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* , < ) ) |