Theorem reslmhm2b 21022

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 2734	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2734	. . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
3		eqid 2734	. . 3 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈)
4		eqid 2734	. . 3 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
5		eqid 2734	. . 3 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
6		eqid 2734	. . 3 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
7		lmhmlmod1 21001	. . . 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 1191	. . . 4 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑇 ∈ LMod)
10		simpl2 1192	. . . 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 20927	. . . 4 ⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝑈 ∈ LMod)
14	9, 10, 13	syl2anc 584	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑈 ∈ LMod)
15		eqid 2734	. . . . . 6 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
16	11, 15	resssca 17360	. . . . 5 ⊢ (𝑋 ∈ 𝐿 → (Scalar‘𝑇) = (Scalar‘𝑈))
17	16	3ad2ant2 1134	. . . 4 ⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) → (Scalar‘𝑇) = (Scalar‘𝑈))
18	4, 15	lmhmsca 20998	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
19	17, 18	sylan9req 2790	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (Scalar‘𝑈) = (Scalar‘𝑆))
20		lmghm 20999	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
21	12	lsssubg 20924	. . . . . 6 ⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝑋 ∈ (SubGrp‘𝑇))
22	11	resghm2b 19222	. . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
23	21, 22	stoic3 1775	. . . . 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 593	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑈))
26		eqid 2734	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
27	4, 6, 1, 2, 26	lmhmlin 21003	. . . . . 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 1120	. . . . 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 714	. . . 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 1213	. . . . 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 17361	. . . . . 6 ⊢ (𝑋 ∈ 𝐿 → ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑈))
32	31	oveqd 7430	. . . . 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 2769	. . 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 21007	. 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 1191	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝑇 ∈ LMod)
38		simpl2 1192	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝑋 ∈ 𝐿)
39	11, 12	reslmhm2 21021	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝐹 ∈ (𝑆 LMHom 𝑇))
40	36, 37, 38, 39	syl3anc 1372	. 2 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
41	35, 40	impbida 800	1 ⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ 𝐹 ∈ (𝑆 LMHom 𝑈)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ⊆ wss 3931  ran crn 5666  ‘cfv 6541  (class class class)co 7413  Basecbs 17230   ↾_s cress 17253  Scalarcsca 17277   ·_𝑠 cvsca 17278  SubGrpcsubg 19108   GrpHom cghm 19200  LModclmod 20827  LSubSpclss 20898   LMHom clmhm 20987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-er 8727  df-map 8850  df-en 8968  df-dom 8969  df-sdom 8970  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-sets 17184  df-slot 17202  df-ndx 17214  df-base 17231  df-ress 17254  df-plusg 17287  df-sca 17290  df-vsca 17291  df-0g 17458  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-mhm 18766  df-submnd 18767  df-grp 18924  df-minusg 18925  df-sbg 18926  df-subg 19111  df-ghm 19201  df-mgp 20107  df-ur 20148  df-ring 20201  df-lmod 20829  df-lss 20899  df-lmhm 20990

This theorem is referenced by:  pj1lmhm2  21069  frlmsplit2  21748  dimkerim  33618  algextdeglem2  33703