frlmup3 - Metamath Proof Explorer

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Theorem frlmup3 20361

Description: The range of such an evaluation map is the finite linear combinations of the target vectors and also the span of the target vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.)

**Hypotheses**
Ref	Expression
frlmup.f	⊢ 𝐹 = (𝑅 freeLMod 𝐼)
frlmup.b	⊢ 𝐵 = (Base‘𝐹)
frlmup.c	⊢ 𝐶 = (Base‘𝑇)
frlmup.v	⊢ · = ( ·_𝑠 ‘𝑇)
frlmup.e	⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σ_g (𝑥 ∘_𝑓 · 𝐴)))
frlmup.t	⊢ (𝜑 → 𝑇 ∈ LMod)
frlmup.i	⊢ (𝜑 → 𝐼 ∈ 𝑋)
frlmup.r	⊢ (𝜑 → 𝑅 = (Scalar‘𝑇))
frlmup.a	⊢ (𝜑 → 𝐴:𝐼⟶𝐶)
frlmup.k	⊢ 𝐾 = (LSpan‘𝑇)

**Assertion**
Ref	Expression
frlmup3	⊢ (𝜑 → ran 𝐸 = (𝐾‘ran 𝐴))

Distinct variable groups: 𝑥,𝑅 𝑥,𝐼 𝑥,𝐹 𝑥,𝐵 𝑥,𝐶 𝑥, · 𝑥,𝐴 𝑥,𝑋 𝑥,𝐾 𝜑,𝑥 𝑥,𝑇 Allowed substitution hint: 𝐸(𝑥)

Proof of Theorem frlmup3 Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frlmup.f	. . . 4 ⊢ 𝐹 = (𝑅 freeLMod 𝐼)
2		frlmup.b	. . . 4 ⊢ 𝐵 = (Base‘𝐹)
3		frlmup.c	. . . 4 ⊢ 𝐶 = (Base‘𝑇)
4		frlmup.v	. . . 4 ⊢ · = ( ·_𝑠 ‘𝑇)
5		frlmup.e	. . . 4 ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σ_g (𝑥 ∘_𝑓 · 𝐴)))
6		frlmup.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ LMod)
7		frlmup.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑋)
8		frlmup.r	. . . 4 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇))
9		frlmup.a	. . . 4 ⊢ (𝜑 → 𝐴:𝐼⟶𝐶)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	frlmup1 20359	. . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇))
11		eqid 2760	. . . . . . . 8 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
12	11	lmodring 19093	. . . . . . 7 ⊢ (𝑇 ∈ LMod → (Scalar‘𝑇) ∈ Ring)
13	6, 12	syl 17	. . . . . 6 ⊢ (𝜑 → (Scalar‘𝑇) ∈ Ring)
14	8, 13	eqeltrd 2839	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
15		eqid 2760	. . . . . 6 ⊢ (𝑅 unitVec 𝐼) = (𝑅 unitVec 𝐼)
16	15, 1, 2	uvcff 20352	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → (𝑅 unitVec 𝐼):𝐼⟶𝐵)
17	14, 7, 16	syl2anc 696	. . . 4 ⊢ (𝜑 → (𝑅 unitVec 𝐼):𝐼⟶𝐵)
18		frn 6214	. . . 4 ⊢ ((𝑅 unitVec 𝐼):𝐼⟶𝐵 → ran (𝑅 unitVec 𝐼) ⊆ 𝐵)
19	17, 18	syl 17	. . 3 ⊢ (𝜑 → ran (𝑅 unitVec 𝐼) ⊆ 𝐵)
20		eqid 2760	. . . 4 ⊢ (LSpan‘𝐹) = (LSpan‘𝐹)
21		frlmup.k	. . . 4 ⊢ 𝐾 = (LSpan‘𝑇)
22	2, 20, 21	lmhmlsp 19271	. . 3 ⊢ ((𝐸 ∈ (𝐹 LMHom 𝑇) ∧ ran (𝑅 unitVec 𝐼) ⊆ 𝐵) → (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼))) = (𝐾‘(𝐸 “ ran (𝑅 unitVec 𝐼))))
23	10, 19, 22	syl2anc 696	. 2 ⊢ (𝜑 → (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼))) = (𝐾‘(𝐸 “ ran (𝑅 unitVec 𝐼))))
24	2, 3	lmhmf 19256	. . . . . 6 ⊢ (𝐸 ∈ (𝐹 LMHom 𝑇) → 𝐸:𝐵⟶𝐶)
25	10, 24	syl 17	. . . . 5 ⊢ (𝜑 → 𝐸:𝐵⟶𝐶)
26		ffn 6206	. . . . 5 ⊢ (𝐸:𝐵⟶𝐶 → 𝐸 Fn 𝐵)
27	25, 26	syl 17	. . . 4 ⊢ (𝜑 → 𝐸 Fn 𝐵)
28		fnima 6171	. . . 4 ⊢ (𝐸 Fn 𝐵 → (𝐸 “ 𝐵) = ran 𝐸)
29	27, 28	syl 17	. . 3 ⊢ (𝜑 → (𝐸 “ 𝐵) = ran 𝐸)
30		eqid 2760	. . . . . . . 8 ⊢ (LBasis‘𝐹) = (LBasis‘𝐹)
31	1, 15, 30	frlmlbs 20358	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘𝐹))
32	14, 7, 31	syl2anc 696	. . . . . 6 ⊢ (𝜑 → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘𝐹))
33	2, 30, 20	lbssp 19301	. . . . . 6 ⊢ (ran (𝑅 unitVec 𝐼) ∈ (LBasis‘𝐹) → ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼)) = 𝐵)
34	32, 33	syl 17	. . . . 5 ⊢ (𝜑 → ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼)) = 𝐵)
35	34	eqcomd 2766	. . . 4 ⊢ (𝜑 → 𝐵 = ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼)))
36	35	imaeq2d 5624	. . 3 ⊢ (𝜑 → (𝐸 “ 𝐵) = (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼))))
37	29, 36	eqtr3d 2796	. 2 ⊢ (𝜑 → ran 𝐸 = (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼))))
38		imaco 5801	. . . 4 ⊢ ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼) = (𝐸 “ ((𝑅 unitVec 𝐼) “ 𝐼))
39		ffn 6206	. . . . . . . 8 ⊢ (𝐴:𝐼⟶𝐶 → 𝐴 Fn 𝐼)
40	9, 39	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 Fn 𝐼)
41		ffn 6206	. . . . . . . . 9 ⊢ ((𝑅 unitVec 𝐼):𝐼⟶𝐵 → (𝑅 unitVec 𝐼) Fn 𝐼)
42	17, 41	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑅 unitVec 𝐼) Fn 𝐼)
43		fnco 6160	. . . . . . . 8 ⊢ ((𝐸 Fn 𝐵 ∧ (𝑅 unitVec 𝐼) Fn 𝐼 ∧ ran (𝑅 unitVec 𝐼) ⊆ 𝐵) → (𝐸 ∘ (𝑅 unitVec 𝐼)) Fn 𝐼)
44	27, 42, 19, 43	syl3anc 1477	. . . . . . 7 ⊢ (𝜑 → (𝐸 ∘ (𝑅 unitVec 𝐼)) Fn 𝐼)
45		fvco2 6436	. . . . . . . . 9 ⊢ (((𝑅 unitVec 𝐼) Fn 𝐼 ∧ 𝑢 ∈ 𝐼) → ((𝐸 ∘ (𝑅 unitVec 𝐼))‘𝑢) = (𝐸‘((𝑅 unitVec 𝐼)‘𝑢)))
46	42, 45	sylan 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → ((𝐸 ∘ (𝑅 unitVec 𝐼))‘𝑢) = (𝐸‘((𝑅 unitVec 𝐼)‘𝑢)))
47	6	adantr 472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑇 ∈ LMod)
48	7	adantr 472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝐼 ∈ 𝑋)
49	8	adantr 472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑅 = (Scalar‘𝑇))
50	9	adantr 472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝐴:𝐼⟶𝐶)
51		simpr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑢 ∈ 𝐼)
52	1, 2, 3, 4, 5, 47, 48, 49, 50, 51, 15	frlmup2 20360	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → (𝐸‘((𝑅 unitVec 𝐼)‘𝑢)) = (𝐴‘𝑢))
53	46, 52	eqtr2d 2795	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → (𝐴‘𝑢) = ((𝐸 ∘ (𝑅 unitVec 𝐼))‘𝑢))
54	40, 44, 53	eqfnfvd 6478	. . . . . 6 ⊢ (𝜑 → 𝐴 = (𝐸 ∘ (𝑅 unitVec 𝐼)))
55	54	imaeq1d 5623	. . . . 5 ⊢ (𝜑 → (𝐴 “ 𝐼) = ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼))
56		fnima 6171	. . . . . 6 ⊢ (𝐴 Fn 𝐼 → (𝐴 “ 𝐼) = ran 𝐴)
57	40, 56	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 “ 𝐼) = ran 𝐴)
58	55, 57	eqtr3d 2796	. . . 4 ⊢ (𝜑 → ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼) = ran 𝐴)
59		fnima 6171	. . . . . 6 ⊢ ((𝑅 unitVec 𝐼) Fn 𝐼 → ((𝑅 unitVec 𝐼) “ 𝐼) = ran (𝑅 unitVec 𝐼))
60	42, 59	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑅 unitVec 𝐼) “ 𝐼) = ran (𝑅 unitVec 𝐼))
61	60	imaeq2d 5624	. . . 4 ⊢ (𝜑 → (𝐸 “ ((𝑅 unitVec 𝐼) “ 𝐼)) = (𝐸 “ ran (𝑅 unitVec 𝐼)))
62	38, 58, 61	3eqtr3a 2818	. . 3 ⊢ (𝜑 → ran 𝐴 = (𝐸 “ ran (𝑅 unitVec 𝐼)))
63	62	fveq2d 6357	. 2 ⊢ (𝜑 → (𝐾‘ran 𝐴) = (𝐾‘(𝐸 “ ran (𝑅 unitVec 𝐼))))
64	23, 37, 63	3eqtr4d 2804	1 ⊢ (𝜑 → ran 𝐸 = (𝐾‘ran 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ⊆ wss 3715 ↦ cmpt 4881 ran crn 5267 “ cima 5269 ∘ ccom 5270 Fn wfn 6044 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 ∘_𝑓 cof 7061 Basecbs 16079 Scalarcsca 16166 ·_𝑠 cvsca 16167 Σ_g cgsu 16323 Ringcrg 18767 LModclmod 19085 LSpanclspn 19193 LMHom clmhm 19241 LBasisclbs 19296 freeLMod cfrlm 20312 unitVec cuvc 20343

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-om 7232 df-1st 7334 df-2nd 7335 df-supp 7465 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-map 8027 df-ixp 8077 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-fsupp 8443 df-sup 8515 df-oi 8582 df-card 8975 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-fz 12540 df-fzo 12680 df-seq 13016 df-hash 13332 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-sca 16179 df-vsca 16180 df-ip 16181 df-tset 16182 df-ple 16183 df-ds 16186 df-hom 16188 df-cco 16189 df-0g 16324 df-gsum 16325 df-prds 16330 df-pws 16332 df-mre 16468 df-mrc 16469 df-acs 16471 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-mhm 17556 df-submnd 17557 df-grp 17646 df-minusg 17647 df-sbg 17648 df-mulg 17762 df-subg 17812 df-ghm 17879 df-cntz 17970 df-cmn 18415 df-abl 18416 df-mgp 18710 df-ur 18722 df-ring 18769 df-subrg 19000 df-lmod 19087 df-lss 19155 df-lsp 19194 df-lmhm 19244 df-lbs 19297 df-sra 19394 df-rgmod 19395 df-nzr 19480 df-dsmm 20298 df-frlm 20313 df-uvc 20344

This theorem is referenced by: ellspd 20363 indlcim 20401 lnrfg 38209

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