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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 6735 | . . . . . 6 ⊢ (𝐹:𝐼⟶𝐵 → 𝐹 Fn 𝐼) | |
| 3 | fnima 6697 | . . . . . 6 ⊢ (𝐹 Fn 𝐼 → (𝐹 “ 𝐼) = ran 𝐹) | |
| 4 | 1, 2, 3 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐼) = ran 𝐹) | 
| 5 | 4 | fveq2d 6909 | . . . 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 5967 | . . . . 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 21821 | . . . . 5 ⊢ (𝜑 → ran (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σg (𝑓 ∘f · 𝐹))) = (𝑁‘ran 𝐹)) | 
| 18 | 7, 17 | eqtr3id 2790 | . . . 4 ⊢ (𝜑 → {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))} = (𝑁‘ran 𝐹)) | 
| 19 | 5, 18 | eqtr4d 2779 | . . 3 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))}) | 
| 20 | 19 | eleq2d 2826 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ 𝑋 ∈ {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))})) | 
| 21 | ovex 7465 | . . . . . 6 ⊢ (𝑀 Σg (𝑓 ∘f · 𝐹)) ∈ V | |
| 22 | eleq1 2828 | . . . . . 6 ⊢ (𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) → (𝑋 ∈ V ↔ (𝑀 Σg (𝑓 ∘f · 𝐹)) ∈ V)) | |
| 23 | 21, 22 | mpbiri 258 | . . . . 5 ⊢ (𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) → 𝑋 ∈ V) | 
| 24 | 23 | rexlimivw 3150 | . . . 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 3178 | . . . 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 3685 | . . 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 6919 | . . . . . . 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 21776 | . . . . . . 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 2742 | . . . . 5 ⊢ (𝜑 → (Base‘(𝑆 freeLMod 𝐼)) = {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 }) | 
| 35 | 34 | rexeqdv 3326 | . . . 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 5145 | . . . . 5 ⊢ (𝑎 = 𝑓 → (𝑎 finSupp 0 ↔ 𝑓 finSupp 0 )) | |
| 37 | 36 | rexrab 3701 | . . . 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 287 | . . 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 283 | . 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 279 | 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 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {cab 2713 ∃wrex 3069 {crab 3435 Vcvv 3479 class class class wbr 5142 ↦ cmpt 5224 ran crn 5685 “ cima 5687 Fn wfn 6555 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∘f cof 7696 ↑m cmap 8867 finSupp cfsupp 9402 Basecbs 17248 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17485 Σg cgsu 17486 LModclmod 20859 LSpanclspn 20970 freeLMod cfrlm 21767 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 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 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-fzo 13696 df-seq 14044 df-hash 14371 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-ghm 19232 df-cntz 19336 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-nzr 20514 df-subrg 20571 df-lmod 20861 df-lss 20931 df-lsp 20971 df-lmhm 21022 df-lbs 21075 df-sra 21173 df-rgmod 21174 df-dsmm 21753 df-frlm 21768 df-uvc 21804 | 
| This theorem is referenced by: elfilspd 21824 islindf4 21859 ellspds 33397 ply1degltdimlem 33674 fedgmul 33683 | 
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