ellspd - Metamath Proof Explorer

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Theorem ellspd 19960

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 5958	. . . . . 6 ⊢ (𝐹:𝐼⟶𝐵 → 𝐹 Fn 𝐼)
3		fnima 5923	. . . . . 6 ⊢ (𝐹 Fn 𝐼 → (𝐹 “ 𝐼) = ran 𝐹)
4	1, 2, 3	3syl 18	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐼) = ran 𝐹)
5	4	fveq2d 6107	. . . 4 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = (𝑁‘ran 𝐹))
6		eqid 2610	. . . . . 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 5292	. . . . 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 2610	. . . . . 6 ⊢ (𝑆 freeLMod 𝐼) = (𝑆 freeLMod 𝐼)
9		eqid 2610	. . . . . 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 19958	. . . . 5 ⊢ (𝜑 → ran (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹))) = (𝑁‘ran 𝐹))
18	7, 17	syl5eqr 2658	. . . 4 ⊢ (𝜑 → {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹))} = (𝑁‘ran 𝐹))
19	5, 18	eqtr4d 2647	. . 3 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹))})
20	19	eleq2d 2673	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ 𝑋 ∈ {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹))}))
21		ovex 6577	. . . . . 6 ⊢ (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹)) ∈ V
22		eleq1 2676	. . . . . 6 ⊢ (𝑋 = (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹)) → (𝑋 ∈ V ↔ (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹)) ∈ V))
23	21, 22	mpbiri 247	. . . . 5 ⊢ (𝑋 = (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹)) → 𝑋 ∈ V)
24	23	rexlimivw 3011	. . . 4 ⊢ (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹)) → 𝑋 ∈ V)
25		eqeq1 2614	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 = (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹)) ↔ 𝑋 = (𝑀 Σ_g (𝑓 ∘_𝑓 · 𝐹))))
26	25	rexbidv 3034	. . . 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 3327	. . 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 6113	. . . . . . . 8 ⊢ (Scalar‘𝑀) ∈ V
29	14, 28	eqeltri 2684	. . . . . . 7 ⊢ 𝑆 ∈ V
30		ellspd.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝑆)
31		ellspd.z	. . . . . . . 8 ⊢ 0 = (0_g‘𝑆)
32		eqid 2610	. . . . . . . 8 ⊢ {𝑎 ∈ (𝐾 ↑_𝑚 𝐼) ∣ 𝑎 finSupp 0 } = {𝑎 ∈ (𝐾 ↑_𝑚 𝐼) ∣ 𝑎 finSupp 0 }
33	8, 30, 31, 32	frlmbas 19918	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐼 ∈ V) → {𝑎 ∈ (𝐾 ↑_𝑚 𝐼) ∣ 𝑎 finSupp 0 } = (Base‘(𝑆 freeLMod 𝐼)))
34	29, 13, 33	sylancr 694	. . . . . 6 ⊢ (𝜑 → {𝑎 ∈ (𝐾 ↑_𝑚 𝐼) ∣ 𝑎 finSupp 0 } = (Base‘(𝑆 freeLMod 𝐼)))
35	34	eqcomd 2616	. . . . 5 ⊢ (𝜑 → (Base‘(𝑆 freeLMod 𝐼)) = {𝑎 ∈ (𝐾 ↑_𝑚 𝐼) ∣ 𝑎 finSupp 0 })
36	35	rexeqdv 3122	. . . 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 4586	. . . . 5 ⊢ (𝑎 = 𝑓 → (𝑎 finSupp 0 ↔ 𝑓 finSupp 0 ))
38	37	rexrab 3337	. . . 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 275	. . 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 271	. 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 267	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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 ∃wrex 2897 {crab 2900 Vcvv 3173 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 “ cima 5041 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 ↑_𝑚 cmap 7744 finSupp cfsupp 8158 Basecbs 15695 Scalarcsca 15771 ·_𝑠 cvsca 15772 0_gc0g 15923 Σ_g cgsu 15924 LModclmod 18686 LSpanclspn 18792 freeLMod cfrlm 19909

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-0g 15925 df-gsum 15926 df-prds 15931 df-pws 15933 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lmhm 18843 df-lbs 18896 df-sra 18993 df-rgmod 18994 df-nzr 19079 df-dsmm 19895 df-frlm 19910 df-uvc 19941

This theorem is referenced by: elfilspd 19961 islindf4 19996

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