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Theorem dvhset 40463
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 40462 . . . 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 6886 . . 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 2778 . 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 6884 . . . . . . . 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 2784 . . . . . . 7 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = 𝑇)
9 fveq2 6884 . . . . . . . 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 2784 . . . . . . 7 (𝑀 = π‘Š β†’ ((TEndoβ€˜πΎ)β€˜π‘€) = 𝐸)
128, 11xpeq12d 5700 . . . . . 6 (𝑀 = π‘Š β†’ (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€)) = (𝑇 Γ— 𝐸))
1312opeq2d 4875 . . . . 5 (𝑀 = π‘Š β†’ ⟨(Baseβ€˜ndx), (((LTrnβ€˜πΎ)β€˜π‘€) Γ— ((TEndoβ€˜πΎ)β€˜π‘€))⟩ = ⟨(Baseβ€˜ndx), (𝑇 Γ— 𝐸)⟩)
148mpteq1d 5236 . . . . . . . 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 4875 . . . . . . 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 7479 . . . . . 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 4875 . . . . 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 6884 . . . . . . 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 2784 . . . . . 6 (𝑀 = π‘Š β†’ ((EDRingβ€˜πΎ)β€˜π‘€) = 𝐷)
2120opeq2d 4875 . . . . 5 (𝑀 = π‘Š β†’ ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩ = ⟨(Scalarβ€˜ndx), 𝐷⟩)
2213, 17, 21tpeq123d 4747 . . . 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 2727 . . . . . . 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 7479 . . . . . 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 4875 . . . . 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 4635 . . . 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 4159 . . 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 2726 . . 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 7730 . . . 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 5424 . . . 4 {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩} ∈ V
3129, 30unex 7729 . . 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 6991 . 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 2786 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 395   = wceq 1533   ∈ wcel 2098   βˆͺ cun 3941  {csn 4623  {ctp 4627  βŸ¨cop 4629   ↦ cmpt 5224   Γ— cxp 5667   ∘ ccom 5673  β€˜cfv 6536   ∈ cmpo 7406  1st c1st 7969  2nd c2nd 7970  ndxcnx 17133  Basecbs 17151  +gcplusg 17204  Scalarcsca 17207   ·𝑠 cvsca 17208  LHypclh 39366  LTrncltrn 39483  TEndoctendo 40134  EDRingcedring 40135  DVecHcdvh 40460
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-dvech 40461
This theorem is referenced by:  dvhsca  40464  dvhvbase  40469  dvhfvadd  40473  dvhfvsca  40482
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