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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvavbase | Structured version Visualization version GIF version |
Description: The vectors (vector base set) of the constructed partial vector space A are all translations (for a fiducial co-atom 𝑊). (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.) |
Ref | Expression |
---|---|
dvavbase.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dvavbase.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dvavbase.u | ⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊) |
dvavbase.v | ⊢ 𝑉 = (Base‘𝑈) |
Ref | Expression |
---|---|
dvavbase | ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑉 = 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvavbase.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dvavbase.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | eqid 2825 | . . . 4 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊) | |
4 | eqid 2825 | . . . 4 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊) | |
5 | dvavbase.u | . . . 4 ⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | dvaset 37075 | . . 3 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑈 = ({〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉, 〈(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))〉})) |
7 | 6 | fveq2d 6441 | . 2 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (Base‘({〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉, 〈(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))〉}))) |
8 | dvavbase.v | . 2 ⊢ 𝑉 = (Base‘𝑈) | |
9 | 2 | fvexi 6451 | . . 3 ⊢ 𝑇 ∈ V |
10 | eqid 2825 | . . . 4 ⊢ ({〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉, 〈(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))〉}) = ({〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉, 〈(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))〉}) | |
11 | 10 | lmodbase 16384 | . . 3 ⊢ (𝑇 ∈ V → 𝑇 = (Base‘({〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉, 〈(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))〉}))) |
12 | 9, 11 | ax-mp 5 | . 2 ⊢ 𝑇 = (Base‘({〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉, 〈(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))〉})) |
13 | 7, 8, 12 | 3eqtr4g 2886 | 1 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑉 = 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∪ cun 3796 {csn 4399 {ctp 4403 〈cop 4405 ∘ ccom 5350 ‘cfv 6127 ↦ cmpt2 6912 ndxcnx 16226 Basecbs 16229 +gcplusg 16312 Scalarcsca 16315 ·𝑠 cvsca 16316 LHypclh 36054 LTrncltrn 36171 TEndoctendo 36822 EDRingcedring 36823 DVecAcdveca 37072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-plusg 16325 df-sca 16328 df-vsca 16329 df-dveca 37073 |
This theorem is referenced by: dvalveclem 37095 dva0g 37097 dialss 37116 diassdvaN 37130 dia1dim2 37132 dia1dimid 37133 dia2dimlem5 37138 dvadiaN 37198 |
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