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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualvs | Structured version Visualization version GIF version |
Description: Scalar product operation value (which is a functional) for the dual of a vector space. (Contributed by NM, 18-Oct-2014.) |
Ref | Expression |
---|---|
ldualfvs.f | ⊢ 𝐹 = (LFnl‘𝑊) |
ldualfvs.v | ⊢ 𝑉 = (Base‘𝑊) |
ldualfvs.r | ⊢ 𝑅 = (Scalar‘𝑊) |
ldualfvs.k | ⊢ 𝐾 = (Base‘𝑅) |
ldualfvs.t | ⊢ × = (.r‘𝑅) |
ldualfvs.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualfvs.s | ⊢ ∙ = ( ·𝑠 ‘𝐷) |
ldualfvs.w | ⊢ (𝜑 → 𝑊 ∈ 𝑌) |
ldualvs.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
ldualvs.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
ldualvs | ⊢ (𝜑 → (𝑋 ∙ 𝐺) = (𝐺 ∘f × (𝑉 × {𝑋}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldualfvs.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
2 | ldualfvs.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
3 | ldualfvs.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
4 | ldualfvs.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
5 | ldualfvs.t | . . . 4 ⊢ × = (.r‘𝑅) | |
6 | ldualfvs.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
7 | ldualfvs.s | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝐷) | |
8 | ldualfvs.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑌) | |
9 | eqid 2818 | . . . 4 ⊢ (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) = (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | ldualfvs 36152 | . . 3 ⊢ (𝜑 → ∙ = (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))) |
11 | 10 | oveqd 7162 | . 2 ⊢ (𝜑 → (𝑋 ∙ 𝐺) = (𝑋(𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))𝐺)) |
12 | ldualvs.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
13 | ldualvs.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
14 | sneq 4567 | . . . . . 6 ⊢ (𝑘 = 𝑋 → {𝑘} = {𝑋}) | |
15 | 14 | xpeq2d 5578 | . . . . 5 ⊢ (𝑘 = 𝑋 → (𝑉 × {𝑘}) = (𝑉 × {𝑋})) |
16 | 15 | oveq2d 7161 | . . . 4 ⊢ (𝑘 = 𝑋 → (𝑓 ∘f × (𝑉 × {𝑘})) = (𝑓 ∘f × (𝑉 × {𝑋}))) |
17 | oveq1 7152 | . . . 4 ⊢ (𝑓 = 𝐺 → (𝑓 ∘f × (𝑉 × {𝑋})) = (𝐺 ∘f × (𝑉 × {𝑋}))) | |
18 | ovex 7178 | . . . 4 ⊢ (𝐺 ∘f × (𝑉 × {𝑋})) ∈ V | |
19 | 16, 17, 9, 18 | ovmpo 7299 | . . 3 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝐺 ∈ 𝐹) → (𝑋(𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))𝐺) = (𝐺 ∘f × (𝑉 × {𝑋}))) |
20 | 12, 13, 19 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑋(𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))𝐺) = (𝐺 ∘f × (𝑉 × {𝑋}))) |
21 | 11, 20 | eqtrd 2853 | 1 ⊢ (𝜑 → (𝑋 ∙ 𝐺) = (𝐺 ∘f × (𝑉 × {𝑋}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {csn 4557 × cxp 5546 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ∘f cof 7396 Basecbs 16471 .rcmulr 16554 Scalarcsca 16556 ·𝑠 cvsca 16557 LFnlclfn 36073 LDualcld 36139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 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-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-sca 16569 df-vsca 16570 df-ldual 36140 |
This theorem is referenced by: ldualvsval 36154 ldualvscl 36155 ldualvsass 36157 ldualvsdi1 36159 ldualvsdi2 36160 lduallmodlem 36168 eqlkr4 36181 ldual1dim 36182 ldualkrsc 36183 lkrss 36184 |
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