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Mirrors > Home > MPE Home > Th. List > lspsneli | Structured version Visualization version GIF version |
Description: A scalar product with a vector belongs to the span of its singleton. (spansnmul 28940 analog.) (Contributed by NM, 2-Jul-2014.) |
Ref | Expression |
---|---|
lspsnvsel.v | ⊢ 𝑉 = (Base‘𝑊) |
lspsnvsel.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lspsnvsel.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lspsnvsel.k | ⊢ 𝐾 = (Base‘𝐹) |
lspsnvsel.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspsnvsel.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lspsnvsel.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
lspsnvsel.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
lspsneli | ⊢ (𝜑 → (𝐴 · 𝑋) ∈ (𝑁‘{𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsnvsel.w | . 2 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lspsnvsel.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | lspsnvsel.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
4 | eqid 2797 | . . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
5 | lspsnvsel.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
6 | 3, 4, 5 | lspsncl 19295 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
7 | 1, 2, 6 | syl2anc 580 | . 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
8 | lspsnvsel.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
9 | 3, 5 | lspsnid 19311 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
10 | 1, 2, 9 | syl2anc 580 | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋})) |
11 | lspsnvsel.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
12 | lspsnvsel.t | . . 3 ⊢ · = ( ·𝑠 ‘𝑊) | |
13 | lspsnvsel.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
14 | 11, 12, 13, 4 | lssvscl 19273 | . 2 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ (𝑁‘{𝑋}))) → (𝐴 · 𝑋) ∈ (𝑁‘{𝑋})) |
15 | 1, 7, 8, 10, 14 | syl22anc 868 | 1 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ (𝑁‘{𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 {csn 4366 ‘cfv 6099 (class class class)co 6876 Basecbs 16181 Scalarcsca 16267 ·𝑠 cvsca 16268 LModclmod 19178 LSubSpclss 19247 LSpanclspn 19289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-plusg 16277 df-0g 16414 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-grp 17738 df-minusg 17739 df-sbg 17740 df-mgp 18803 df-ur 18815 df-ring 18862 df-lmod 19180 df-lss 19248 df-lsp 19290 |
This theorem is referenced by: lspsnvsi 19322 lsmspsn 19402 lsppreli 19408 lspexch 19448 lvecindp 19457 lvecindp2 19458 lshpdisj 35000 lkrlsp 35115 lshpsmreu 35122 lshpkrlem5 35127 baerlem3lem2 37723 baerlem5alem2 37724 baerlem5blem2 37725 |
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