ellspd - Metamath Proof Explorer

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Theorem ellspd 21989

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 6712	. . . . . 6 ⊢ (𝐹:𝐼⟶𝐵 → 𝐹 Fn 𝐼)
3		fnima 6672	. . . . . 6 ⊢ (𝐹 Fn 𝐼 → (𝐹 “ 𝐼) = ran 𝐹)
4	1, 2, 3	3syl 19	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐼) = ran 𝐹)
5	4	fveq2d 6892	. . . 4 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = (𝑁‘ran 𝐹))
6		eqid 2766	. . . . . 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 5952	. . . . 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 2766	. . . . . 6 ⊢ (𝑆 freeLMod 𝐼) = (𝑆 freeLMod 𝐼)
9		eqid 2766	. . . . . 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 21987	. . . . 5 ⊢ (𝜑 → ran (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σ_g (𝑓 ∘_f · 𝐹))) = (𝑁‘ran 𝐹))
18	7, 17	eqtr3id 2815	. . . 4 ⊢ (𝜑 → {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))} = (𝑁‘ran 𝐹))
19	5, 18	eqtr4d 2804	. . 3 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))})
20	19	eleq2d 2852	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ 𝑋 ∈ {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))}))
21		ovex 7456	. . . . . 6 ⊢ (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) ∈ V
22		eleq1 2854	. . . . . 6 ⊢ (𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) → (𝑋 ∈ V ↔ (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) ∈ V))
23	21, 22	mpbiri 261	. . . . 5 ⊢ (𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) → 𝑋 ∈ V)
24	23	rexlimivw 3165	. . . 4 ⊢ (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) → 𝑋 ∈ V)
25		eqeq1 2770	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) ↔ 𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))))
26	25	rexbidv 3192	. . . 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 3648	. . 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 6902	. . . . . . 7 ⊢ 𝑆 ∈ V
29		ellspd.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝑆)
30		ellspd.z	. . . . . . . 8 ⊢ 0 = (0_g‘𝑆)
31		eqid 2766	. . . . . . . 8 ⊢ {𝑎 ∈ (𝐾 ↑_m 𝐼) ∣ 𝑎 finSupp 0 } = {𝑎 ∈ (𝐾 ↑_m 𝐼) ∣ 𝑎 finSupp 0 }
32	8, 29, 30, 31	frlmbas 21942	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐼 ∈ 𝑉) → {𝑎 ∈ (𝐾 ↑_m 𝐼) ∣ 𝑎 finSupp 0 } = (Base‘(𝑆 freeLMod 𝐼)))
33	28, 13, 32	sylancr 599	. . . . . 6 ⊢ (𝜑 → {𝑎 ∈ (𝐾 ↑_m 𝐼) ∣ 𝑎 finSupp 0 } = (Base‘(𝑆 freeLMod 𝐼)))
34	33	eqcomd 2772	. . . . 5 ⊢ (𝜑 → (Base‘(𝑆 freeLMod 𝐼)) = {𝑎 ∈ (𝐾 ↑_m 𝐼) ∣ 𝑎 finSupp 0 })
35	34	rexeqdv 3327	. . . 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 5117	. . . . 5 ⊢ (𝑎 = 𝑓 → (𝑎 finSupp 0 ↔ 𝑓 finSupp 0 ))
37	36	rexrab 3662	. . . 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 290	. . 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 286	. 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 282	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 {cab 2744 ∃wrex 3092 {crab 3419 Vcvv 3458 class class class wbr 5114 ↦ cmpt 5197 ran crn 5667 “ cima 5669 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7423 ∘_f cof 7685 ↑_m cmap 8833 finSupp cfsupp 9331 Basecbs 17294 Scalarcsca 17338 ·_𝑠 cvsca 17339 0_gc0g 17517 Σ_g cgsu 17518 LModclmod 21018 LSpanclspn 21129 freeLMod cfrlm 21933

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 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 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-sup 9412 df-oi 9482 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-fz 13554 df-fzo 13702 df-seq 14058 df-hash 14387 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-sca 17351 df-vsca 17352 df-ip 17353 df-tset 17354 df-ple 17355 df-ds 17357 df-hom 17359 df-cco 17360 df-0g 17519 df-gsum 17520 df-prds 17525 df-pws 17527 df-mre 17663 df-mrc 17664 df-acs 17666 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-mhm 18872 df-submnd 18873 df-grp 19034 df-minusg 19035 df-sbg 19036 df-mulg 19165 df-subg 19220 df-ghm 19315 df-cntz 19418 df-cmn 19883 df-abl 19884 df-mgp 20248 df-rng 20262 df-ur 20295 df-ring 20348 df-nzr 20647 df-subrg 20706 df-lmod 21020 df-lss 21090 df-lsp 21130 df-lmhm 21180 df-lbs 21233 df-sra 21331 df-rgmod 21332 df-dsmm 21919 df-frlm 21934 df-uvc 21970

This theorem is used by: elfilspd 21990 islindf4 22025 ellspds 33714 ply1degltdimlem 34043 fedgmul 34052

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