ellspd - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ellspd		Structured version Visualization version GIF version

Theorem ellspd 21009

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 6600	. . . . . 6 ⊢ (𝐹:𝐼⟶𝐵 → 𝐹 Fn 𝐼)
3		fnima 6563	. . . . . 6 ⊢ (𝐹 Fn 𝐼 → (𝐹 “ 𝐼) = ran 𝐹)
4	1, 2, 3	3syl 18	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐼) = ran 𝐹)
5	4	fveq2d 6778	. . . 4 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = (𝑁‘ran 𝐹))
6		eqid 2738	. . . . . 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 5864	. . . . 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 2738	. . . . . 6 ⊢ (𝑆 freeLMod 𝐼) = (𝑆 freeLMod 𝐼)
9		eqid 2738	. . . . . 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 21007	. . . . 5 ⊢ (𝜑 → ran (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σ_g (𝑓 ∘_f · 𝐹))) = (𝑁‘ran 𝐹))
18	7, 17	eqtr3id 2792	. . . 4 ⊢ (𝜑 → {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))} = (𝑁‘ran 𝐹))
19	5, 18	eqtr4d 2781	. . 3 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))})
20	19	eleq2d 2824	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ 𝑋 ∈ {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))}))
21		ovex 7308	. . . . . 6 ⊢ (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) ∈ V
22		eleq1 2826	. . . . . 6 ⊢ (𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) → (𝑋 ∈ V ↔ (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) ∈ V))
23	21, 22	mpbiri 257	. . . . 5 ⊢ (𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) → 𝑋 ∈ V)
24	23	rexlimivw 3211	. . . 4 ⊢ (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) → 𝑋 ∈ V)
25		eqeq1 2742	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹)) ↔ 𝑋 = (𝑀 Σ_g (𝑓 ∘_f · 𝐹))))
26	25	rexbidv 3226	. . . 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 3617	. . 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 6788	. . . . . . 7 ⊢ 𝑆 ∈ V
29		ellspd.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝑆)
30		ellspd.z	. . . . . . . 8 ⊢ 0 = (0_g‘𝑆)
31		eqid 2738	. . . . . . . 8 ⊢ {𝑎 ∈ (𝐾 ↑_m 𝐼) ∣ 𝑎 finSupp 0 } = {𝑎 ∈ (𝐾 ↑_m 𝐼) ∣ 𝑎 finSupp 0 }
32	8, 29, 30, 31	frlmbas 20962	. . . . . . 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 2744	. . . . 5 ⊢ (𝜑 → (Base‘(𝑆 freeLMod 𝐼)) = {𝑎 ∈ (𝐾 ↑_m 𝐼) ∣ 𝑎 finSupp 0 })
35	34	rexeqdv 3349	. . . 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 5077	. . . . 5 ⊢ (𝑎 = 𝑓 → (𝑎 finSupp 0 ↔ 𝑓 finSupp 0 ))
37	36	rexrab 3633	. . . 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 282	. 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 278	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 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {cab 2715 ∃wrex 3065 {crab 3068 Vcvv 3432 class class class wbr 5074 ↦ cmpt 5157 ran crn 5590 “ cima 5592 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∘_f cof 7531 ↑_m cmap 8615 finSupp cfsupp 9128 Basecbs 16912 Scalarcsca 16965 ·_𝑠 cvsca 16966 0_gc0g 17150 Σ_g cgsu 17151 LModclmod 20123 LSpanclspn 20233 freeLMod cfrlm 20953

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-hom 16986 df-cco 16987 df-0g 17152 df-gsum 17153 df-prds 17158 df-pws 17160 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-ghm 18832 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-subrg 20022 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lmhm 20284 df-lbs 20337 df-sra 20434 df-rgmod 20435 df-nzr 20529 df-dsmm 20939 df-frlm 20954 df-uvc 20990

This theorem is referenced by: elfilspd 21010 islindf4 21045 ellspds 31564 fedgmul 31712

W3C validator