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Mirrors > Home > MPE Home > Th. List > elfilspd | Structured version Visualization version GIF version |
Description: Simplified version of ellspd 20141 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 3212 | . . . 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 20141 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑𝑚 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘𝑓 · 𝐹))))) |
13 | elmapi 7879 | . . . . . 6 ⊢ (𝑓 ∈ (𝐾 ↑𝑚 𝐼) → 𝑓:𝐼⟶𝐾) | |
14 | 13 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐾 ↑𝑚 𝐼)) → 𝑓:𝐼⟶𝐾) |
15 | 9 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐾 ↑𝑚 𝐼)) → 𝐼 ∈ Fin) |
16 | fvex 6201 | . . . . . . 7 ⊢ (0g‘𝑆) ∈ V | |
17 | 5, 16 | eqeltri 2697 | . . . . . 6 ⊢ 0 ∈ V |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐾 ↑𝑚 𝐼)) → 0 ∈ V) |
19 | 14, 15, 18 | fdmfifsupp 8285 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐾 ↑𝑚 𝐼)) → 𝑓 finSupp 0 ) |
20 | 19 | biantrurd 529 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐾 ↑𝑚 𝐼)) → (𝑋 = (𝑀 Σg (𝑓 ∘𝑓 · 𝐹)) ↔ (𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘𝑓 · 𝐹))))) |
21 | 20 | rexbidva 3049 | . 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 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 Vcvv 3200 class class class wbr 4653 “ cima 5117 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 ↑𝑚 cmap 7857 Fincfn 7955 finSupp cfsupp 8275 Basecbs 15857 Scalarcsca 15944 ·𝑠 cvsca 15945 0gc0g 16100 Σg cgsu 16101 LModclmod 18863 LSpanclspn 18971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lmhm 19022 df-lbs 19075 df-sra 19172 df-rgmod 19173 df-nzr 19258 df-dsmm 20076 df-frlm 20091 df-uvc 20122 |
This theorem is referenced by: matunitlindflem2 33406 |
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