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Mirrors > Home > MPE Home > Th. List > elfilspd | Structured version Visualization version GIF version |
Description: Simplified version of ellspd 20363 when the spanning set is finite: all linear combinations are then acceptable. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
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) |
elfilspd.f | ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) |
elfilspd.m | ⊢ (𝜑 → 𝑀 ∈ LMod) |
elfilspd.i | ⊢ (𝜑 → 𝐼 ∈ Fin) |
Ref | Expression |
---|---|
elfilspd | ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑𝑚 𝐼)𝑋 = (𝑀 Σg (𝑓 ∘𝑓 · 𝐹)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ellspd.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑀) | |
2 | ellspd.v | . . 3 ⊢ 𝐵 = (Base‘𝑀) | |
3 | ellspd.k | . . 3 ⊢ 𝐾 = (Base‘𝑆) | |
4 | ellspd.s | . . 3 ⊢ 𝑆 = (Scalar‘𝑀) | |
5 | ellspd.z | . . 3 ⊢ 0 = (0g‘𝑆) | |
6 | ellspd.t | . . 3 ⊢ · = ( ·𝑠 ‘𝑀) | |
7 | elfilspd.f | . . 3 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) | |
8 | elfilspd.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ LMod) | |
9 | elfilspd.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ Fin) | |
10 | elex 3352 | . . . 4 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ V) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 11 | ellspd 20363 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑𝑚 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘𝑓 · 𝐹))))) |
13 | elmapi 8047 | . . . . . 6 ⊢ (𝑓 ∈ (𝐾 ↑𝑚 𝐼) → 𝑓:𝐼⟶𝐾) | |
14 | 13 | adantl 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐾 ↑𝑚 𝐼)) → 𝑓:𝐼⟶𝐾) |
15 | 9 | adantr 472 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐾 ↑𝑚 𝐼)) → 𝐼 ∈ Fin) |
16 | fvex 6363 | . . . . . . 7 ⊢ (0g‘𝑆) ∈ V | |
17 | 5, 16 | eqeltri 2835 | . . . . . 6 ⊢ 0 ∈ V |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐾 ↑𝑚 𝐼)) → 0 ∈ V) |
19 | 14, 15, 18 | fdmfifsupp 8452 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐾 ↑𝑚 𝐼)) → 𝑓 finSupp 0 ) |
20 | 19 | biantrurd 530 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐾 ↑𝑚 𝐼)) → (𝑋 = (𝑀 Σg (𝑓 ∘𝑓 · 𝐹)) ↔ (𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘𝑓 · 𝐹))))) |
21 | 20 | rexbidva 3187 | . 2 ⊢ (𝜑 → (∃𝑓 ∈ (𝐾 ↑𝑚 𝐼)𝑋 = (𝑀 Σg (𝑓 ∘𝑓 · 𝐹)) ↔ ∃𝑓 ∈ (𝐾 ↑𝑚 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘𝑓 · 𝐹))))) |
22 | 12, 21 | bitr4d 271 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑𝑚 𝐼)𝑋 = (𝑀 Σg (𝑓 ∘𝑓 · 𝐹)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∃wrex 3051 Vcvv 3340 class class class wbr 4804 “ cima 5269 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 ∘𝑓 cof 7061 ↑𝑚 cmap 8025 Fincfn 8123 finSupp cfsupp 8442 Basecbs 16079 Scalarcsca 16166 ·𝑠 cvsca 16167 0gc0g 16322 Σg cgsu 16323 LModclmod 19085 LSpanclspn 19193 |
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 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
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 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-om 7232 df-1st 7334 df-2nd 7335 df-supp 7465 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-map 8027 df-ixp 8077 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-fsupp 8443 df-sup 8515 df-oi 8582 df-card 8975 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-fz 12540 df-fzo 12680 df-seq 13016 df-hash 13332 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-sca 16179 df-vsca 16180 df-ip 16181 df-tset 16182 df-ple 16183 df-ds 16186 df-hom 16188 df-cco 16189 df-0g 16324 df-gsum 16325 df-prds 16330 df-pws 16332 df-mre 16468 df-mrc 16469 df-acs 16471 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-mhm 17556 df-submnd 17557 df-grp 17646 df-minusg 17647 df-sbg 17648 df-mulg 17762 df-subg 17812 df-ghm 17879 df-cntz 17970 df-cmn 18415 df-abl 18416 df-mgp 18710 df-ur 18722 df-ring 18769 df-subrg 19000 df-lmod 19087 df-lss 19155 df-lsp 19194 df-lmhm 19244 df-lbs 19297 df-sra 19394 df-rgmod 19395 df-nzr 19480 df-dsmm 20298 df-frlm 20313 df-uvc 20344 |
This theorem is referenced by: matunitlindflem2 33737 |
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