ellspd - Metamath Proof Explorer

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Theorem ellspd 21741

Description: The elements of the span of an indexed collection of basic vectors are those vectors which can be written as finite linear combinations of basic vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by AV, 24-Jun-2019.) (Revised by AV, 11-Apr-2024.)

**Hypotheses**
Ref	Expression
ellspd.n	⊢ 𝑁 = (LSpan‘𝑀)
ellspd.v	⊢ 𝐵 = (Base‘𝑀)
ellspd.k	⊢ 𝐾 = (Base‘𝑆)
ellspd.s	⊢ 𝑆 = (Scalar‘𝑀)
ellspd.z	⊢ 0 = (0_g‘𝑆)
ellspd.t	⊢ · = ( ·_𝑠 ‘𝑀)
ellspd.f	⊢ (𝜑 → 𝐹:𝐼⟶𝐵)
ellspd.m	⊢ (𝜑 → 𝑀 ∈ LMod)
ellspd.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)

**Assertion**
Ref	Expression
ellspd	⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑_m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)))))

Distinct variable groups: 𝐵,𝑓 𝑓,𝐹 𝑓,𝐼 𝑓,𝐾 𝑓,𝑀 𝑓,𝑁 𝑆,𝑓 𝑓,𝑋 0 ,𝑓 · ,𝑓 𝜑,𝑓 𝑓,𝑉

Proof of Theorem ellspd Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ellspd.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)
2		ffn 6656	. . . . . 6 ⊢ (𝐹:𝐼⟶𝐵 → 𝐹 Fn 𝐼)
3		fnima 6616	. . . . . 6 ⊢ (𝐹 Fn 𝐼 → (𝐹 “ 𝐼) = ran 𝐹)
4	1, 2, 3	3syl 18	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐼) = ran 𝐹)
5	4	fveq2d 6832	. . . 4 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = (𝑁‘ran 𝐹))
6		eqid 2733	. . . . . 6 ⊢ (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σ_g (𝑓 ∘_f · 𝐹))) = (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σ_g (𝑓 ∘_f · 𝐹)))
7	6	rnmpt 5901	. . . . 5 ⊢ ran (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σ_g (𝑓 ∘_f · 𝐹))) = {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))}
8		eqid 2733	. . . . . 6 ⊢ (𝑆 freeLMod 𝐼) = (𝑆 freeLMod 𝐼)
9		eqid 2733	. . . . . 6 ⊢ (Base‘(𝑆 freeLMod 𝐼)) = (Base‘(𝑆 freeLMod 𝐼))
10		ellspd.v	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
11		ellspd.t	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑀)
12		ellspd.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ LMod)
13		ellspd.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
14		ellspd.s	. . . . . . 7 ⊢ 𝑆 = (Scalar‘𝑀)
15	14	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑆 = (Scalar‘𝑀))
16		ellspd.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑀)
17	8, 9, 10, 11, 6, 12, 13, 15, 1, 16	frlmup3 21739	. . . . 5 ⊢ (𝜑 → ran (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σ_g (𝑓 ∘_f · 𝐹))) = (𝑁‘ran 𝐹))
18	7, 17	eqtr3id 2782	. . . 4 ⊢ (𝜑 → {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))} = (𝑁‘ran 𝐹))
19	5, 18	eqtr4d 2771	. . 3 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))})
20	19	eleq2d 2819	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ 𝑋 ∈ {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))}))
21		ovex 7385	. . . . . 6 ⊢ (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) ∈ V
22		eleq1 2821	. . . . . 6 ⊢ (𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) → (𝑋 ∈ V ↔ (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) ∈ V))
23	21, 22	mpbiri 258	. . . . 5 ⊢ (𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) → 𝑋 ∈ V)
24	23	rexlimivw 3130	. . . 4 ⊢ (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) → 𝑋 ∈ V)
25		eqeq1 2737	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) ↔ 𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))))
26	25	rexbidv 3157	. . . 4 ⊢ (𝑎 = 𝑋 → (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) ↔ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))))
27	24, 26	elab3 3638	. . 3 ⊢ (𝑋 ∈ {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))} ↔ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)))
28	14	fvexi 6842	. . . . . . 7 ⊢ 𝑆 ∈ V
29		ellspd.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝑆)
30		ellspd.z	. . . . . . . 8 ⊢ 0 = (0_g‘𝑆)
31		eqid 2733	. . . . . . . 8 ⊢ {𝑎 ∈ (𝐾 ↑_m 𝐼) ∣ 𝑎 finSupp 0 } = {𝑎 ∈ (𝐾 ↑_m 𝐼) ∣ 𝑎 finSupp 0 }
32	8, 29, 30, 31	frlmbas 21694	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐼 ∈ 𝑉) → {𝑎 ∈ (𝐾 ↑_m 𝐼) ∣ 𝑎 finSupp 0 } = (Base‘(𝑆 freeLMod 𝐼)))
33	28, 13, 32	sylancr 587	. . . . . 6 ⊢ (𝜑 → {𝑎 ∈ (𝐾 ↑_m 𝐼) ∣ 𝑎 finSupp 0 } = (Base‘(𝑆 freeLMod 𝐼)))
34	33	eqcomd 2739	. . . . 5 ⊢ (𝜑 → (Base‘(𝑆 freeLMod 𝐼)) = {𝑎 ∈ (𝐾 ↑_m 𝐼) ∣ 𝑎 finSupp 0 })
35	34	rexeqdv 3294	. . . 4 ⊢ (𝜑 → (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) ↔ ∃𝑓 ∈ {𝑎 ∈ (𝐾 ↑_m 𝐼) ∣ 𝑎 finSupp 0 }𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))))
36		breq1 5096	. . . . 5 ⊢ (𝑎 = 𝑓 → (𝑎 finSupp 0 ↔ 𝑓 finSupp 0 ))
37	36	rexrab 3651	. . . 4 ⊢ (∃𝑓 ∈ {𝑎 ∈ (𝐾 ↑_m 𝐼) ∣ 𝑎 finSupp 0 }𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) ↔ ∃𝑓 ∈ (𝐾 ↑_m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))))
38	35, 37	bitrdi 287	. . 3 ⊢ (𝜑 → (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) ↔ ∃𝑓 ∈ (𝐾 ↑_m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)))))
39	27, 38	bitrid 283	. 2 ⊢ (𝜑 → (𝑋 ∈ {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))} ↔ ∃𝑓 ∈ (𝐾 ↑_m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)))))
40	20, 39	bitrd 279	1 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑_m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 {cab 2711 ∃wrex 3057 {crab 3396 Vcvv 3437 class class class wbr 5093 ↦ cmpt 5174 ran crn 5620 “ cima 5622 Fn wfn 6481 ⟶wf 6482 ‘cfv 6486 (class class class)co 7352 ∘_f cof 7614 ↑_m cmap 8756 finSupp cfsupp 9252 Basecbs 17122 Scalarcsca 17166 ·_𝑠 cvsca 17167 0_gc0g 17345 Σ_g cgsu 17346 LModclmod 20795 LSpanclspn 20906 freeLMod cfrlm 21685

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-sup 9333 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-fz 13410 df-fzo 13557 df-seq 13911 df-hash 14240 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-sca 17179 df-vsca 17180 df-ip 17181 df-tset 17182 df-ple 17183 df-ds 17185 df-hom 17187 df-cco 17188 df-0g 17347 df-gsum 17348 df-prds 17353 df-pws 17355 df-mre 17490 df-mrc 17491 df-acs 17493 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-mhm 18693 df-submnd 18694 df-grp 18851 df-minusg 18852 df-sbg 18853 df-mulg 18983 df-subg 19038 df-ghm 19127 df-cntz 19231 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-nzr 20430 df-subrg 20487 df-lmod 20797 df-lss 20867 df-lsp 20907 df-lmhm 20958 df-lbs 21011 df-sra 21109 df-rgmod 21110 df-dsmm 21671 df-frlm 21686 df-uvc 21722

This theorem is referenced by: elfilspd 21742 islindf4 21777 ellspds 33340 ply1degltdimlem 33656 fedgmul 33665

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