Theorem resmhm2b 13773

Description: Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)

Hypothesis

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

Assertion

Ref	Expression
resmhm2b	⊢ ((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))

Proof of Theorem resmhm2b

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

Step	Hyp	Ref	Expression
1		mhmrcl1 13747	. . . 4 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
2	1	adantl 277	. . 3 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Mnd)
3		resmhm2.u	. . . . 5 ⊢ 𝑈 = (𝑇 ↾s 𝑋)
4	3	submmnd 13764	. . . 4 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → 𝑈 ∈ Mnd)
5	4	ad2antrr 492	. . 3 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑈 ∈ Mnd)
6		eqid 2238	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
7		eqid 2238	. . . . . . . . 9 ⊢ (Base‘𝑇) = (Base‘𝑇)
8	6, 7	mhmf 13749	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
9	8	adantl 277	. . . . . . 7 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
10	9	ffnd 5529	. . . . . 6 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
11		simplr 533	. . . . . 6 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → ran 𝐹 ⊆ 𝑋)
12		df-f 5376	. . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶𝑋 ↔ (𝐹 Fn (Base‘𝑆) ∧ ran 𝐹 ⊆ 𝑋))
13	10, 11, 12	sylanbrc 421	. . . . 5 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶𝑋)
14	3	submbas 13765	. . . . . . 7 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → 𝑋 = (Base‘𝑈))
15	14	ad2antrr 492	. . . . . 6 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑋 = (Base‘𝑈))
16	15	feq3d 5517	. . . . 5 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹:(Base‘𝑆)⟶𝑋 ↔ 𝐹:(Base‘𝑆)⟶(Base‘𝑈)))
17	13, 16	mpbid 147	. . . 4 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑈))
18		eqid 2238	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
19		eqid 2238	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
20	6, 18, 19	mhmlin 13751	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
21	20	3expb 1235	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
22	21	adantll 480	. . . . . 6 ⊢ ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
23	3	a1i 9	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → 𝑈 = (𝑇 ↾s 𝑋))
24		eqidd 2239	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → (+g‘𝑇) = (+g‘𝑇))
25		id 19	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → 𝑋 ∈ (SubMnd‘𝑇))
26		submrcl 13755	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → 𝑇 ∈ Mnd)
27	23, 24, 25, 26	ressplusgd 13460	. . . . . . . 8 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → (+g‘𝑇) = (+g‘𝑈))
28	27	ad3antrrr 496	. . . . . . 7 ⊢ ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (+g‘𝑇) = (+g‘𝑈))
29	28	oveqd 6092	. . . . . 6 ⊢ ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)))
30	22, 29	eqtrd 2271	. . . . 5 ⊢ ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)))
31	30	ralrimivva 2632	. . . 4 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)))
32		eqid 2238	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
33		eqid 2238	. . . . . . 7 ⊢ (0g‘𝑇) = (0g‘𝑇)
34	32, 33	mhm0 13752	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
35	34	adantl 277	. . . . 5 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
36	3, 33	subm0 13766	. . . . . 6 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → (0g‘𝑇) = (0g‘𝑈))
37	36	ad2antrr 492	. . . . 5 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (0g‘𝑇) = (0g‘𝑈))
38	35, 37	eqtrd 2271	. . . 4 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑈))
39	17, 31, 38	3jca 1208	. . 3 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹:(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑈)))
40		eqid 2238	. . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈)
41		eqid 2238	. . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈)
42		eqid 2238	. . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈)
43	6, 40, 18, 41, 32, 42	ismhm 13745	. . 3 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑈) ↔ ((𝑆 ∈ Mnd ∧ 𝑈 ∈ Mnd) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑈))))
44	2, 5, 39, 43	syl21anbrc 1213	. 2 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑈))
45	3	resmhm2 13772	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
46	45	ancoms 268	. . 3 ⊢ ((𝑋 ∈ (SubMnd‘𝑇) ∧ 𝐹 ∈ (𝑆 MndHom 𝑈)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
47	46	adantlr 481	. 2 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑈)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
48	44, 47	impbida 604	1 ⊢ ((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220  ran crn 4770   Fn wfn 5367  ⟶wf 5368  ‘cfv 5372  (class class class)co 6075  Basecbs 13330   ↾_s cress 13331  +_gcplusg 13408  0_gc0g 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:  resghm2b  14042  resrhm2b  14530  lgseisenlem4  16106