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 |
1 2 3 4 5 6
|
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* , < ) ) ) |
11 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑛 ∈ ℕ ) → 𝑥 = 𝑋 ) |
12 |
11
|
eqeq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝐹 ‘ ( 1st ‘ ( ℎ ‘ 1 ) ) ) = 𝑥 ↔ ( 𝐹 ‘ ( 1st ‘ ( ℎ ‘ 1 ) ) ) = 𝑋 ) ) |
13 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑛 ∈ ℕ ) → 𝑦 = 𝑌 ) |
14 |
13
|
eqeq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝐹 ‘ ( 2nd ‘ ( ℎ ‘ 𝑛 ) ) ) = 𝑦 ↔ ( 𝐹 ‘ ( 2nd ‘ ( ℎ ‘ 𝑛 ) ) ) = 𝑌 ) ) |
15 |
12 14
|
3anbi12d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑛 ∈ ℕ ) → ( ( ( 𝐹 ‘ ( 1st ‘ ( ℎ ‘ 1 ) ) ) = 𝑥 ∧ ( 𝐹 ‘ ( 2nd ‘ ( ℎ ‘ 𝑛 ) ) ) = 𝑦 ∧ ∀ 𝑖 ∈ ( 1 ... ( 𝑛 − 1 ) ) ( 𝐹 ‘ ( 2nd ‘ ( ℎ ‘ 𝑖 ) ) ) = ( 𝐹 ‘ ( 1st ‘ ( ℎ ‘ ( 𝑖 + 1 ) ) ) ) ) ↔ ( ( 𝐹 ‘ ( 1st ‘ ( ℎ ‘ 1 ) ) ) = 𝑋 ∧ ( 𝐹 ‘ ( 2nd ‘ ( ℎ ‘ 𝑛 ) ) ) = 𝑌 ∧ ∀ 𝑖 ∈ ( 1 ... ( 𝑛 − 1 ) ) ( 𝐹 ‘ ( 2nd ‘ ( ℎ ‘ 𝑖 ) ) ) = ( 𝐹 ‘ ( 1st ‘ ( ℎ ‘ ( 𝑖 + 1 ) ) ) ) ) ) ) |
16 |
15
|
rabbidv |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑛 ∈ ℕ ) → { 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 ) ) ) ) ) } = { 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 ) ) ) ) ) } ) |
17 |
16 9
|
eqtr4di |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑛 ∈ ℕ ) → { 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 ) ) ) ) ) } = 𝑆 ) |
18 |
17
|
mpteq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑛 ∈ ℕ ) → ( 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 ) ) ) = ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝐸 ∘ 𝑔 ) ) ) ) |
19 |
18
|
rneqd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 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 ) ) ) = ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝐸 ∘ 𝑔 ) ) ) ) |
20 |
19
|
iuneq2dv |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ∪ 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 ) ) ) = ∪ 𝑛 ∈ ℕ ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝐸 ∘ 𝑔 ) ) ) ) |
21 |
20
|
infeq1d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → 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* , < ) = inf ( ∪ 𝑛 ∈ ℕ ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝐸 ∘ 𝑔 ) ) ) , ℝ* , < ) ) |
22 |
|
xrltso |
⊢ < Or ℝ* |
23 |
22
|
infex |
⊢ inf ( ∪ 𝑛 ∈ ℕ ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝐸 ∘ 𝑔 ) ) ) , ℝ* , < ) ∈ V |
24 |
23
|
a1i |
⊢ ( 𝜑 → inf ( ∪ 𝑛 ∈ ℕ ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝐸 ∘ 𝑔 ) ) ) , ℝ* , < ) ∈ V ) |
25 |
10 21 7 8 24
|
ovmpod |
⊢ ( 𝜑 → ( 𝑋 𝐷 𝑌 ) = inf ( ∪ 𝑛 ∈ ℕ ran ( 𝑔 ∈ 𝑆 ↦ ( ℝ*𝑠 Σg ( 𝐸 ∘ 𝑔 ) ) ) , ℝ* , < ) ) |