Theorem reslmhm2b 21309

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 2761	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2761	. . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
3		eqid 2761	. . 3 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈)
4		eqid 2761	. . 3 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
5		eqid 2761	. . 3 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
6		eqid 2761	. . 3 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
7		lmhmlmod1 21288	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
8	7	adantl 487	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ LMod)
9		simpl1 1210	. . . 4 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑇 ∈ LMod)
10		simpl2 1211	. . . 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 21215	. . . 4 ⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝑈 ∈ LMod)
14	9, 10, 13	syl2anc 596	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑈 ∈ LMod)
15		eqid 2761	. . . . . 6 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
16	11, 15	resssca 17494	. . . . 5 ⊢ (𝑋 ∈ 𝐿 → (Scalar‘𝑇) = (Scalar‘𝑈))
17	16	3ad2ant2 1152	. . . 4 ⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) → (Scalar‘𝑇) = (Scalar‘𝑈))
18	4, 15	lmhmsca 21285	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
19	17, 18	sylan9req 2817	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (Scalar‘𝑈) = (Scalar‘𝑆))
20		lmghm 21286	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
21	12	lsssubg 21212	. . . . . 6 ⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝑋 ∈ (SubGrp‘𝑇))
22	11	resghm2b 19428	. . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
23	21, 22	stoic3 1809	. . . . 5 ⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
24	23	biimpa 482	. . . 4 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑈))
25	20, 24	sylan2 605	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑈))
26		eqid 2761	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
27	4, 6, 1, 2, 26	lmhmlin 21290	. . . . . 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 1138	. . . . 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 727	. . . 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 1232	. . . . 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 17495	. . . . . 6 ⊢ (𝑋 ∈ 𝐿 → ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑈))
32	31	oveqd 7429	. . . . 5 ⊢ (𝑋 ∈ 𝐿 → (𝑥( ·𝑠 ‘𝑇)(𝐹‘𝑦)) = (𝑥( ·𝑠 ‘𝑈)(𝐹‘𝑦)))
33	30, 32	syl 18	. . . 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 2796	. . 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 21294	. 2 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝐹 ∈ (𝑆 LMHom 𝑈))
36		simpr 490	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝐹 ∈ (𝑆 LMHom 𝑈))
37		simpl1 1210	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝑇 ∈ LMod)
38		simpl2 1211	. . 3 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝑋 ∈ 𝐿)
39	11, 12	reslmhm2 21308	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝐹 ∈ (𝑆 LMHom 𝑇))
40	36, 37, 38, 39	syl3anc 1398	. 2 ⊢ (((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
41	35, 40	impbida 813	1 ⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ 𝐹 ∈ (𝑆 LMHom 𝑈)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  ran crn 5652  ‘cfv 6531  (class class class)co 7412  Basecbs 17367   ↾_s cress 17388  Scalarcsca 17411   ·_𝑠 cvsca 17412  SubGrpcsubg 19310   GrpHom cghm 19407  LModclmod 21115  LSubSpclss 21186   LMHom clmhm 21274

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8701  df-map 8833  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-3 12387  df-4 12388  df-5 12389  df-6 12390  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-ress 17389  df-plusg 17421  df-sca 17424  df-vsca 17425  df-0g 17592  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-mhm 18958  df-submnd 18959  df-grp 19127  df-minusg 19128  df-sbg 19129  df-subg 19313  df-ghm 19408  df-mgp 20341  df-ur 20388  df-ring 20441  df-lmod 21117  df-lss 21187  df-lmhm 21277

This theorem is used by:  pj1lmhm2  21356  frlmsplit2  22059  dimkerim  34241  algextdeglem2  34332