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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 6730 | . . . . . 6 ⊢ (𝐹:𝐼⟶𝐵 → 𝐹 Fn 𝐼) | |
3 | fnima 6693 | . . . . . 6 ⊢ (𝐹 Fn 𝐼 → (𝐹 “ 𝐼) = ran 𝐹) | |
4 | 1, 2, 3 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐼) = ran 𝐹) |
5 | 4 | fveq2d 6907 | . . . 4 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = (𝑁‘ran 𝐹)) |
6 | eqid 2726 | . . . . . 6 ⊢ (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σg (𝑓 ∘f · 𝐹))) = (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σg (𝑓 ∘f · 𝐹))) | |
7 | 6 | rnmpt 5963 | . . . . 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 2726 | . . . . . 6 ⊢ (𝑆 freeLMod 𝐼) = (𝑆 freeLMod 𝐼) | |
9 | eqid 2726 | . . . . . 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 21800 | . . . . 5 ⊢ (𝜑 → ran (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σg (𝑓 ∘f · 𝐹))) = (𝑁‘ran 𝐹)) |
18 | 7, 17 | eqtr3id 2780 | . . . 4 ⊢ (𝜑 → {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))} = (𝑁‘ran 𝐹)) |
19 | 5, 18 | eqtr4d 2769 | . . 3 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))}) |
20 | 19 | eleq2d 2812 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ 𝑋 ∈ {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))})) |
21 | ovex 7459 | . . . . . 6 ⊢ (𝑀 Σg (𝑓 ∘f · 𝐹)) ∈ V | |
22 | eleq1 2814 | . . . . . 6 ⊢ (𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) → (𝑋 ∈ V ↔ (𝑀 Σg (𝑓 ∘f · 𝐹)) ∈ V)) | |
23 | 21, 22 | mpbiri 257 | . . . . 5 ⊢ (𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) → 𝑋 ∈ V) |
24 | 23 | rexlimivw 3141 | . . . 4 ⊢ (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) → 𝑋 ∈ V) |
25 | eqeq1 2730 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹)) ↔ 𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)))) | |
26 | 25 | rexbidv 3169 | . . . 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 3674 | . . 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 6917 | . . . . . . 7 ⊢ 𝑆 ∈ V |
29 | ellspd.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝑆) | |
30 | ellspd.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑆) | |
31 | eqid 2726 | . . . . . . . 8 ⊢ {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 } = {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 } | |
32 | 8, 29, 30, 31 | frlmbas 21755 | . . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐼 ∈ 𝑉) → {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 } = (Base‘(𝑆 freeLMod 𝐼))) |
33 | 28, 13, 32 | sylancr 585 | . . . . . 6 ⊢ (𝜑 → {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 } = (Base‘(𝑆 freeLMod 𝐼))) |
34 | 33 | eqcomd 2732 | . . . . 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 5158 | . . . . 5 ⊢ (𝑎 = 𝑓 → (𝑎 finSupp 0 ↔ 𝑓 finSupp 0 )) | |
37 | 36 | rexrab 3690 | . . . 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 286 | . . 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 | bitrid 282 | . 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 278 | 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 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 {cab 2703 ∃wrex 3060 {crab 3419 Vcvv 3462 class class class wbr 5155 ↦ cmpt 5238 ran crn 5685 “ cima 5687 Fn wfn 6551 ⟶wf 6552 ‘cfv 6556 (class class class)co 7426 ∘f cof 7690 ↑m cmap 8857 finSupp cfsupp 9407 Basecbs 17215 Scalarcsca 17271 ·𝑠 cvsca 17272 0gc0g 17456 Σg cgsu 17457 LModclmod 20838 LSpanclspn 20950 freeLMod cfrlm 21746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-iin 5006 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-se 5640 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-isom 6565 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8005 df-2nd 8006 df-supp 8177 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-2o 8499 df-er 8736 df-map 8859 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9408 df-sup 9487 df-oi 9555 df-card 9984 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12613 df-dec 12732 df-uz 12877 df-fz 13541 df-fzo 13684 df-seq 14024 df-hash 14350 df-struct 17151 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-ress 17245 df-plusg 17281 df-mulr 17282 df-sca 17284 df-vsca 17285 df-ip 17286 df-tset 17287 df-ple 17288 df-ds 17290 df-hom 17292 df-cco 17293 df-0g 17458 df-gsum 17459 df-prds 17464 df-pws 17466 df-mre 17601 df-mrc 17602 df-acs 17604 df-mgm 18635 df-sgrp 18714 df-mnd 18730 df-mhm 18775 df-submnd 18776 df-grp 18933 df-minusg 18934 df-sbg 18935 df-mulg 19064 df-subg 19119 df-ghm 19209 df-cntz 19313 df-cmn 19782 df-abl 19783 df-mgp 20120 df-rng 20138 df-ur 20167 df-ring 20220 df-nzr 20497 df-subrg 20555 df-lmod 20840 df-lss 20911 df-lsp 20951 df-lmhm 21002 df-lbs 21055 df-sra 21153 df-rgmod 21154 df-dsmm 21732 df-frlm 21747 df-uvc 21783 |
This theorem is referenced by: elfilspd 21803 islindf4 21838 ellspds 33245 ply1degltdimlem 33519 fedgmul 33528 |
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