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Mirrors > Home > ILE Home > Th. List > lspsneli | GIF version |
Description: A scalar product with a vector belongs to the span of its singleton. (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 2189 | . . . 4 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
5 | lspsnvsel.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
6 | 3, 4, 5 | lspsncl 13705 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
7 | 1, 2, 6 | syl2anc 411 | . 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
8 | lspsnvsel.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
9 | 3, 5 | lspsnid 13720 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
10 | 1, 2, 9 | syl2anc 411 | . 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 13688 | . 2 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ (𝑁‘{𝑋}))) → (𝐴 · 𝑋) ∈ (𝑁‘{𝑋})) |
15 | 1, 7, 8, 10, 14 | syl22anc 1250 | 1 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ (𝑁‘{𝑋})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 {csn 3607 ‘cfv 5235 (class class class)co 5895 Basecbs 12511 Scalarcsca 12589 ·𝑠 cvsca 12590 LModclmod 13600 LSubSpclss 13665 LSpanclspn 13699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-addcom 7940 ax-addass 7942 ax-i2m1 7945 ax-0lt1 7946 ax-0id 7948 ax-rnegex 7949 ax-pre-ltirr 7952 ax-pre-ltadd 7956 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-pnf 8023 df-mnf 8024 df-ltxr 8026 df-inn 8949 df-2 9007 df-3 9008 df-4 9009 df-5 9010 df-6 9011 df-ndx 12514 df-slot 12515 df-base 12517 df-sets 12518 df-plusg 12599 df-mulr 12600 df-sca 12602 df-vsca 12603 df-0g 12760 df-mgm 12829 df-sgrp 12862 df-mnd 12875 df-grp 12945 df-minusg 12946 df-sbg 12947 df-mgp 13272 df-ur 13311 df-ring 13349 df-lmod 13602 df-lssm 13666 df-lsp 13700 |
This theorem is referenced by: lspsnvsi 13731 |
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