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Theorem imasbas 17527
Description: The base set 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 (𝜑𝑈 = (𝐹s 𝑅))
imasbas.v (𝜑𝑉 = (Base‘𝑅))
imasbas.f (𝜑𝐹:𝑉onto𝐵)
imasbas.r (𝜑𝑅𝑍)
Assertion
Ref Expression
imasbas (𝜑𝐵 = (Base‘𝑈))

Proof of Theorem imasbas
Dummy variables 𝑔 𝑖 𝑛 𝑝 𝑞 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasbas.u . . . 4 (𝜑𝑈 = (𝐹s 𝑅))
2 imasbas.v . . . 4 (𝜑𝑉 = (Base‘𝑅))
3 eqid 2726 . . . 4 (+g𝑅) = (+g𝑅)
4 eqid 2726 . . . 4 (.r𝑅) = (.r𝑅)
5 eqid 2726 . . . 4 (Scalar‘𝑅) = (Scalar‘𝑅)
6 eqid 2726 . . . 4 (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
7 eqid 2726 . . . 4 ( ·𝑠𝑅) = ( ·𝑠𝑅)
8 eqid 2726 . . . 4 (·𝑖𝑅) = (·𝑖𝑅)
9 eqid 2726 . . . 4 (TopOpen‘𝑅) = (TopOpen‘𝑅)
10 eqid 2726 . . . 4 (dist‘𝑅) = (dist‘𝑅)
11 eqid 2726 . . . 4 (le‘𝑅) = (le‘𝑅)
12 eqidd 2727 . . . 4 (𝜑 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩} = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩})
13 eqidd 2727 . . . 4 (𝜑 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩} = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩})
14 eqidd 2727 . . . 4 (𝜑 𝑞𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞))) = 𝑞𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞))))
15 eqidd 2727 . . . 4 (𝜑 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩} = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩})
16 eqidd 2727 . . . 4 (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((TopOpen‘𝑅) qTop 𝐹))
17 eqidd 2727 . . . 4 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ inf( 𝑛 ∈ ℕ ran (𝑔 ∈ { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑅) ∘ 𝑔))), ℝ*, < )) = (𝑥𝐵, 𝑦𝐵 ↦ inf( 𝑛 ∈ ℕ ran (𝑔 ∈ { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑅) ∘ 𝑔))), ℝ*, < )))
18 eqidd 2727 . . . 4 (𝜑 → ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹) = ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹))
19 imasbas.f . . . 4 (𝜑𝐹:𝑉onto𝐵)
20 imasbas.r . . . 4 (𝜑𝑅𝑍)
211, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20imasval 17526 . . 3 (𝜑𝑈 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ inf( 𝑛 ∈ ℕ ran (𝑔 ∈ { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑅) ∘ 𝑔))), ℝ*, < ))⟩}))
22 eqid 2726 . . . 4 (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ inf( 𝑛 ∈ ℕ ran (𝑔 ∈ { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑅) ∘ 𝑔))), ℝ*, < ))⟩}) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ inf( 𝑛 ∈ ℕ ran (𝑔 ∈ { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑅) ∘ 𝑔))), ℝ*, < ))⟩})
2322imasvalstr 17466 . . 3 (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ inf( 𝑛 ∈ ℕ ran (𝑔 ∈ { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑅) ∘ 𝑔))), ℝ*, < ))⟩}) Struct ⟨1, 12⟩
24 baseid 17216 . . 3 Base = Slot (Base‘ndx)
25 snsstp1 4825 . . . . 5 {⟨(Base‘ndx), 𝐵⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩}⟩}
26 ssun1 4173 . . . . 5 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩}⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩})
2725, 26sstri 3989 . . . 4 {⟨(Base‘ndx), 𝐵⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩})
28 ssun1 4173 . . . 4 ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ inf( 𝑛 ∈ ℕ ran (𝑔 ∈ { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑅) ∘ 𝑔))), ℝ*, < ))⟩})
2927, 28sstri 3989 . . 3 {⟨(Base‘ndx), 𝐵⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ inf( 𝑛 ∈ ℕ ran (𝑔 ∈ { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑅) ∘ 𝑔))), ℝ*, < ))⟩})
30 fvex 6914 . . . . 5 (Base‘𝑅) ∈ V
312, 30eqeltrdi 2834 . . . 4 (𝜑𝑉 ∈ V)
32 focdmex 7969 . . . 4 (𝑉 ∈ V → (𝐹:𝑉onto𝐵𝐵 ∈ V))
3331, 19, 32sylc 65 . . 3 (𝜑𝐵 ∈ V)
34 eqid 2726 . . 3 (Base‘𝑈) = (Base‘𝑈)
3521, 23, 24, 29, 33, 34strfv3 17207 . 2 (𝜑 → (Base‘𝑈) = 𝐵)
3635eqcomd 2732 1 (𝜑𝐵 = (Base‘𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1534  wcel 2099  wral 3051  {crab 3419  Vcvv 3462  cun 3945  {csn 4633  {ctp 4637  cop 4639   ciun 5001  cmpt 5236   × cxp 5680  ccnv 5681  ran crn 5683  ccom 5686  ontowfo 6552  cfv 6554  (class class class)co 7424  cmpo 7426  1st c1st 8001  2nd c2nd 8002  m cmap 8855  infcinf 9484  1c1 11159   + caddc 11161  *cxr 11297   < clt 11298  cmin 11494  cn 12264  2c2 12319  cdc 12729  ...cfz 13538  ndxcnx 17195  Basecbs 17213  +gcplusg 17266  .rcmulr 17267  Scalarcsca 17269   ·𝑠 cvsca 17270  ·𝑖cip 17271  TopSetcts 17272  lecple 17273  distcds 17275  TopOpenctopn 17436   Σg cgsu 17455  *𝑠cxrs 17515   qTop cqtop 17518  s cimas 17519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9485  df-inf 9486  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12611  df-dec 12730  df-uz 12875  df-fz 13539  df-struct 17149  df-slot 17184  df-ndx 17196  df-base 17214  df-plusg 17279  df-mulr 17280  df-sca 17282  df-vsca 17283  df-ip 17284  df-tset 17285  df-ple 17286  df-ds 17288  df-imas 17523
This theorem is referenced by:  qusbas  17560  xpsbas  17587  imasmnd2  18764  imasgrp2  19049  imasabl  19874  imasrng  20160  imasring  20309  imastopn  23715  imastps  23716  imasf1oxms  24489  imasf1oms  24490  imaslmod  33228  imasmhm  33229  imasghm  33230  imasrhm  33231  r1pquslmic  33478  algextdeglem8  33591  circtopn  33652  imasgim  42761  isnumbasgrplem1  42762
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