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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lbslsp | Structured version Visualization version GIF version |
Description: Any element of a left module 𝑀 can be expressed as a linear combination of the elements of a basis 𝑉 of 𝑀. (Contributed by Thierry Arnoux, 3-Aug-2023.) |
Ref | Expression |
---|---|
lbslsp.v | ⊢ 𝐵 = (Base‘𝑀) |
lbslsp.k | ⊢ 𝐾 = (Base‘𝑆) |
lbslsp.s | ⊢ 𝑆 = (Scalar‘𝑀) |
lbslsp.z | ⊢ 0 = (0g‘𝑆) |
lbslsp.t | ⊢ · = ( ·𝑠 ‘𝑀) |
lbslsp.m | ⊢ (𝜑 → 𝑀 ∈ LMod) |
lbslsp.1 | ⊢ (𝜑 → 𝑉 ∈ (LBasis‘𝑀)) |
Ref | Expression |
---|---|
lbslsp | ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ ∃𝑎 ∈ (𝐾 ↑m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lbslsp.1 | . . . 4 ⊢ (𝜑 → 𝑉 ∈ (LBasis‘𝑀)) | |
2 | lbslsp.v | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
3 | eqid 2728 | . . . . 5 ⊢ (LBasis‘𝑀) = (LBasis‘𝑀) | |
4 | eqid 2728 | . . . . 5 ⊢ (LSpan‘𝑀) = (LSpan‘𝑀) | |
5 | 2, 3, 4 | lbssp 20963 | . . . 4 ⊢ (𝑉 ∈ (LBasis‘𝑀) → ((LSpan‘𝑀)‘𝑉) = 𝐵) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝜑 → ((LSpan‘𝑀)‘𝑉) = 𝐵) |
7 | 6 | eleq2d 2815 | . 2 ⊢ (𝜑 → (𝑋 ∈ ((LSpan‘𝑀)‘𝑉) ↔ 𝑋 ∈ 𝐵)) |
8 | lbslsp.k | . . 3 ⊢ 𝐾 = (Base‘𝑆) | |
9 | lbslsp.s | . . 3 ⊢ 𝑆 = (Scalar‘𝑀) | |
10 | lbslsp.z | . . 3 ⊢ 0 = (0g‘𝑆) | |
11 | lbslsp.t | . . 3 ⊢ · = ( ·𝑠 ‘𝑀) | |
12 | lbslsp.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ LMod) | |
13 | 2, 3 | lbsss 20961 | . . . 4 ⊢ (𝑉 ∈ (LBasis‘𝑀) → 𝑉 ⊆ 𝐵) |
14 | 1, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝑉 ⊆ 𝐵) |
15 | 4, 2, 8, 9, 10, 11, 12, 14 | ellspds 33080 | . 2 ⊢ (𝜑 → (𝑋 ∈ ((LSpan‘𝑀)‘𝑉) ↔ ∃𝑎 ∈ (𝐾 ↑m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))))) |
16 | 7, 15 | bitr3d 281 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ ∃𝑎 ∈ (𝐾 ↑m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3067 ⊆ wss 3947 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6548 (class class class)co 7420 ↑m cmap 8844 finSupp cfsupp 9385 Basecbs 17179 Scalarcsca 17235 ·𝑠 cvsca 17236 0gc0g 17420 Σg cgsu 17421 LModclmod 20742 LSpanclspn 20854 LBasisclbs 20958 |
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 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18892 df-minusg 18893 df-sbg 18894 df-mulg 19023 df-subg 19077 df-ghm 19167 df-cntz 19267 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-nzr 20451 df-subrg 20507 df-lmod 20744 df-lss 20815 df-lsp 20855 df-lmhm 20906 df-lbs 20959 df-sra 21057 df-rgmod 21058 df-dsmm 21665 df-frlm 21680 df-uvc 21716 |
This theorem is referenced by: extdg1id 33351 |
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