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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 2728 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2728 | . . . 4 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊)) | |
3 | eqid 2728 | . . . 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 2728 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
7 | eqid 2728 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
8 | eqid 2728 | . . . 4 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊)) | |
9 | eqid 2728 | . . . 4 ⊢ (oppr‘(Scalar‘𝑊)) = (oppr‘(Scalar‘𝑊)) | |
10 | eqid 2728 | . . . 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 38629 | . . 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 6906 | . 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 6916 | . . 3 ⊢ 𝐹 ∈ V |
16 | eqid 2728 | . . . 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 17314 | . . 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 2793 | 1 ⊢ (𝜑 → 𝑉 = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ∪ cun 3947 {csn 4632 {ctp 4636 ⟨cop 4638 × cxp 5680 ↾ cres 5684 ‘cfv 6553 (class class class)co 7426 ∈ cmpo 7428 ∘f cof 7689 ndxcnx 17169 Basecbs 17187 +gcplusg 17240 .rcmulr 17241 Scalarcsca 17243 ·𝑠 cvsca 17244 opprcoppr 20279 LFnlclfn 38561 LDualcld 38627 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-sca 17256 df-vsca 17257 df-ldual 38628 |
This theorem is referenced by: ldualelvbase 38631 ldualgrplem 38649 lduallmodlem 38656 lclkr 41038 lclkrs 41044 lcfrvalsnN 41046 lcfrlem4 41050 lcfrlem5 41051 lcfrlem6 41052 lcfrlem16 41063 lcfr 41090 lcdvbase 41098 mapdunirnN 41155 |
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