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Metamath Proof Explorer

Theorem imasdsval2

Description: The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015) (Revised by AV, 6-Oct-2020)

Ref Expression
Hypotheses 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 )
imasdsval.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
imasdsval.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
imasdsval.s ⊢ 𝑆 = { 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 ) ) ) ) ) }
imasds.u ⊢ 𝑇 = ( 𝐸 ↾ ( 𝑉 × 𝑉 ) )
Assertion imasdsval2 ( 𝜑 → ( 𝑋 𝐷 𝑌 ) = inf ( ∪ 𝑛 ∈ ℕ ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝑇 ∘ 𝑔 ) ) ) , ℝ* , < ) )

Proof

Step Hyp Ref Expression
1 imasbas.u ⊢ ( 𝜑 → 𝑈 = ( 𝐹 “s 𝑅 ) )
2 imasbas.v ⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝑅 ) )
3 imasbas.f ⊢ ( 𝜑 → 𝐹 : 𝑉 –onto→ 𝐵 )
4 imasbas.r ⊢ ( 𝜑 → 𝑅 ∈ 𝑍 )
5 imasds.e ⊢ 𝐸 = ( dist ‘ 𝑅 )
6 imasds.d ⊢ 𝐷 = ( dist ‘ 𝑈 )
7 imasdsval.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
8 imasdsval.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
9 imasdsval.s ⊢ 𝑆 = { 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 ) ) ) ) ) }
10 imasds.u ⊢ 𝑇 = ( 𝐸 ↾ ( 𝑉 × 𝑉 ) )
11 1 2 3 4 5 6 7 8 9 imasdsval ⊢ ( 𝜑 → ( 𝑋 𝐷 𝑌 ) = inf ( ∪ 𝑛 ∈ ℕ ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝐸 ∘ 𝑔 ) ) ) , ℝ* , < ) )
12 10 coeq1i ⊢ ( 𝑇 ∘ 𝑔 ) = ( ( 𝐸 ↾ ( 𝑉 × 𝑉 ) ) ∘ 𝑔 )
13 9 ssrab3 ⊢ 𝑆 ⊆ ( ( 𝑉 × 𝑉 ) ↑m ( 1 ... 𝑛 ) )
14 13 sseli ⊢ ( 𝑔 ∈ 𝑆 → 𝑔 ∈ ( ( 𝑉 × 𝑉 ) ↑m ( 1 ... 𝑛 ) ) )
15 elmapi ⊢ ( 𝑔 ∈ ( ( 𝑉 × 𝑉 ) ↑m ( 1 ... 𝑛 ) ) → 𝑔 : ( 1 ... 𝑛 ) ⟶ ( 𝑉 × 𝑉 ) )
16 frn ⊢ ( 𝑔 : ( 1 ... 𝑛 ) ⟶ ( 𝑉 × 𝑉 ) → ran 𝑔 ⊆ ( 𝑉 × 𝑉 ) )
17 cores ⊢ ( ran 𝑔 ⊆ ( 𝑉 × 𝑉 ) → ( ( 𝐸 ↾ ( 𝑉 × 𝑉 ) ) ∘ 𝑔 ) = ( 𝐸 ∘ 𝑔 ) )
18 14 15 16 17 4syl ⊢ ( 𝑔 ∈ 𝑆 → ( ( 𝐸 ↾ ( 𝑉 × 𝑉 ) ) ∘ 𝑔 ) = ( 𝐸 ∘ 𝑔 ) )
19 12 18 eqtrid ⊢ ( 𝑔 ∈ 𝑆 → ( 𝑇 ∘ 𝑔 ) = ( 𝐸 ∘ 𝑔 ) )
20 19 oveq2d ⊢ ( 𝑔 ∈ 𝑆 → ( ℝ*𝑠 Σg ( 𝑇 ∘ 𝑔 ) ) = ( ℝ*𝑠 Σg ( 𝐸 ∘ 𝑔 ) ) )
21 20 mpteq2ia ⊢ ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝑇 ∘ 𝑔 ) ) ) = ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝐸 ∘ 𝑔 ) ) )
22 21 rneqi ⊢ ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝑇 ∘ 𝑔 ) ) ) = ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝐸 ∘ 𝑔 ) ) )
23 22 a1i ⊢ ( 𝑛 ∈ ℕ → ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝑇 ∘ 𝑔 ) ) ) = ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝐸 ∘ 𝑔 ) ) ) )
24 23 iuneq2i ⊢ ∪ 𝑛 ∈ ℕ ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝑇 ∘ 𝑔 ) ) ) = ∪ 𝑛 ∈ ℕ ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝐸 ∘ 𝑔 ) ) )
25 24 infeq1i ⊢ inf ( ∪ 𝑛 ∈ ℕ ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝑇 ∘ 𝑔 ) ) ) , ℝ* , < ) = inf ( ∪ 𝑛 ∈ ℕ ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝐸 ∘ 𝑔 ) ) ) , ℝ* , < )
26 11 25 eqtr4di ⊢ ( 𝜑 → ( 𝑋 𝐷 𝑌 ) = inf ( ∪ 𝑛 ∈ ℕ ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝑇 ∘ 𝑔 ) ) ) , ℝ* , < ) )