ellspd - Metamath Proof Explorer

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Theorem ellspd 20364

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.)

**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: 𝑓,𝑀 𝐵,𝑓 𝑓,𝑁 𝑓,𝐾 𝑆,𝑓 0 ,𝑓 · ,𝑓 𝑓,𝐹 𝑓,𝐼 𝑓,𝑋 𝜑,𝑓

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 6207	. . . . . 6 ⊢ (𝐹:𝐼⟶𝐵 → 𝐹 Fn 𝐼)
3		fnima 6172	. . . . . 6 ⊢ (𝐹 Fn 𝐼 → (𝐹 “ 𝐼) = ran 𝐹)
4	1, 2, 3	3syl 18	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐼) = ran 𝐹)
5	4	fveq2d 6358	. . . 4 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = (𝑁‘ran 𝐹))
6		eqid 2761	. . . . . 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 5527	. . . . 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 2761	. . . . . 6 ⊢ (𝑆 freeLMod 𝐼) = (𝑆 freeLMod 𝐼)
9		eqid 2761	. . . . . 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 20362	. . . . 5 ⊢ (𝜑 → ran (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹))) = (𝑁‘ran 𝐹))
18	7, 17	syl5eqr 2809	. . . 4 ⊢ (𝜑 → {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹))} = (𝑁‘ran 𝐹))
19	5, 18	eqtr4d 2798	. . 3 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹))})
20	19	eleq2d 2826	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ 𝑋 ∈ {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹))}))
21		ovex 6843	. . . . . 6 ⊢ (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹)) ∈ V
22		eleq1 2828	. . . . . 6 ⊢ (𝑋 = (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹)) → (𝑋 ∈ V ↔ (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹)) ∈ V))
23	21, 22	mpbiri 248	. . . . 5 ⊢ (𝑋 = (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹)) → 𝑋 ∈ V)
24	23	rexlimivw 3168	. . . 4 ⊢ (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹)) → 𝑋 ∈ V)
25		eqeq1 2765	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 = (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹)) ↔ 𝑋 = (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹))))
26	25	rexbidv 3191	. . . 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 3499	. . 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		fvex 6364	. . . . . . . 8 ⊢ (Scalar‘𝑀) ∈ V
29	14, 28	eqeltri 2836	. . . . . . 7 ⊢ 𝑆 ∈ V
30		ellspd.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝑆)
31		ellspd.z	. . . . . . . 8 ⊢ 0 = (0_g‘𝑆)
32		eqid 2761	. . . . . . . 8 ⊢ {𝑎 ∈ (𝐾 ↑_𝑚 𝐼) ∣ 𝑎 finSupp 0 } = {𝑎 ∈ (𝐾 ↑_𝑚 𝐼) ∣ 𝑎 finSupp 0 }
33	8, 30, 31, 32	frlmbas 20322	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐼 ∈ V) → {𝑎 ∈ (𝐾 ↑_𝑚 𝐼) ∣ 𝑎 finSupp 0 } = (Base‘(𝑆 freeLMod 𝐼)))
34	29, 13, 33	sylancr 698	. . . . . 6 ⊢ (𝜑 → {𝑎 ∈ (𝐾 ↑_𝑚 𝐼) ∣ 𝑎 finSupp 0 } = (Base‘(𝑆 freeLMod 𝐼)))
35	34	eqcomd 2767	. . . . 5 ⊢ (𝜑 → (Base‘(𝑆 freeLMod 𝐼)) = {𝑎 ∈ (𝐾 ↑_𝑚 𝐼) ∣ 𝑎 finSupp 0 })
36	35	rexeqdv 3285	. . . 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))))
37		breq1 4808	. . . . 5 ⊢ (𝑎 = 𝑓 → (𝑎 finSupp 0 ↔ 𝑓 finSupp 0 ))
38	37	rexrab 3512	. . . 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))))
39	36, 38	syl6bb 276	. . 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)))))
40	27, 39	syl5bb 272	. 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)))))
41	20, 40	bitrd 268	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 196 ∧ wa 383 = wceq 1632 ∈ wcel 2140 {cab 2747 ∃wrex 3052 {crab 3055 Vcvv 3341 class class class wbr 4805 ↦ cmpt 4882 ran crn 5268 “ cima 5270 Fn wfn 6045 ⟶wf 6046 ‘cfv 6050 (class class class)co 6815 ∘_𝑓 cof 7062 ↑_𝑚 cmap 8026 finSupp cfsupp 8443 Basecbs 16080 Scalarcsca 16167 ·_𝑠 cvsca 16168 0_gc0g 16323 Σ_g cgsu 16324 LModclmod 19086 LSpanclspn 19194 freeLMod cfrlm 20313

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 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-rep 4924 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 ax-inf2 8714 ax-cnex 10205 ax-resscn 10206 ax-1cn 10207 ax-icn 10208 ax-addcl 10209 ax-addrcl 10210 ax-mulcl 10211 ax-mulrcl 10212 ax-mulcom 10213 ax-addass 10214 ax-mulass 10215 ax-distr 10216 ax-i2m1 10217 ax-1ne0 10218 ax-1rid 10219 ax-rnegex 10220 ax-rrecex 10221 ax-cnre 10222 ax-pre-lttri 10223 ax-pre-lttrn 10224 ax-pre-ltadd 10225 ax-pre-mulgt0 10226

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 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-tp 4327 df-op 4329 df-uni 4590 df-int 4629 df-iun 4675 df-iin 4676 df-br 4806 df-opab 4866 df-mpt 4883 df-tr 4906 df-id 5175 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-se 5227 df-we 5228 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-pred 5842 df-ord 5888 df-on 5889 df-lim 5890 df-suc 5891 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-isom 6059 df-riota 6776 df-ov 6818 df-oprab 6819 df-mpt2 6820 df-of 7064 df-om 7233 df-1st 7335 df-2nd 7336 df-supp 7466 df-wrecs 7578 df-recs 7639 df-rdg 7677 df-1o 7731 df-oadd 7735 df-er 7914 df-map 8028 df-ixp 8078 df-en 8125 df-dom 8126 df-sdom 8127 df-fin 8128 df-fsupp 8444 df-sup 8516 df-oi 8583 df-card 8976 df-pnf 10289 df-mnf 10290 df-xr 10291 df-ltxr 10292 df-le 10293 df-sub 10481 df-neg 10482 df-nn 11234 df-2 11292 df-3 11293 df-4 11294 df-5 11295 df-6 11296 df-7 11297 df-8 11298 df-9 11299 df-n0 11506 df-z 11591 df-dec 11707 df-uz 11901 df-fz 12541 df-fzo 12681 df-seq 13017 df-hash 13333 df-struct 16082 df-ndx 16083 df-slot 16084 df-base 16086 df-sets 16087 df-ress 16088 df-plusg 16177 df-mulr 16178 df-sca 16180 df-vsca 16181 df-ip 16182 df-tset 16183 df-ple 16184 df-ds 16187 df-hom 16189 df-cco 16190 df-0g 16325 df-gsum 16326 df-prds 16331 df-pws 16333 df-mre 16469 df-mrc 16470 df-acs 16472 df-mgm 17464 df-sgrp 17506 df-mnd 17517 df-mhm 17557 df-submnd 17558 df-grp 17647 df-minusg 17648 df-sbg 17649 df-mulg 17763 df-subg 17813 df-ghm 17880 df-cntz 17971 df-cmn 18416 df-abl 18417 df-mgp 18711 df-ur 18723 df-ring 18770 df-subrg 19001 df-lmod 19088 df-lss 19156 df-lsp 19195 df-lmhm 19245 df-lbs 19298 df-sra 19395 df-rgmod 19396 df-nzr 19481 df-dsmm 20299 df-frlm 20314 df-uvc 20345

This theorem is referenced by: elfilspd 20365 islindf4 20400

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