Theorem reslmhm2b 20231

Description: Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)

Hypotheses

Ref	Expression
reslmhm2.u	⊢ 𝑈 = (𝑇 ↾s 𝑋)
reslmhm2.l	⊢ 𝐿 = (LSubSp‘𝑇)

Assertion

Ref	Expression
reslmhm2b	⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ 𝐹 ∈ (𝑆 LMHom 𝑈)))

Proof of Theorem reslmhm2b

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

Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2738	. . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
3		eqid 2738	. . 3 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈)
4		eqid 2738	. . 3 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
5		eqid 2738	. . 3 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
6		eqid 2738	. . 3 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
7		lmhmlmod1 20210	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
8	7	adantl 481	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ LMod)
9		simpl1 1189	. . . 4 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑇 ∈ LMod)
10		simpl2 1190	. . . 4 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑋 ∈ 𝐿)
11		reslmhm2.u	. . . . 5 ⊢ 𝑈 = (𝑇 ↾s 𝑋)
12		reslmhm2.l	. . . . 5 ⊢ 𝐿 = (LSubSp‘𝑇)
13	11, 12	lsslmod 20137	. . . 4 ⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝑈 ∈ LMod)
14	9, 10, 13	syl2anc 583	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑈 ∈ LMod)
15		eqid 2738	. . . . . 6 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
16	11, 15	resssca 16978	. . . . 5 ⊢ (𝑋 ∈ 𝐿 → (Scalar‘𝑇) = (Scalar‘𝑈))
17	16	3ad2ant2 1132	. . . 4 ⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) → (Scalar‘𝑇) = (Scalar‘𝑈))
18	4, 15	lmhmsca 20207	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
19	17, 18	sylan9req 2800	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (Scalar‘𝑈) = (Scalar‘𝑆))
20		lmghm 20208	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
21	12	lsssubg 20134	. . . . . 6 ⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝑋 ∈ (SubGrp‘𝑇))
22	11	resghm2b 18767	. . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
23	21, 22	stoic3 1780	. . . . 5 ⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
24	23	biimpa 476	. . . 4 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑈))
25	20, 24	sylan2 592	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑈))
26		eqid 2738	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
27	4, 6, 1, 2, 26	lmhmlin 20212	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)) = (𝑥( ·𝑠 ‘𝑇)(𝐹‘𝑦)))
28	27	3expb 1118	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)) = (𝑥( ·𝑠 ‘𝑇)(𝐹‘𝑦)))
29	28	adantll 710	. . . 4 ⊢ ((((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)) = (𝑥( ·𝑠 ‘𝑇)(𝐹‘𝑦)))
30		simpll2 1211	. . . . 5 ⊢ ((((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑋 ∈ 𝐿)
31	11, 26	ressvsca 16979	. . . . . 6 ⊢ (𝑋 ∈ 𝐿 → ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑈))
32	31	oveqd 7272	. . . . 5 ⊢ (𝑋 ∈ 𝐿 → (𝑥( ·𝑠 ‘𝑇)(𝐹‘𝑦)) = (𝑥( ·𝑠 ‘𝑈)(𝐹‘𝑦)))
33	30, 32	syl 17	. . . 4 ⊢ ((((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥( ·𝑠 ‘𝑇)(𝐹‘𝑦)) = (𝑥( ·𝑠 ‘𝑈)(𝐹‘𝑦)))
34	29, 33	eqtrd 2778	. . 3 ⊢ ((((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)) = (𝑥( ·𝑠 ‘𝑈)(𝐹‘𝑦)))
35	1, 2, 3, 4, 5, 6, 8, 14, 19, 25, 34	islmhmd 20216	. 2 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝐹 ∈ (𝑆 LMHom 𝑈))
36		simpr 484	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝐹 ∈ (𝑆 LMHom 𝑈))
37		simpl1 1189	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝑇 ∈ LMod)
38		simpl2 1190	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝑋 ∈ 𝐿)
39	11, 12	reslmhm2 20230	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝐹 ∈ (𝑆 LMHom 𝑇))
40	36, 37, 38, 39	syl3anc 1369	. 2 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
41	35, 40	impbida 797	1 ⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ 𝐹 ∈ (𝑆 LMHom 𝑈)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  ran crn 5581  ‘cfv 6418  (class class class)co 7255  Basecbs 16840   ↾_s cress 16867  Scalarcsca 16891   ·_𝑠 cvsca 16892  SubGrpcsubg 18664   GrpHom cghm 18746  LModclmod 20038  LSubSpclss 20108   LMHom clmhm 20196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-sca 16904  df-vsca 16905  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-subg 18667  df-ghm 18747  df-mgp 19636  df-ur 19653  df-ring 19700  df-lmod 20040  df-lss 20109  df-lmhm 20199

This theorem is referenced by:  pj1lmhm2  20278  frlmsplit2  20890  dimkerim  31610