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Mirrors > Home > MPE Home > Th. List > ellspd | Structured version Visualization version GIF version |
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.) |
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 = (0g‘𝑆) |
ellspd.t | ⊢ · = ( ·𝑠 ‘𝑀) |
ellspd.f | ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) |
ellspd.m | ⊢ (𝜑 → 𝑀 ∈ LMod) |
ellspd.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
Ref | Expression |
---|---|
ellspd | ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ellspd.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) | |
2 | ffn 6523 | . . . . . 6 ⊢ (𝐹:𝐼⟶𝐵 → 𝐹 Fn 𝐼) | |
3 | fnima 6486 | . . . . . 6 ⊢ (𝐹 Fn 𝐼 → (𝐹 “ 𝐼) = ran 𝐹) | |
4 | 1, 2, 3 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐼) = ran 𝐹) |
5 | 4 | fveq2d 6699 | . . . 4 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = (𝑁‘ran 𝐹)) |
6 | eqid 2736 | . . . . . 6 ⊢ (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σg (𝑓 ∘f · 𝐹))) = (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σg (𝑓 ∘f · 𝐹))) | |
7 | 6 | rnmpt 5809 | . . . . 5 ⊢ ran (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σg (𝑓 ∘f · 𝐹))) = {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))} |
8 | eqid 2736 | . . . . . 6 ⊢ (𝑆 freeLMod 𝐼) = (𝑆 freeLMod 𝐼) | |
9 | eqid 2736 | . . . . . 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 20716 | . . . . 5 ⊢ (𝜑 → ran (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σg (𝑓 ∘f · 𝐹))) = (𝑁‘ran 𝐹)) |
18 | 7, 17 | eqtr3id 2785 | . . . 4 ⊢ (𝜑 → {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))} = (𝑁‘ran 𝐹)) |
19 | 5, 18 | eqtr4d 2774 | . . 3 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))}) |
20 | 19 | eleq2d 2816 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ 𝑋 ∈ {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))})) |
21 | ovex 7224 | . . . . . 6 ⊢ (𝑀 Σg (𝑓 ∘f · 𝐹)) ∈ V | |
22 | eleq1 2818 | . . . . . 6 ⊢ (𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) → (𝑋 ∈ V ↔ (𝑀 Σg (𝑓 ∘f · 𝐹)) ∈ V)) | |
23 | 21, 22 | mpbiri 261 | . . . . 5 ⊢ (𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) → 𝑋 ∈ V) |
24 | 23 | rexlimivw 3191 | . . . 4 ⊢ (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) → 𝑋 ∈ V) |
25 | eqeq1 2740 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹)) ↔ 𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)))) | |
26 | 25 | rexbidv 3206 | . . . 4 ⊢ (𝑎 = 𝑋 → (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹)) ↔ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)))) |
27 | 24, 26 | elab3 3584 | . . 3 ⊢ (𝑋 ∈ {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))} ↔ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹))) |
28 | 14 | fvexi 6709 | . . . . . . 7 ⊢ 𝑆 ∈ V |
29 | ellspd.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝑆) | |
30 | ellspd.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑆) | |
31 | eqid 2736 | . . . . . . . 8 ⊢ {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 } = {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 } | |
32 | 8, 29, 30, 31 | frlmbas 20671 | . . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐼 ∈ 𝑉) → {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 } = (Base‘(𝑆 freeLMod 𝐼))) |
33 | 28, 13, 32 | sylancr 590 | . . . . . 6 ⊢ (𝜑 → {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 } = (Base‘(𝑆 freeLMod 𝐼))) |
34 | 33 | eqcomd 2742 | . . . . 5 ⊢ (𝜑 → (Base‘(𝑆 freeLMod 𝐼)) = {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 }) |
35 | 34 | rexeqdv 3316 | . . . 4 ⊢ (𝜑 → (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) ↔ ∃𝑓 ∈ {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 }𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)))) |
36 | breq1 5042 | . . . . 5 ⊢ (𝑎 = 𝑓 → (𝑎 finSupp 0 ↔ 𝑓 finSupp 0 )) | |
37 | 36 | rexrab 3598 | . . . 4 ⊢ (∃𝑓 ∈ {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 }𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) ↔ ∃𝑓 ∈ (𝐾 ↑m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)))) |
38 | 35, 37 | bitrdi 290 | . . 3 ⊢ (𝜑 → (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) ↔ ∃𝑓 ∈ (𝐾 ↑m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹))))) |
39 | 27, 38 | syl5bb 286 | . 2 ⊢ (𝜑 → (𝑋 ∈ {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))} ↔ ∃𝑓 ∈ (𝐾 ↑m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹))))) |
40 | 20, 39 | bitrd 282 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {cab 2714 ∃wrex 3052 {crab 3055 Vcvv 3398 class class class wbr 5039 ↦ cmpt 5120 ran crn 5537 “ cima 5539 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ∘f cof 7445 ↑m cmap 8486 finSupp cfsupp 8963 Basecbs 16666 Scalarcsca 16752 ·𝑠 cvsca 16753 0gc0g 16898 Σg cgsu 16899 LModclmod 19853 LSpanclspn 19962 freeLMod cfrlm 20662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-sup 9036 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-hom 16773 df-cco 16774 df-0g 16900 df-gsum 16901 df-prds 16906 df-pws 16908 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-ghm 18574 df-cntz 18665 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-subrg 19752 df-lmod 19855 df-lss 19923 df-lsp 19963 df-lmhm 20013 df-lbs 20066 df-sra 20163 df-rgmod 20164 df-nzr 20250 df-dsmm 20648 df-frlm 20663 df-uvc 20699 |
This theorem is referenced by: elfilspd 20719 islindf4 20754 ellspds 31232 fedgmul 31380 |
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