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Theorem dvaset 40387
Description: The constructed partial vector space A for a lattice 𝐾. (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Hypotheses
Ref Expression
dvaset.h 𝐻 = (LHypβ€˜πΎ)
dvaset.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dvaset.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
dvaset.d 𝐷 = ((EDRingβ€˜πΎ)β€˜π‘Š)
dvaset.u π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dvaset ((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ = ({⟨(Baseβ€˜ndx), π‘‡βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (π‘ β€˜π‘“))⟩}))
Distinct variable groups:   𝑓,𝑔,𝑠,𝐾   𝑓,π‘Š,𝑔,𝑠
Allowed substitution hints:   𝐷(𝑓,𝑔,𝑠)   𝑇(𝑓,𝑔,𝑠)   π‘ˆ(𝑓,𝑔,𝑠)   𝐸(𝑓,𝑔,𝑠)   𝐻(𝑓,𝑔,𝑠)   𝑋(𝑓,𝑔,𝑠)

Proof of Theorem dvaset
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 dvaset.u . 2 π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)
2 dvaset.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
32dvafset 40386 . . . 4 (𝐾 ∈ 𝑋 β†’ (DVecAβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ ({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (π‘ β€˜π‘“))⟩})))
43fveq1d 6886 . . 3 (𝐾 ∈ 𝑋 β†’ ((DVecAβ€˜πΎ)β€˜π‘Š) = ((𝑀 ∈ 𝐻 ↦ ({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (π‘ β€˜π‘“))⟩}))β€˜π‘Š))
5 fveq2 6884 . . . . . . . 8 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘Š))
6 dvaset.t . . . . . . . 8 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
75, 6eqtr4di 2784 . . . . . . 7 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = 𝑇)
87opeq2d 4875 . . . . . 6 (𝑀 = π‘Š β†’ ⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩ = ⟨(Baseβ€˜ndx), π‘‡βŸ©)
9 eqidd 2727 . . . . . . . 8 (𝑀 = π‘Š β†’ (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
107, 7, 9mpoeq123dv 7479 . . . . . . 7 (𝑀 = π‘Š β†’ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔)))
1110opeq2d 4875 . . . . . 6 (𝑀 = π‘Š β†’ ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩ = ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩)
12 fveq2 6884 . . . . . . . 8 (𝑀 = π‘Š β†’ ((EDRingβ€˜πΎ)β€˜π‘€) = ((EDRingβ€˜πΎ)β€˜π‘Š))
13 dvaset.d . . . . . . . 8 𝐷 = ((EDRingβ€˜πΎ)β€˜π‘Š)
1412, 13eqtr4di 2784 . . . . . . 7 (𝑀 = π‘Š β†’ ((EDRingβ€˜πΎ)β€˜π‘€) = 𝐷)
1514opeq2d 4875 . . . . . 6 (𝑀 = π‘Š β†’ ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩ = ⟨(Scalarβ€˜ndx), 𝐷⟩)
168, 11, 15tpeq123d 4747 . . . . 5 (𝑀 = π‘Š β†’ {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩} = {⟨(Baseβ€˜ndx), π‘‡βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩})
17 fveq2 6884 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((TEndoβ€˜πΎ)β€˜π‘€) = ((TEndoβ€˜πΎ)β€˜π‘Š))
18 dvaset.e . . . . . . . . 9 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
1917, 18eqtr4di 2784 . . . . . . . 8 (𝑀 = π‘Š β†’ ((TEndoβ€˜πΎ)β€˜π‘€) = 𝐸)
20 eqidd 2727 . . . . . . . 8 (𝑀 = π‘Š β†’ (π‘ β€˜π‘“) = (π‘ β€˜π‘“))
2119, 7, 20mpoeq123dv 7479 . . . . . . 7 (𝑀 = π‘Š β†’ (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (π‘ β€˜π‘“)) = (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (π‘ β€˜π‘“)))
2221opeq2d 4875 . . . . . 6 (𝑀 = π‘Š β†’ ⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (π‘ β€˜π‘“))⟩ = ⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (π‘ β€˜π‘“))⟩)
2322sneqd 4635 . . . . 5 (𝑀 = π‘Š β†’ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (π‘ β€˜π‘“))⟩} = {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (π‘ β€˜π‘“))⟩})
2416, 23uneq12d 4159 . . . 4 (𝑀 = π‘Š β†’ ({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (π‘ β€˜π‘“))⟩}) = ({⟨(Baseβ€˜ndx), π‘‡βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (π‘ β€˜π‘“))⟩}))
25 eqid 2726 . . . 4 (𝑀 ∈ 𝐻 ↦ ({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (π‘ β€˜π‘“))⟩})) = (𝑀 ∈ 𝐻 ↦ ({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (π‘ β€˜π‘“))⟩}))
26 tpex 7730 . . . . 5 {⟨(Baseβ€˜ndx), π‘‡βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩} ∈ V
27 snex 5424 . . . . 5 {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (π‘ β€˜π‘“))⟩} ∈ V
2826, 27unex 7729 . . . 4 ({⟨(Baseβ€˜ndx), π‘‡βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (π‘ β€˜π‘“))⟩}) ∈ V
2924, 25, 28fvmpt 6991 . . 3 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ ({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (π‘ β€˜π‘“))⟩}))β€˜π‘Š) = ({⟨(Baseβ€˜ndx), π‘‡βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (π‘ β€˜π‘“))⟩}))
304, 29sylan9eq 2786 . 2 ((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) β†’ ((DVecAβ€˜πΎ)β€˜π‘Š) = ({⟨(Baseβ€˜ndx), π‘‡βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (π‘ β€˜π‘“))⟩}))
311, 30eqtrid 2778 1 ((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ = ({⟨(Baseβ€˜ndx), π‘‡βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (π‘ β€˜π‘“))⟩}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   βˆͺ cun 3941  {csn 4623  {ctp 4627  βŸ¨cop 4629   ↦ cmpt 5224   ∘ ccom 5673  β€˜cfv 6536   ∈ cmpo 7406  ndxcnx 17133  Basecbs 17151  +gcplusg 17204  Scalarcsca 17207   ·𝑠 cvsca 17208  LHypclh 39366  LTrncltrn 39483  TEndoctendo 40134  EDRingcedring 40135  DVecAcdveca 40384
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-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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-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-nul 4318  df-if 4524  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-id 5567  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-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-oprab 7408  df-mpo 7409  df-dveca 40385
This theorem is referenced by:  dvasca  40388  dvavbase  40395  dvafvadd  40396  dvafvsca  40398  dvaabl  40406
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