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Theorem imassca 17472
Description: The scalar field of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
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)
imassca.g 𝐺 = (Scalarβ€˜π‘…)
Assertion
Ref Expression
imassca (πœ‘ β†’ 𝐺 = (Scalarβ€˜π‘ˆ))

Proof of Theorem imassca
Dummy variables 𝑔 β„Ž 𝑖 𝑛 𝑝 π‘ž π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imassca.g . . . 4 𝐺 = (Scalarβ€˜π‘…)
21fvexi 6905 . . 3 𝐺 ∈ V
3 eqid 2731 . . . . 5 (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ˆ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ˆ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), 𝐺⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜πΊ), 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), (distβ€˜π‘ˆ)⟩}) = (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ˆ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ˆ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), 𝐺⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜πΊ), 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), (distβ€˜π‘ˆ)⟩})
43imasvalstr 17404 . . . 4 (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ˆ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ˆ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), 𝐺⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜πΊ), 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), (distβ€˜π‘ˆ)⟩}) Struct ⟨1, 12⟩
5 scaid 17267 . . . 4 Scalar = Slot (Scalarβ€˜ndx)
6 snsstp1 4819 . . . . . 6 {⟨(Scalarβ€˜ndx), 𝐺⟩} βŠ† {⟨(Scalarβ€˜ndx), 𝐺⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜πΊ), 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)⟩}⟩}
7 ssun2 4173 . . . . . 6 {⟨(Scalarβ€˜ndx), 𝐺⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜πΊ), 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)⟩}⟩} βŠ† ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ˆ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ˆ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), 𝐺⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜πΊ), 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)⟩}⟩})
86, 7sstri 3991 . . . . 5 {⟨(Scalarβ€˜ndx), 𝐺⟩} βŠ† ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ˆ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ˆ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), 𝐺⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜πΊ), 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)⟩}⟩})
9 ssun1 4172 . . . . 5 ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ˆ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ˆ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), 𝐺⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜πΊ), 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)⟩}⟩}) βŠ† (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ˆ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ˆ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), 𝐺⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜πΊ), 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), (distβ€˜π‘ˆ)⟩})
108, 9sstri 3991 . . . 4 {⟨(Scalarβ€˜ndx), 𝐺⟩} βŠ† (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ˆ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ˆ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), 𝐺⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜πΊ), 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), (distβ€˜π‘ˆ)⟩})
114, 5, 10strfv 17144 . . 3 (𝐺 ∈ V β†’ 𝐺 = (Scalarβ€˜(({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ˆ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ˆ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), 𝐺⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜πΊ), 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), (distβ€˜π‘ˆ)⟩})))
122, 11ax-mp 5 . 2 𝐺 = (Scalarβ€˜(({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ˆ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ˆ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), 𝐺⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜πΊ), 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), (distβ€˜π‘ˆ)⟩}))
13 imasbas.u . . . 4 (πœ‘ β†’ π‘ˆ = (𝐹 β€œs 𝑅))
14 imasbas.v . . . 4 (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘…))
15 eqid 2731 . . . 4 (+gβ€˜π‘…) = (+gβ€˜π‘…)
16 eqid 2731 . . . 4 (.rβ€˜π‘…) = (.rβ€˜π‘…)
17 eqid 2731 . . . 4 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
18 eqid 2731 . . . 4 ( ·𝑠 β€˜π‘…) = ( ·𝑠 β€˜π‘…)
19 eqid 2731 . . . 4 (Β·π‘–β€˜π‘…) = (Β·π‘–β€˜π‘…)
20 eqid 2731 . . . 4 (TopOpenβ€˜π‘…) = (TopOpenβ€˜π‘…)
21 eqid 2731 . . . 4 (distβ€˜π‘…) = (distβ€˜π‘…)
22 eqid 2731 . . . 4 (leβ€˜π‘…) = (leβ€˜π‘…)
23 imasbas.f . . . . 5 (πœ‘ β†’ 𝐹:𝑉–onto→𝐡)
24 imasbas.r . . . . 5 (πœ‘ β†’ 𝑅 ∈ 𝑍)
25 eqid 2731 . . . . 5 (+gβ€˜π‘ˆ) = (+gβ€˜π‘ˆ)
2613, 14, 23, 24, 15, 25imasplusg 17470 . . . 4 (πœ‘ β†’ (+gβ€˜π‘ˆ) = βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(+gβ€˜π‘…)π‘ž))⟩})
27 eqid 2731 . . . . 5 (.rβ€˜π‘ˆ) = (.rβ€˜π‘ˆ)
2813, 14, 23, 24, 16, 27imasmulr 17471 . . . 4 (πœ‘ β†’ (.rβ€˜π‘ˆ) = βˆͺ 𝑝 ∈ 𝑉 βˆͺ π‘ž ∈ 𝑉 {⟨⟨(πΉβ€˜π‘), (πΉβ€˜π‘ž)⟩, (πΉβ€˜(𝑝(.rβ€˜π‘…)π‘ž))⟩})
29 eqidd 2732 . . . 4 (πœ‘ β†’ βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜πΊ), π‘₯ ∈ {(πΉβ€˜π‘ž)} ↦ (πΉβ€˜(𝑝( ·𝑠 β€˜π‘…)π‘ž))) = βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜πΊ), π‘₯ ∈ {(πΉβ€˜π‘ž)} ↦ (πΉβ€˜(𝑝( ·𝑠 β€˜π‘…)π‘ž))))
30 eqidd 2732 . . . 4 (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)⟩})
31 eqidd 2732 . . . 4 (πœ‘ β†’ ((TopOpenβ€˜π‘…) qTop 𝐹) = ((TopOpenβ€˜π‘…) qTop 𝐹))
32 eqid 2731 . . . . 5 (distβ€˜π‘ˆ) = (distβ€˜π‘ˆ)
3313, 14, 23, 24, 21, 32imasds 17466 . . . 4 (πœ‘ β†’ (distβ€˜π‘ˆ) = (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 ((distβ€˜π‘…) ∘ 𝑔))), ℝ*, < )))
34 eqidd 2732 . . . 4 (πœ‘ β†’ ((𝐹 ∘ (leβ€˜π‘…)) ∘ ◑𝐹) = ((𝐹 ∘ (leβ€˜π‘…)) ∘ ◑𝐹))
3513, 14, 15, 16, 1, 17, 18, 19, 20, 21, 22, 26, 28, 29, 30, 31, 33, 34, 23, 24imasval 17464 . . 3 (πœ‘ β†’ π‘ˆ = (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ˆ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ˆ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), 𝐺⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜πΊ), 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), (distβ€˜π‘ˆ)⟩}))
3635fveq2d 6895 . 2 (πœ‘ β†’ (Scalarβ€˜π‘ˆ) = (Scalarβ€˜(({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ˆ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ˆ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), 𝐺⟩, ⟨( ·𝑠 β€˜ndx), βˆͺ π‘ž ∈ 𝑉 (𝑝 ∈ (Baseβ€˜πΊ), 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), (distβ€˜π‘ˆ)⟩})))
3712, 36eqtr4id 2790 1 (πœ‘ β†’ 𝐺 = (Scalarβ€˜π‘ˆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1540   ∈ wcel 2105  Vcvv 3473   βˆͺ cun 3946  {csn 4628  {ctp 4632  βŸ¨cop 4634  βˆͺ ciun 4997  β—‘ccnv 5675   ∘ ccom 5680  β€“ontoβ†’wfo 6541  β€˜cfv 6543  (class class class)co 7412   ∈ cmpo 7414  1c1 11117  2c2 12274  cdc 12684  ndxcnx 17133  Basecbs 17151  +gcplusg 17204  .rcmulr 17205  Scalarcsca 17207   ·𝑠 cvsca 17208  Β·π‘–cip 17209  TopSetcts 17210  lecple 17211  distcds 17213  TopOpenctopn 17374   qTop cqtop 17456   β€œs cimas 17457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9443  df-inf 9444  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-fz 13492  df-struct 17087  df-slot 17122  df-ndx 17134  df-base 17152  df-plusg 17217  df-mulr 17218  df-sca 17220  df-vsca 17221  df-ip 17222  df-tset 17223  df-ple 17224  df-ds 17226  df-imas 17461
This theorem is referenced by:  quss  17499  xpssca  17529  imaslmod  32906  imaslmhm  32910  algextdeglem8  33237
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