Theorem imasds 17468

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	⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
imasbas.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasbas.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasbas.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)
imasds.e	⊢ 𝐸 = (dist‘𝑅)
imasds.d	⊢ 𝐷 = (dist‘𝑈)

Assertion

Ref	Expression
imasds	⊢ (𝜑 → 𝐷 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < )))

Distinct variable groups:   𝑔,ℎ,𝑖,𝑛,𝑥,𝑦,𝐹   𝑅,𝑔,ℎ,𝑖,𝑛,𝑥,𝑦   𝑥,𝑈   𝑥,𝐵,𝑦   𝑥,𝐸,𝑦   𝜑,𝑔,ℎ,𝑖,𝑛,𝑥,𝑦   𝑔,𝑉,ℎ

Allowed substitution hints:   𝐵(𝑔,ℎ,𝑖,𝑛)   𝐷(𝑥,𝑦,𝑔,ℎ,𝑖,𝑛)   𝑈(𝑦,𝑔,ℎ,𝑖,𝑛)   𝐸(𝑔,ℎ,𝑖,𝑛)   𝑉(𝑥,𝑦,𝑖,𝑛)   𝑍(𝑥,𝑦,𝑔,ℎ,𝑖,𝑛)

Proof of Theorem imasds

Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasbas.u	. . 3 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
2		imasbas.v	. . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		eqid 2739	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2739	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5		eqid 2739	. . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
6		eqid 2739	. . 3 ⊢ (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
7		eqid 2739	. . 3 ⊢ ( ·𝑠 ‘𝑅) = ( ·𝑠 ‘𝑅)
8		eqid 2739	. . 3 ⊢ (·𝑖‘𝑅) = (·𝑖‘𝑅)
9		eqid 2739	. . 3 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
10		imasds.e	. . 3 ⊢ 𝐸 = (dist‘𝑅)
11		eqid 2739	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
12		eqidd 2740	. . 3 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩})
13		eqidd 2740	. . 3 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩})
14		eqidd 2740	. . 3 ⊢ (𝜑 → ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞))) = ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞))))
15		eqidd 2740	. . 3 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩} = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩})
16		eqidd 2740	. . 3 ⊢ (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((TopOpen‘𝑅) qTop 𝐹))
17		eqidd 2740	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < )) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < )))
18		eqidd 2740	. . 3 ⊢ (𝜑 → ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) = ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹))
19		imasbas.f	. . 3 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
20		imasbas.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
21	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20	imasval 17466	. 2 ⊢ (𝜑 → 𝑈 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ))⟩}))
22		eqid 2739	. . 3 ⊢ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ))⟩}) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ))⟩})
23	22	imasvalstr 17405	. 2 ⊢ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ))⟩}) Struct ⟨1, ;12⟩
24		dsid 17340	. 2 ⊢ dist = Slot (dist‘ndx)
25		snsstp3 4749	. . 3 ⊢ {⟨(dist‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ))⟩} ⊆ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ))⟩}
26		ssun2 4108	. . 3 ⊢ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ))⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ))⟩})
27	25, 26	sstri 3924	. 2 ⊢ {⟨(dist‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ))⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑉 (𝑝 ∈ (Base‘(Scalar‘𝑅)), 𝑥 ∈ {(𝐹‘𝑞)} ↦ (𝐹‘(𝑝( ·𝑠 ‘𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝑝(·𝑖‘𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)⟩, ⟨(dist‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ))⟩})
28		fvex 6840	. . . . 5 ⊢ (Base‘𝑅) ∈ V
29	2, 28	eqeltrdi 2847	. . . 4 ⊢ (𝜑 → 𝑉 ∈ V)
30		focdmex 7898	. . . 4 ⊢ (𝑉 ∈ V → (𝐹:𝑉–onto→𝐵 → 𝐵 ∈ V))
31	29, 19, 30	sylc 65	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
32		mpoexga 8019	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < )) ∈ V)
33	31, 31, 32	syl2anc 590	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < )) ∈ V)
34		imasds.d	. 2 ⊢ 𝐷 = (dist‘𝑈)
35	21, 23, 24, 27, 33, 34	strfv3 17165	1 ⊢ (𝜑 → 𝐷 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < )))

Syntax hints:   → wi 4   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  {crab 3391  Vcvv 3431   ∪ cun 3881  {csn 4555  {ctp 4559  ⟨cop 4561  ∪ ciun 4921   ↦ cmpt 5153   × cxp 5616  ^◡ccnv 5617  ran crn 5619   ∘ ccom 5622  –onto→wfo 6483  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930   ↑_m cmap 8763  infcinf 9344  1c1 11030   + caddc 11032  ℝ^*cxr 11169   < clt 11170   − cmin 11368  ℕcn 12165  2c2 12227  ;cdc 12635  ...cfz 13452  ndxcnx 17154  Basecbs 17170  +_gcplusg 17211  ._rcmulr 17212  Scalarcsca 17214   ·_𝑠 cvsca 17215  ·_𝑖cip 17216  TopSetcts 17217  lecple 17218  distcds 17220  TopOpenctopn 17375   Σ_g cgsu 17394  ℝ^*_𝑠cxrs 17455   qTop cqtop 17458   “_s cimas 17459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-slot 17143  df-ndx 17155  df-base 17171  df-plusg 17224  df-mulr 17225  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-imas 17463

This theorem is referenced by:  imasdsfn  17469  imasdsval  17470  imasplusg  17472  imasmulr  17473  imassca  17474  imasvsca  17475  imasip  17476  imastset  17477  imasle  17478