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Theorem imasds 17396
Description: The distance function of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 6-Oct-2020.)
Hypotheses
Ref Expression
imasbas.u (πœ‘ β†’ π‘ˆ = (𝐹 β€œs 𝑅))
imasbas.v (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘…))
imasbas.f (πœ‘ β†’ 𝐹:𝑉–onto→𝐡)
imasbas.r (πœ‘ β†’ 𝑅 ∈ 𝑍)
imasds.e 𝐸 = (distβ€˜π‘…)
imasds.d 𝐷 = (distβ€˜π‘ˆ)
Assertion
Ref Expression
imasds (πœ‘ β†’ 𝐷 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < )))
Distinct variable groups:   𝑔,β„Ž,𝑖,𝑛,π‘₯,𝑦,𝐹   𝑅,𝑔,β„Ž,𝑖,𝑛,π‘₯,𝑦   π‘₯,π‘ˆ   π‘₯,𝐡,𝑦   π‘₯,𝐸,𝑦   πœ‘,𝑔,β„Ž,𝑖,𝑛,π‘₯,𝑦   𝑔,𝑉,β„Ž
Allowed substitution hints:   𝐡(𝑔,β„Ž,𝑖,𝑛)   𝐷(π‘₯,𝑦,𝑔,β„Ž,𝑖,𝑛)   π‘ˆ(𝑦,𝑔,β„Ž,𝑖,𝑛)   𝐸(𝑔,β„Ž,𝑖,𝑛)   𝑉(π‘₯,𝑦,𝑖,𝑛)   𝑍(π‘₯,𝑦,𝑔,β„Ž,𝑖,𝑛)

Proof of Theorem imasds
Dummy variables 𝑝 π‘ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasbas.u . . 3 (πœ‘ β†’ π‘ˆ = (𝐹 β€œs 𝑅))
2 imasbas.v . . 3 (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘…))
3 eqid 2737 . . 3 (+gβ€˜π‘…) = (+gβ€˜π‘…)
4 eqid 2737 . . 3 (.rβ€˜π‘…) = (.rβ€˜π‘…)
5 eqid 2737 . . 3 (Scalarβ€˜π‘…) = (Scalarβ€˜π‘…)
6 eqid 2737 . . 3 (Baseβ€˜(Scalarβ€˜π‘…)) = (Baseβ€˜(Scalarβ€˜π‘…))
7 eqid 2737 . . 3 ( ·𝑠 β€˜π‘…) = ( ·𝑠 β€˜π‘…)
8 eqid 2737 . . 3 (Β·π‘–β€˜π‘…) = (Β·π‘–β€˜π‘…)
9 eqid 2737 . . 3 (TopOpenβ€˜π‘…) = (TopOpenβ€˜π‘…)
10 imasds.e . . 3 𝐸 = (distβ€˜π‘…)
11 eqid 2737 . . 3 (leβ€˜π‘…) = (leβ€˜π‘…)
12 eqidd 2738 . . 3 (πœ‘ β†’ βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(+gβ€˜π‘…)π‘ž))⟩} = βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(+gβ€˜π‘…)π‘ž))⟩})
13 eqidd 2738 . . 3 (πœ‘ β†’ βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(.rβ€˜π‘…)π‘ž))⟩} = βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(.rβ€˜π‘…)π‘ž))⟩})
14 eqidd 2738 . . 3 (πœ‘ β†’ βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜(Scalarβ€˜π‘…)), π‘₯ ∈ {(πΉβ€˜π‘ž)} ↦ (πΉβ€˜(𝑝( ·𝑠 β€˜π‘…)π‘ž))) = βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜(Scalarβ€˜π‘…)), π‘₯ ∈ {(πΉβ€˜π‘ž)} ↦ (πΉβ€˜(𝑝( ·𝑠 β€˜π‘…)π‘ž))))
15 eqidd 2738 . . 3 (πœ‘ β†’ βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (𝑝(Β·π‘–β€˜π‘…)π‘ž)⟩} = βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (𝑝(Β·π‘–β€˜π‘…)π‘ž)⟩})
16 eqidd 2738 . . 3 (πœ‘ β†’ ((TopOpenβ€˜π‘…) qTop 𝐹) = ((TopOpenβ€˜π‘…) qTop 𝐹))
17 eqidd 2738 . . 3 (πœ‘ β†’ (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < )) = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < )))
18 eqidd 2738 . . 3 (πœ‘ β†’ ((𝐹 ∘ (leβ€˜π‘…)) ∘ ◑𝐹) = ((𝐹 ∘ (leβ€˜π‘…)) ∘ ◑𝐹))
19 imasbas.f . . 3 (πœ‘ β†’ 𝐹:𝑉–onto→𝐡)
20 imasbas.r . . 3 (πœ‘ β†’ 𝑅 ∈ 𝑍)
211, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20imasval 17394 . 2 (πœ‘ β†’ π‘ˆ = (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(+gβ€˜π‘…)π‘ž))⟩}⟩, ⟨(.rβ€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(.rβ€˜π‘…)π‘ž))⟩}⟩} βˆͺ {⟨(Scalarβ€˜ndx), (Scalarβ€˜π‘…)⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜(Scalarβ€˜π‘…)), π‘₯ ∈ {(πΉβ€˜π‘ž)} ↦ (πΉβ€˜(𝑝( ·𝑠 β€˜π‘…)π‘ž)))⟩, ⟨(Β·π‘–β€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (𝑝(Β·π‘–β€˜π‘…)π‘ž)⟩}⟩}) βˆͺ {⟨(TopSetβ€˜ndx), ((TopOpenβ€˜π‘…) qTop 𝐹)⟩, ⟨(leβ€˜ndx), ((𝐹 ∘ (leβ€˜π‘…)) ∘ ◑𝐹)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < ))⟩}))
22 eqid 2737 . . 3 (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(+gβ€˜π‘…)π‘ž))⟩}⟩, ⟨(.rβ€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(.rβ€˜π‘…)π‘ž))⟩}⟩} βˆͺ {⟨(Scalarβ€˜ndx), (Scalarβ€˜π‘…)⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜(Scalarβ€˜π‘…)), π‘₯ ∈ {(πΉβ€˜π‘ž)} ↦ (πΉβ€˜(𝑝( ·𝑠 β€˜π‘…)π‘ž)))⟩, ⟨(Β·π‘–β€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (𝑝(Β·π‘–β€˜π‘…)π‘ž)⟩}⟩}) βˆͺ {⟨(TopSetβ€˜ndx), ((TopOpenβ€˜π‘…) qTop 𝐹)⟩, ⟨(leβ€˜ndx), ((𝐹 ∘ (leβ€˜π‘…)) ∘ ◑𝐹)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < ))⟩}) = (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(+gβ€˜π‘…)π‘ž))⟩}⟩, ⟨(.rβ€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(.rβ€˜π‘…)π‘ž))⟩}⟩} βˆͺ {⟨(Scalarβ€˜ndx), (Scalarβ€˜π‘…)⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜(Scalarβ€˜π‘…)), π‘₯ ∈ {(πΉβ€˜π‘ž)} ↦ (πΉβ€˜(𝑝( ·𝑠 β€˜π‘…)π‘ž)))⟩, ⟨(Β·π‘–β€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (𝑝(Β·π‘–β€˜π‘…)π‘ž)⟩}⟩}) βˆͺ {⟨(TopSetβ€˜ndx), ((TopOpenβ€˜π‘…) qTop 𝐹)⟩, ⟨(leβ€˜ndx), ((𝐹 ∘ (leβ€˜π‘…)) ∘ ◑𝐹)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < ))⟩})
2322imasvalstr 17334 . 2 (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(+gβ€˜π‘…)π‘ž))⟩}⟩, ⟨(.rβ€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(.rβ€˜π‘…)π‘ž))⟩}⟩} βˆͺ {⟨(Scalarβ€˜ndx), (Scalarβ€˜π‘…)⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜(Scalarβ€˜π‘…)), π‘₯ ∈ {(πΉβ€˜π‘ž)} ↦ (πΉβ€˜(𝑝( ·𝑠 β€˜π‘…)π‘ž)))⟩, ⟨(Β·π‘–β€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (𝑝(Β·π‘–β€˜π‘…)π‘ž)⟩}⟩}) βˆͺ {⟨(TopSetβ€˜ndx), ((TopOpenβ€˜π‘…) qTop 𝐹)⟩, ⟨(leβ€˜ndx), ((𝐹 ∘ (leβ€˜π‘…)) ∘ ◑𝐹)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < ))⟩}) Struct ⟨1, 12⟩
24 dsid 17268 . 2 dist = Slot (distβ€˜ndx)
25 snsstp3 4779 . . 3 {⟨(distβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < ))⟩} βŠ† {⟨(TopSetβ€˜ndx), ((TopOpenβ€˜π‘…) qTop 𝐹)⟩, ⟨(leβ€˜ndx), ((𝐹 ∘ (leβ€˜π‘…)) ∘ ◑𝐹)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < ))⟩}
26 ssun2 4134 . . 3 {⟨(TopSetβ€˜ndx), ((TopOpenβ€˜π‘…) qTop 𝐹)⟩, ⟨(leβ€˜ndx), ((𝐹 ∘ (leβ€˜π‘…)) ∘ ◑𝐹)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < ))⟩} βŠ† (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(+gβ€˜π‘…)π‘ž))⟩}⟩, ⟨(.rβ€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(.rβ€˜π‘…)π‘ž))⟩}⟩} βˆͺ {⟨(Scalarβ€˜ndx), (Scalarβ€˜π‘…)⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜(Scalarβ€˜π‘…)), π‘₯ ∈ {(πΉβ€˜π‘ž)} ↦ (πΉβ€˜(𝑝( ·𝑠 β€˜π‘…)π‘ž)))⟩, ⟨(Β·π‘–β€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (𝑝(Β·π‘–β€˜π‘…)π‘ž)⟩}⟩}) βˆͺ {⟨(TopSetβ€˜ndx), ((TopOpenβ€˜π‘…) qTop 𝐹)⟩, ⟨(leβ€˜ndx), ((𝐹 ∘ (leβ€˜π‘…)) ∘ ◑𝐹)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < ))⟩})
2725, 26sstri 3954 . 2 {⟨(distβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < ))⟩} βŠ† (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(+gβ€˜π‘…)π‘ž))⟩}⟩, ⟨(.rβ€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(.rβ€˜π‘…)π‘ž))⟩}⟩} βˆͺ {⟨(Scalarβ€˜ndx), (Scalarβ€˜π‘…)⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜(Scalarβ€˜π‘…)), π‘₯ ∈ {(πΉβ€˜π‘ž)} ↦ (πΉβ€˜(𝑝( ·𝑠 β€˜π‘…)π‘ž)))⟩, ⟨(Β·π‘–β€˜ndx), βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (𝑝(Β·π‘–β€˜π‘…)π‘ž)⟩}⟩}) βˆͺ {⟨(TopSetβ€˜ndx), ((TopOpenβ€˜π‘…) qTop 𝐹)⟩, ⟨(leβ€˜ndx), ((𝐹 ∘ (leβ€˜π‘…)) ∘ ◑𝐹)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < ))⟩})
28 fvex 6856 . . . . 5 (Baseβ€˜π‘…) ∈ V
292, 28eqeltrdi 2846 . . . 4 (πœ‘ β†’ 𝑉 ∈ V)
30 focdmex 7889 . . . 4 (𝑉 ∈ V β†’ (𝐹:𝑉–onto→𝐡 β†’ 𝐡 ∈ V))
3129, 19, 30sylc 65 . . 3 (πœ‘ β†’ 𝐡 ∈ V)
32 mpoexga 8011 . . 3 ((𝐡 ∈ V ∧ 𝐡 ∈ V) β†’ (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < )) ∈ V)
3331, 31, 32syl2anc 585 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < )) ∈ V)
34 imasds.d . 2 𝐷 = (distβ€˜π‘ˆ)
3521, 23, 24, 27, 33, 34strfv3 17078 1 (πœ‘ β†’ 𝐷 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < )))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3408  Vcvv 3446   βˆͺ cun 3909  {csn 4587  {ctp 4591  βŸ¨cop 4593  βˆͺ ciun 4955   ↦ cmpt 5189   Γ— cxp 5632  β—‘ccnv 5633  ran crn 5635   ∘ ccom 5638  β€“ontoβ†’wfo 6495  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1st c1st 7920  2nd c2nd 7921   ↑m cmap 8766  infcinf 9378  1c1 11053   + caddc 11055  β„*cxr 11189   < clt 11190   βˆ’ cmin 11386  β„•cn 12154  2c2 12209  cdc 12619  ...cfz 13425  ndxcnx 17066  Basecbs 17084  +gcplusg 17134  .rcmulr 17135  Scalarcsca 17137   ·𝑠 cvsca 17138  Β·π‘–cip 17139  TopSetcts 17140  lecple 17141  distcds 17143  TopOpenctopn 17304   Ξ£g cgsu 17323  β„*𝑠cxrs 17383   qTop cqtop 17386   β€œs cimas 17387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8649  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-sup 9379  df-inf 9380  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-nn 12155  df-2 12217  df-3 12218  df-4 12219  df-5 12220  df-6 12221  df-7 12222  df-8 12223  df-9 12224  df-n0 12415  df-z 12501  df-dec 12620  df-uz 12765  df-fz 13426  df-struct 17020  df-slot 17055  df-ndx 17067  df-base 17085  df-plusg 17147  df-mulr 17148  df-sca 17150  df-vsca 17151  df-ip 17152  df-tset 17153  df-ple 17154  df-ds 17156  df-imas 17391
This theorem is referenced by:  imasdsfn  17397  imasdsval  17398  imasplusg  17400  imasmulr  17401  imassca  17402  imasvsca  17403  imasip  17404  imastset  17405  imasle  17406
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