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 2820 | . . . . 5 ⊢ (LBasis‘𝑀) = (LBasis‘𝑀) | |
4 | eqid 2820 | . . . . 5 ⊢ (LSpan‘𝑀) = (LSpan‘𝑀) | |
5 | 2, 3, 4 | lbssp 19844 | . . . 4 ⊢ (𝑉 ∈ (LBasis‘𝑀) → ((LSpan‘𝑀)‘𝑉) = 𝐵) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝜑 → ((LSpan‘𝑀)‘𝑉) = 𝐵) |
7 | 6 | eleq2d 2897 | . 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 19842 | . . . 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 30952 | . 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 283 | 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 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3138 ⊆ wss 3929 class class class wbr 5059 ↦ cmpt 5139 ‘cfv 6348 (class class class)co 7149 ↑m cmap 8399 finSupp cfsupp 8826 Basecbs 16476 Scalarcsca 16561 ·𝑠 cvsca 16562 0gc0g 16706 Σg cgsu 16707 LModclmod 19627 LSpanclspn 19736 LBasisclbs 19839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-sup 8899 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12890 df-fzo 13031 df-seq 13367 df-hash 13688 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-mulr 16572 df-sca 16574 df-vsca 16575 df-ip 16576 df-tset 16577 df-ple 16578 df-ds 16580 df-hom 16582 df-cco 16583 df-0g 16708 df-gsum 16709 df-prds 16714 df-pws 16716 df-mre 16850 df-mrc 16851 df-acs 16853 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-mhm 17949 df-submnd 17950 df-grp 18099 df-minusg 18100 df-sbg 18101 df-mulg 18218 df-subg 18269 df-ghm 18349 df-cntz 18440 df-cmn 18901 df-abl 18902 df-mgp 19233 df-ur 19245 df-ring 19292 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lmhm 19787 df-lbs 19840 df-sra 19937 df-rgmod 19938 df-nzr 20024 df-dsmm 20869 df-frlm 20884 df-uvc 20920 |
This theorem is referenced by: extdg1id 31075 |
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