frlmup3 - Metamath Proof Explorer

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Theorem frlmup3 21728

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 21726	. . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇))
11		eqid 2728	. . . . . . . 8 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
12	11	lmodring 20745	. . . . . . 7 ⊢ (𝑇 ∈ LMod → (Scalar‘𝑇) ∈ Ring)
13	6, 12	syl 17	. . . . . 6 ⊢ (𝜑 → (Scalar‘𝑇) ∈ Ring)
14	8, 13	eqeltrd 2829	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
15		eqid 2728	. . . . . 6 ⊢ (𝑅 unitVec 𝐼) = (𝑅 unitVec 𝐼)
16	15, 1, 2	uvcff 21719	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → (𝑅 unitVec 𝐼):𝐼⟶𝐵)
17	14, 7, 16	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝑅 unitVec 𝐼):𝐼⟶𝐵)
18	17	frnd 6725	. . 3 ⊢ (𝜑 → ran (𝑅 unitVec 𝐼) ⊆ 𝐵)
19		eqid 2728	. . . 4 ⊢ (LSpan‘𝐹) = (LSpan‘𝐹)
20		frlmup.k	. . . 4 ⊢ 𝐾 = (LSpan‘𝑇)
21	2, 19, 20	lmhmlsp 20928	. . 3 ⊢ ((𝐸 ∈ (𝐹 LMHom 𝑇) ∧ ran (𝑅 unitVec 𝐼) ⊆ 𝐵) → (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼))) = (𝐾‘(𝐸 “ ran (𝑅 unitVec 𝐼))))
22	10, 18, 21	syl2anc 583	. 2 ⊢ (𝜑 → (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼))) = (𝐾‘(𝐸 “ ran (𝑅 unitVec 𝐼))))
23	2, 3	lmhmf 20913	. . . . . 6 ⊢ (𝐸 ∈ (𝐹 LMHom 𝑇) → 𝐸:𝐵⟶𝐶)
24	10, 23	syl 17	. . . . 5 ⊢ (𝜑 → 𝐸:𝐵⟶𝐶)
25	24	ffnd 6718	. . . 4 ⊢ (𝜑 → 𝐸 Fn 𝐵)
26		fnima 6680	. . . 4 ⊢ (𝐸 Fn 𝐵 → (𝐸 “ 𝐵) = ran 𝐸)
27	25, 26	syl 17	. . 3 ⊢ (𝜑 → (𝐸 “ 𝐵) = ran 𝐸)
28		eqid 2728	. . . . . . . 8 ⊢ (LBasis‘𝐹) = (LBasis‘𝐹)
29	1, 15, 28	frlmlbs 21725	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘𝐹))
30	14, 7, 29	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘𝐹))
31	2, 28, 19	lbssp 20958	. . . . . 6 ⊢ (ran (𝑅 unitVec 𝐼) ∈ (LBasis‘𝐹) → ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼)) = 𝐵)
32	30, 31	syl 17	. . . . 5 ⊢ (𝜑 → ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼)) = 𝐵)
33	32	eqcomd 2734	. . . 4 ⊢ (𝜑 → 𝐵 = ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼)))
34	33	imaeq2d 6058	. . 3 ⊢ (𝜑 → (𝐸 “ 𝐵) = (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼))))
35	27, 34	eqtr3d 2770	. 2 ⊢ (𝜑 → ran 𝐸 = (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼))))
36		imaco 6250	. . . 4 ⊢ ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼) = (𝐸 “ ((𝑅 unitVec 𝐼) “ 𝐼))
37	9	ffnd 6718	. . . . . . 7 ⊢ (𝜑 → 𝐴 Fn 𝐼)
38	17	ffnd 6718	. . . . . . . 8 ⊢ (𝜑 → (𝑅 unitVec 𝐼) Fn 𝐼)
39		fnco 6667	. . . . . . . 8 ⊢ ((𝐸 Fn 𝐵 ∧ (𝑅 unitVec 𝐼) Fn 𝐼 ∧ ran (𝑅 unitVec 𝐼) ⊆ 𝐵) → (𝐸 ∘ (𝑅 unitVec 𝐼)) Fn 𝐼)
40	25, 38, 18, 39	syl3anc 1369	. . . . . . 7 ⊢ (𝜑 → (𝐸 ∘ (𝑅 unitVec 𝐼)) Fn 𝐼)
41		fvco2 6990	. . . . . . . . 9 ⊢ (((𝑅 unitVec 𝐼) Fn 𝐼 ∧ 𝑢 ∈ 𝐼) → ((𝐸 ∘ (𝑅 unitVec 𝐼))‘𝑢) = (𝐸‘((𝑅 unitVec 𝐼)‘𝑢)))
42	38, 41	sylan 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → ((𝐸 ∘ (𝑅 unitVec 𝐼))‘𝑢) = (𝐸‘((𝑅 unitVec 𝐼)‘𝑢)))
43	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑇 ∈ LMod)
44	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝐼 ∈ 𝑋)
45	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑅 = (Scalar‘𝑇))
46	9	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝐴:𝐼⟶𝐶)
47		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑢 ∈ 𝐼)
48	1, 2, 3, 4, 5, 43, 44, 45, 46, 47, 15	frlmup2 21727	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → (𝐸‘((𝑅 unitVec 𝐼)‘𝑢)) = (𝐴‘𝑢))
49	42, 48	eqtr2d 2769	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → (𝐴‘𝑢) = ((𝐸 ∘ (𝑅 unitVec 𝐼))‘𝑢))
50	37, 40, 49	eqfnfvd 7038	. . . . . 6 ⊢ (𝜑 → 𝐴 = (𝐸 ∘ (𝑅 unitVec 𝐼)))
51	50	imaeq1d 6057	. . . . 5 ⊢ (𝜑 → (𝐴 “ 𝐼) = ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼))
52		fnima 6680	. . . . . 6 ⊢ (𝐴 Fn 𝐼 → (𝐴 “ 𝐼) = ran 𝐴)
53	37, 52	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 “ 𝐼) = ran 𝐴)
54	51, 53	eqtr3d 2770	. . . 4 ⊢ (𝜑 → ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼) = ran 𝐴)
55		fnima 6680	. . . . . 6 ⊢ ((𝑅 unitVec 𝐼) Fn 𝐼 → ((𝑅 unitVec 𝐼) “ 𝐼) = ran (𝑅 unitVec 𝐼))
56	38, 55	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑅 unitVec 𝐼) “ 𝐼) = ran (𝑅 unitVec 𝐼))
57	56	imaeq2d 6058	. . . 4 ⊢ (𝜑 → (𝐸 “ ((𝑅 unitVec 𝐼) “ 𝐼)) = (𝐸 “ ran (𝑅 unitVec 𝐼)))
58	36, 54, 57	3eqtr3a 2792	. . 3 ⊢ (𝜑 → ran 𝐴 = (𝐸 “ ran (𝑅 unitVec 𝐼)))
59	58	fveq2d 6896	. 2 ⊢ (𝜑 → (𝐾‘ran 𝐴) = (𝐾‘(𝐸 “ ran (𝑅 unitVec 𝐼))))
60	22, 35, 59	3eqtr4d 2778	1 ⊢ (𝜑 → ran 𝐸 = (𝐾‘ran 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⊆ wss 3945 ↦ cmpt 5226 ran crn 5674 “ cima 5676 ∘ ccom 5677 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7415 ∘_f cof 7678 Basecbs 17174 Scalarcsca 17230 ·_𝑠 cvsca 17231 Σ_g cgsu 17416 Ringcrg 20167 LModclmod 20737 LSpanclspn 20849 LMHom clmhm 20898 LBasisclbs 20953 freeLMod cfrlm 21674 unitVec cuvc 21710

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7680 df-om 7866 df-1st 7988 df-2nd 7989 df-supp 8161 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-ixp 8911 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-fsupp 9381 df-sup 9460 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-fzo 13655 df-seq 13994 df-hash 14317 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-hom 17251 df-cco 17252 df-0g 17417 df-gsum 17418 df-prds 17423 df-pws 17425 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mhm 18734 df-submnd 18735 df-grp 18887 df-minusg 18888 df-sbg 18889 df-mulg 19018 df-subg 19072 df-ghm 19162 df-cntz 19262 df-cmn 19731 df-abl 19732 df-mgp 20069 df-rng 20087 df-ur 20116 df-ring 20169 df-nzr 20446 df-subrg 20502 df-lmod 20739 df-lss 20810 df-lsp 20850 df-lmhm 20901 df-lbs 20954 df-sra 21052 df-rgmod 21053 df-dsmm 21660 df-frlm 21675 df-uvc 21711

This theorem is referenced by: ellspd 21730 indlcim 21768 lnrfg 42534

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