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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualvbase | Structured version Visualization version GIF version |
Description: The vectors of a dual space are functionals of the original space. (Contributed by NM, 18-Oct-2014.) |
Ref | Expression |
---|---|
ldualvbase.f | ⊢ 𝐹 = (LFnl‘𝑊) |
ldualvbase.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualvbase.v | ⊢ 𝑉 = (Base‘𝐷) |
ldualvbase.w | ⊢ (𝜑 → 𝑊 ∈ 𝑋) |
Ref | Expression |
---|---|
ldualvbase | ⊢ (𝜑 → 𝑉 = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2733 | . . . 4 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊)) | |
3 | eqid 2733 | . . . 4 ⊢ ( ∘f (+g‘(Scalar‘𝑊)) ↾ (𝐹 × 𝐹)) = ( ∘f (+g‘(Scalar‘𝑊)) ↾ (𝐹 × 𝐹)) | |
4 | ldualvbase.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
5 | ldualvbase.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
6 | eqid 2733 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
7 | eqid 2733 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
8 | eqid 2733 | . . . 4 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊)) | |
9 | eqid 2733 | . . . 4 ⊢ (oppr‘(Scalar‘𝑊)) = (oppr‘(Scalar‘𝑊)) | |
10 | eqid 2733 | . . . 4 ⊢ (𝑘 ∈ (Base‘(Scalar‘𝑊)), 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘}))) = (𝑘 ∈ (Base‘(Scalar‘𝑊)), 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘}))) | |
11 | ldualvbase.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | ldualset 37637 | . . 3 ⊢ (𝜑 → 𝐷 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑊)) ↾ (𝐹 × 𝐹))⟩, ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑊))⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑊)), 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘})))⟩})) |
13 | 12 | fveq2d 6850 | . 2 ⊢ (𝜑 → (Base‘𝐷) = (Base‘({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑊)) ↾ (𝐹 × 𝐹))⟩, ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑊))⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑊)), 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘})))⟩}))) |
14 | ldualvbase.v | . 2 ⊢ 𝑉 = (Base‘𝐷) | |
15 | 4 | fvexi 6860 | . . 3 ⊢ 𝐹 ∈ V |
16 | eqid 2733 | . . . 4 ⊢ ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑊)) ↾ (𝐹 × 𝐹))⟩, ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑊))⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑊)), 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘})))⟩}) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑊)) ↾ (𝐹 × 𝐹))⟩, ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑊))⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑊)), 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘})))⟩}) | |
17 | 16 | lmodbase 17215 | . . 3 ⊢ (𝐹 ∈ V → 𝐹 = (Base‘({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑊)) ↾ (𝐹 × 𝐹))⟩, ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑊))⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑊)), 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘})))⟩}))) |
18 | 15, 17 | ax-mp 5 | . 2 ⊢ 𝐹 = (Base‘({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑊)) ↾ (𝐹 × 𝐹))⟩, ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑊))⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑊)), 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘})))⟩})) |
19 | 13, 14, 18 | 3eqtr4g 2798 | 1 ⊢ (𝜑 → 𝑉 = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ∪ cun 3912 {csn 4590 {ctp 4594 ⟨cop 4596 × cxp 5635 ↾ cres 5639 ‘cfv 6500 (class class class)co 7361 ∈ cmpo 7363 ∘f cof 7619 ndxcnx 17073 Basecbs 17091 +gcplusg 17141 .rcmulr 17142 Scalarcsca 17144 ·𝑠 cvsca 17145 opprcoppr 20056 LFnlclfn 37569 LDualcld 37635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-struct 17027 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-sca 17157 df-vsca 17158 df-ldual 37636 |
This theorem is referenced by: ldualelvbase 37639 ldualgrplem 37657 lduallmodlem 37664 lclkr 40046 lclkrs 40052 lcfrvalsnN 40054 lcfrlem4 40058 lcfrlem5 40059 lcfrlem6 40060 lcfrlem16 40071 lcfr 40098 lcdvbase 40106 mapdunirnN 40163 |
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