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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualfvs | Structured version Visualization version GIF version |
Description: Scalar product operation 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) |
ldualfvs.m | ⊢ · = (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) |
Ref | Expression |
---|---|
ldualfvs | ⊢ (𝜑 → ∙ = · ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldualfvs.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | eqid 2798 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
3 | eqid 2798 | . . . 4 ⊢ ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹)) = ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹)) | |
4 | ldualfvs.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
5 | ldualfvs.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
6 | ldualfvs.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
7 | ldualfvs.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
8 | ldualfvs.t | . . . 4 ⊢ × = (.r‘𝑅) | |
9 | eqid 2798 | . . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
10 | eqid 2798 | . . . 4 ⊢ (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) = (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) | |
11 | ldualfvs.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑌) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | ldualset 36421 | . . 3 ⊢ (𝜑 → 𝐷 = ({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹))〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))〉})) |
13 | 12 | fveq2d 6649 | . 2 ⊢ (𝜑 → ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹))〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))〉}))) |
14 | ldualfvs.s | . 2 ⊢ ∙ = ( ·𝑠 ‘𝐷) | |
15 | ldualfvs.m | . . 3 ⊢ · = (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) | |
16 | 7 | fvexi 6659 | . . . . 5 ⊢ 𝐾 ∈ V |
17 | 4 | fvexi 6659 | . . . . 5 ⊢ 𝐹 ∈ V |
18 | 16, 17 | mpoex 7760 | . . . 4 ⊢ (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) ∈ V |
19 | eqid 2798 | . . . . 5 ⊢ ({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹))〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))〉}) = ({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹))〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))〉}) | |
20 | 19 | lmodvsca 16632 | . . . 4 ⊢ ((𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) ∈ V → (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) = ( ·𝑠 ‘({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹))〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))〉}))) |
21 | 18, 20 | ax-mp 5 | . . 3 ⊢ (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) = ( ·𝑠 ‘({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹))〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))〉})) |
22 | 15, 21 | eqtri 2821 | . 2 ⊢ · = ( ·𝑠 ‘({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹))〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))〉})) |
23 | 13, 14, 22 | 3eqtr4g 2858 | 1 ⊢ (𝜑 → ∙ = · ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 {csn 4525 {ctp 4529 〈cop 4531 × cxp 5517 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ∘f cof 7387 ndxcnx 16472 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 Scalarcsca 16560 ·𝑠 cvsca 16561 opprcoppr 19368 LFnlclfn 36353 LDualcld 36419 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 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 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-sca 16573 df-vsca 16574 df-ldual 36420 |
This theorem is referenced by: ldualvs 36433 |
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