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Theorem dvaset 40482
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 40481 . . . 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 6902 . . 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 6900 . . . . . . . 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 2785 . . . . . . 7 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = 𝑇)
87opeq2d 4883 . . . . . 6 (𝑀 = π‘Š β†’ ⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩ = ⟨(Baseβ€˜ndx), π‘‡βŸ©)
9 eqidd 2728 . . . . . . . 8 (𝑀 = π‘Š β†’ (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
107, 7, 9mpoeq123dv 7499 . . . . . . 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 4883 . . . . . 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 6900 . . . . . . . 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 2785 . . . . . . 7 (𝑀 = π‘Š β†’ ((EDRingβ€˜πΎ)β€˜π‘€) = 𝐷)
1514opeq2d 4883 . . . . . 6 (𝑀 = π‘Š β†’ ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩ = ⟨(Scalarβ€˜ndx), 𝐷⟩)
168, 11, 15tpeq123d 4755 . . . . 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 6900 . . . . . . . . 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 2785 . . . . . . . 8 (𝑀 = π‘Š β†’ ((TEndoβ€˜πΎ)β€˜π‘€) = 𝐸)
20 eqidd 2728 . . . . . . . 8 (𝑀 = π‘Š β†’ (π‘ β€˜π‘“) = (π‘ β€˜π‘“))
2119, 7, 20mpoeq123dv 7499 . . . . . . 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 4883 . . . . . 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 4642 . . . . 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 4163 . . . 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 2727 . . . 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 7753 . . . . 5 {⟨(Baseβ€˜ndx), π‘‡βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩} ∈ V
27 snex 5435 . . . . 5 {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (π‘ β€˜π‘“))⟩} ∈ V
2826, 27unex 7752 . . . 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 7008 . . 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 2787 . 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 2779 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 394   = wceq 1533   ∈ wcel 2098   βˆͺ cun 3945  {csn 4630  {ctp 4634  βŸ¨cop 4636   ↦ cmpt 5233   ∘ ccom 5684  β€˜cfv 6551   ∈ cmpo 7426  ndxcnx 17167  Basecbs 17185  +gcplusg 17238  Scalarcsca 17241   ·𝑠 cvsca 17242  LHypclh 39461  LTrncltrn 39578  TEndoctendo 40229  EDRingcedring 40230  DVecAcdveca 40479
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 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-oprab 7428  df-mpo 7429  df-dveca 40480
This theorem is referenced by:  dvasca  40483  dvavbase  40490  dvafvadd  40491  dvafvsca  40493  dvaabl  40501
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