9, 14, 36, 27, 31	ismhm 13605	. 2 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))))
38	3, 35, 37	sylanbrc 417	1 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  ∀wral 2511   ⊆ wss 3201  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028  Basecbs 13143   ↾_s cress 13144  +_gcplusg 13221  0_gc0g 13400  Mndcmnd 13560   MndHom cmhm 13601  SubMndcsubmnd 13602

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-pnf 8259  df-mnf 8260  df-ltxr 8262  df-inn 9187  df-2 9245  df-ndx 13146  df-slot 13147  df-base 13149  df-sets 13150  df-iress 13151  df-plusg 13234  df-0g 13402  df-mgm 13500  df-sgrp 13546  df-mnd 13561  df-mhm 13603  df-submnd 13604

This theorem is referenced by:  resmhm2b  13633  resghm2  13909  lgseisenlem4  15872