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Mathbox for Norm Megill |
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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 2736 | . . . 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 37588 | . . 3 ⊢ (𝜑 → ∙ = (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))) |
11 | 10 | oveqd 7373 | . 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 4596 | . . . . . 6 ⊢ (𝑘 = 𝑋 → {𝑘} = {𝑋}) | |
15 | 14 | xpeq2d 5663 | . . . . 5 ⊢ (𝑘 = 𝑋 → (𝑉 × {𝑘}) = (𝑉 × {𝑋})) |
16 | 15 | oveq2d 7372 | . . . 4 ⊢ (𝑘 = 𝑋 → (𝑓 ∘f × (𝑉 × {𝑘})) = (𝑓 ∘f × (𝑉 × {𝑋}))) |
17 | oveq1 7363 | . . . 4 ⊢ (𝑓 = 𝐺 → (𝑓 ∘f × (𝑉 × {𝑋})) = (𝐺 ∘f × (𝑉 × {𝑋}))) | |
18 | ovex 7389 | . . . 4 ⊢ (𝐺 ∘f × (𝑉 × {𝑋})) ∈ V | |
19 | 16, 17, 9, 18 | ovmpo 7514 | . . 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 2776 | 1 ⊢ (𝜑 → (𝑋 ∙ 𝐺) = (𝐺 ∘f × (𝑉 × {𝑋}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {csn 4586 × cxp 5631 ‘cfv 6496 (class class class)co 7356 ∈ cmpo 7358 ∘f cof 7614 Basecbs 17082 .rcmulr 17133 Scalarcsca 17135 ·𝑠 cvsca 17136 LFnlclfn 37509 LDualcld 37575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-n0 12413 df-z 12499 df-uz 12763 df-fz 13424 df-struct 17018 df-slot 17053 df-ndx 17065 df-base 17083 df-plusg 17145 df-sca 17148 df-vsca 17149 df-ldual 37576 |
This theorem is referenced by: ldualvsval 37590 ldualvscl 37591 ldualvsass 37593 ldualvsdi1 37595 ldualvsdi2 37596 lduallmodlem 37604 eqlkr4 37617 ldual1dim 37618 ldualkrsc 37619 lkrss 37620 |
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