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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 2726 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2726 | . . . 4 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊)) | |
3 | eqid 2726 | . . . 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 2726 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
7 | eqid 2726 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
8 | eqid 2726 | . . . 4 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊)) | |
9 | eqid 2726 | . . . 4 ⊢ (oppr‘(Scalar‘𝑊)) = (oppr‘(Scalar‘𝑊)) | |
10 | eqid 2726 | . . . 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 38507 | . . 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 6888 | . 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 6898 | . . 3 ⊢ 𝐹 ∈ V |
16 | eqid 2726 | . . . 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 17277 | . . 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 2791 | 1 ⊢ (𝜑 → 𝑉 = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ∪ cun 3941 {csn 4623 {ctp 4627 ⟨cop 4629 × cxp 5667 ↾ cres 5671 ‘cfv 6536 (class class class)co 7404 ∈ cmpo 7406 ∘f cof 7664 ndxcnx 17132 Basecbs 17150 +gcplusg 17203 .rcmulr 17204 Scalarcsca 17206 ·𝑠 cvsca 17207 opprcoppr 20232 LFnlclfn 38439 LDualcld 38505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-sca 17219 df-vsca 17220 df-ldual 38506 |
This theorem is referenced by: ldualelvbase 38509 ldualgrplem 38527 lduallmodlem 38534 lclkr 40916 lclkrs 40922 lcfrvalsnN 40924 lcfrlem4 40928 lcfrlem5 40929 lcfrlem6 40930 lcfrlem16 40941 lcfr 40968 lcdvbase 40976 mapdunirnN 41033 |
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