Step |
Hyp |
Ref |
Expression |
1 |
|
prdsbasmpt2.y |
⊢ 𝑌 = ( 𝑆 Xs ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ) |
2 |
|
prdsbasmpt2.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
3 |
|
prdsbasmpt2.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑉 ) |
4 |
|
prdsbasmpt2.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
5 |
|
prdsbasmpt2.r |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 𝑅 ∈ 𝑋 ) |
6 |
|
prdsdsval2.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
7 |
|
prdsdsval2.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐵 ) |
8 |
|
prdsdsval3.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
9 |
|
prdsdsval3.e |
⊢ 𝐸 = ( ( dist ‘ 𝑅 ) ↾ ( 𝐾 × 𝐾 ) ) |
10 |
|
prdsdsval3.d |
⊢ 𝐷 = ( dist ‘ 𝑌 ) |
11 |
|
eqid |
⊢ ( dist ‘ 𝑅 ) = ( dist ‘ 𝑅 ) |
12 |
1 2 3 4 5 6 7 11 10
|
prdsdsval2 |
⊢ ( 𝜑 → ( 𝐹 𝐷 𝐺 ) = sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ 𝑅 ) ( 𝐺 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) |
13 |
|
eqidd |
⊢ ( 𝜑 → 𝐼 = 𝐼 ) |
14 |
1 2 3 4 5 8 6
|
prdsbascl |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ∈ 𝐾 ) |
15 |
1 2 3 4 5 8 7
|
prdsbascl |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 ( 𝐺 ‘ 𝑥 ) ∈ 𝐾 ) |
16 |
9
|
oveqi |
⊢ ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( ( dist ‘ 𝑅 ) ↾ ( 𝐾 × 𝐾 ) ) ( 𝐺 ‘ 𝑥 ) ) |
17 |
|
ovres |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) ∈ 𝐾 ∧ ( 𝐺 ‘ 𝑥 ) ∈ 𝐾 ) → ( ( 𝐹 ‘ 𝑥 ) ( ( dist ‘ 𝑅 ) ↾ ( 𝐾 × 𝐾 ) ) ( 𝐺 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ 𝑅 ) ( 𝐺 ‘ 𝑥 ) ) ) |
18 |
16 17
|
eqtrid |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) ∈ 𝐾 ∧ ( 𝐺 ‘ 𝑥 ) ∈ 𝐾 ) → ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ 𝑅 ) ( 𝐺 ‘ 𝑥 ) ) ) |
19 |
18
|
ex |
⊢ ( ( 𝐹 ‘ 𝑥 ) ∈ 𝐾 → ( ( 𝐺 ‘ 𝑥 ) ∈ 𝐾 → ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ 𝑅 ) ( 𝐺 ‘ 𝑥 ) ) ) ) |
20 |
19
|
ral2imi |
⊢ ( ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ∈ 𝐾 → ( ∀ 𝑥 ∈ 𝐼 ( 𝐺 ‘ 𝑥 ) ∈ 𝐾 → ∀ 𝑥 ∈ 𝐼 ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ 𝑅 ) ( 𝐺 ‘ 𝑥 ) ) ) ) |
21 |
14 15 20
|
sylc |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ 𝑅 ) ( 𝐺 ‘ 𝑥 ) ) ) |
22 |
|
mpteq12 |
⊢ ( ( 𝐼 = 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ 𝑅 ) ( 𝐺 ‘ 𝑥 ) ) ) → ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ 𝑅 ) ( 𝐺 ‘ 𝑥 ) ) ) ) |
23 |
13 21 22
|
syl2anc |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ 𝑅 ) ( 𝐺 ‘ 𝑥 ) ) ) ) |
24 |
23
|
rneqd |
⊢ ( 𝜑 → ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) = ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ 𝑅 ) ( 𝐺 ‘ 𝑥 ) ) ) ) |
25 |
24
|
uneq1d |
⊢ ( 𝜑 → ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) ∪ { 0 } ) = ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ 𝑅 ) ( 𝐺 ‘ 𝑥 ) ) ) ∪ { 0 } ) ) |
26 |
25
|
supeq1d |
⊢ ( 𝜑 → sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) = sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ 𝑅 ) ( 𝐺 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) |
27 |
12 26
|
eqtr4d |
⊢ ( 𝜑 → ( 𝐹 𝐷 𝐺 ) = sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) |