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Theorem dvhset 39482
Description: The constructed full vector space H for a lattice 𝐾. (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Hypotheses
Ref Expression
dvhset.h 𝐻 = (LHypβ€˜πΎ)
dvhset.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dvhset.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
dvhset.d 𝐷 = ((EDRingβ€˜πΎ)β€˜π‘Š)
dvhset.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dvhset ((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ = ({⟨(Baseβ€˜ndx), (𝑇 Γ— 𝐸)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩}))
Distinct variable groups:   𝑓,𝑔,𝐻   𝑓,β„Ž,𝑠,𝐾,𝑔   𝑇,β„Ž   𝑓,π‘Š,𝑔,β„Ž,𝑠   𝑓,𝑋,𝑔
Allowed substitution hints:   𝐷(𝑓,𝑔,β„Ž,𝑠)   𝑇(𝑓,𝑔,𝑠)   π‘ˆ(𝑓,𝑔,β„Ž,𝑠)   𝐸(𝑓,𝑔,β„Ž,𝑠)   𝐻(β„Ž,𝑠)   𝑋(β„Ž,𝑠)

Proof of Theorem dvhset
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 dvhset.u . . 3 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
2 dvhset.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
32dvhfset 39481 . . . 4 (𝐾 ∈ 𝑋 β†’ (DVecHβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ ({⟨(Baseβ€˜ndx), (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€))⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)), 𝑔 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩})))
43fveq1d 6841 . . 3 (𝐾 ∈ 𝑋 β†’ ((DVecHβ€˜πΎ)β€˜π‘Š) = ((𝑀 ∈ 𝐻 ↦ ({⟨(Baseβ€˜ndx), (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€))⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)), 𝑔 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩}))β€˜π‘Š))
51, 4eqtrid 2789 . 2 (𝐾 ∈ 𝑋 β†’ π‘ˆ = ((𝑀 ∈ 𝐻 ↦ ({⟨(Baseβ€˜ndx), (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€))⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)), 𝑔 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩}))β€˜π‘Š))
6 fveq2 6839 . . . . . . . 8 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘Š))
7 dvhset.t . . . . . . . 8 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
86, 7eqtr4di 2795 . . . . . . 7 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = 𝑇)
9 fveq2 6839 . . . . . . . 8 (𝑀 = π‘Š β†’ ((TEndoβ€˜πΎ)β€˜π‘€) = ((TEndoβ€˜πΎ)β€˜π‘Š))
10 dvhset.e . . . . . . . 8 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
119, 10eqtr4di 2795 . . . . . . 7 (𝑀 = π‘Š β†’ ((TEndoβ€˜πΎ)β€˜π‘€) = 𝐸)
128, 11xpeq12d 5662 . . . . . 6 (𝑀 = π‘Š β†’ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) = (𝑇 Γ— 𝐸))
1312opeq2d 4835 . . . . 5 (𝑀 = π‘Š β†’ ⟨(Baseβ€˜ndx), (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€))⟩ = ⟨(Baseβ€˜ndx), (𝑇 Γ— 𝐸)⟩)
148mpteq1d 5198 . . . . . . . 8 (𝑀 = π‘Š β†’ (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž))) = (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž))))
1514opeq2d 4835 . . . . . . 7 (𝑀 = π‘Š β†’ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩ = ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)
1612, 12, 15mpoeq123dv 7426 . . . . . 6 (𝑀 = π‘Š β†’ (𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)), 𝑔 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩) = (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩))
1716opeq2d 4835 . . . . 5 (𝑀 = π‘Š β†’ ⟨(+gβ€˜ndx), (𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)), 𝑔 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩ = ⟨(+gβ€˜ndx), (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩)
18 fveq2 6839 . . . . . . 7 (𝑀 = π‘Š β†’ ((EDRingβ€˜πΎ)β€˜π‘€) = ((EDRingβ€˜πΎ)β€˜π‘Š))
19 dvhset.d . . . . . . 7 𝐷 = ((EDRingβ€˜πΎ)β€˜π‘Š)
2018, 19eqtr4di 2795 . . . . . 6 (𝑀 = π‘Š β†’ ((EDRingβ€˜πΎ)β€˜π‘€) = 𝐷)
2120opeq2d 4835 . . . . 5 (𝑀 = π‘Š β†’ ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩ = ⟨(Scalarβ€˜ndx), 𝐷⟩)
2213, 17, 21tpeq123d 4707 . . . 4 (𝑀 = π‘Š β†’ {⟨(Baseβ€˜ndx), (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€))⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)), 𝑔 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩} = {⟨(Baseβ€˜ndx), (𝑇 Γ— 𝐸)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩})
23 eqidd 2738 . . . . . . 7 (𝑀 = π‘Š β†’ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩ = ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)
2411, 12, 23mpoeq123dv 7426 . . . . . 6 (𝑀 = π‘Š β†’ (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩) = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩))
2524opeq2d 4835 . . . . 5 (𝑀 = π‘Š β†’ ⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩ = ⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩)
2625sneqd 4596 . . . 4 (𝑀 = π‘Š β†’ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩} = {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩})
2722, 26uneq12d 4122 . . 3 (𝑀 = π‘Š β†’ ({⟨(Baseβ€˜ndx), (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€))⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)), 𝑔 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩}) = ({⟨(Baseβ€˜ndx), (𝑇 Γ— 𝐸)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩}))
28 eqid 2737 . . 3 (𝑀 ∈ 𝐻 ↦ ({⟨(Baseβ€˜ndx), (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€))⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)), 𝑔 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩})) = (𝑀 ∈ 𝐻 ↦ ({⟨(Baseβ€˜ndx), (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€))⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)), 𝑔 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩}))
29 tpex 7673 . . . 4 {⟨(Baseβ€˜ndx), (𝑇 Γ— 𝐸)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩} ∈ V
30 snex 5386 . . . 4 {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩} ∈ V
3129, 30unex 7672 . . 3 ({⟨(Baseβ€˜ndx), (𝑇 Γ— 𝐸)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩}) ∈ V
3227, 28, 31fvmpt 6945 . 2 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ ({⟨(Baseβ€˜ndx), (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€))⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)), 𝑔 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩}))β€˜π‘Š) = ({⟨(Baseβ€˜ndx), (𝑇 Γ— 𝐸)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩}))
335, 32sylan9eq 2797 1 ((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ = ({⟨(Baseβ€˜ndx), (𝑇 Γ— 𝐸)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   βˆͺ cun 3906  {csn 4584  {ctp 4588  βŸ¨cop 4590   ↦ cmpt 5186   Γ— cxp 5629   ∘ ccom 5635  β€˜cfv 6493   ∈ cmpo 7353  1st c1st 7911  2nd c2nd 7912  ndxcnx 17025  Basecbs 17043  +gcplusg 17093  Scalarcsca 17096   ·𝑠 cvsca 17097  LHypclh 38385  LTrncltrn 38502  TEndoctendo 39153  EDRingcedring 39154  DVecHcdvh 39479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-oprab 7355  df-mpo 7356  df-dvech 39480
This theorem is referenced by:  dvhsca  39483  dvhvbase  39488  dvhfvadd  39492  dvhfvsca  39501
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