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 |
|
prdsdsval2.e |
⊢ 𝐸 = ( dist ‘ 𝑅 ) |
9 |
|
prdsdsval2.d |
⊢ 𝐷 = ( dist ‘ 𝑌 ) |
10 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) = ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) |
11 |
10
|
fnmpt |
⊢ ( ∀ 𝑥 ∈ 𝐼 𝑅 ∈ 𝑋 → ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) Fn 𝐼 ) |
12 |
5 11
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) Fn 𝐼 ) |
13 |
1 2 3 4 12 6 7 9
|
prdsdsval |
⊢ ( 𝜑 → ( 𝐹 𝐷 𝐺 ) = sup ( ( ran ( 𝑦 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑦 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑦 ) ) ( 𝐺 ‘ 𝑦 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) |
14 |
|
nfcv |
⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝑦 ) |
15 |
|
nfcv |
⊢ Ⅎ 𝑥 dist |
16 |
|
nffvmpt1 |
⊢ Ⅎ 𝑥 ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑦 ) |
17 |
15 16
|
nffv |
⊢ Ⅎ 𝑥 ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑦 ) ) |
18 |
|
nfcv |
⊢ Ⅎ 𝑥 ( 𝐺 ‘ 𝑦 ) |
19 |
14 17 18
|
nfov |
⊢ Ⅎ 𝑥 ( ( 𝐹 ‘ 𝑦 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑦 ) ) ( 𝐺 ‘ 𝑦 ) ) |
20 |
|
nfcv |
⊢ Ⅎ 𝑦 ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) |
21 |
|
2fveq3 |
⊢ ( 𝑦 = 𝑥 → ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑦 ) ) = ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) ) |
22 |
|
fveq2 |
⊢ ( 𝑦 = 𝑥 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑥 ) ) |
23 |
|
fveq2 |
⊢ ( 𝑦 = 𝑥 → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑥 ) ) |
24 |
21 22 23
|
oveq123d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝐹 ‘ 𝑦 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑦 ) ) ( 𝐺 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) |
25 |
19 20 24
|
cbvmpt |
⊢ ( 𝑦 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑦 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑦 ) ) ( 𝐺 ‘ 𝑦 ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) |
26 |
|
eqidd |
⊢ ( 𝜑 → 𝐼 = 𝐼 ) |
27 |
10
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ 𝐼 ∧ 𝑅 ∈ 𝑋 ) → ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) = 𝑅 ) |
28 |
27
|
fveq2d |
⊢ ( ( 𝑥 ∈ 𝐼 ∧ 𝑅 ∈ 𝑋 ) → ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) = ( dist ‘ 𝑅 ) ) |
29 |
28 8
|
eqtr4di |
⊢ ( ( 𝑥 ∈ 𝐼 ∧ 𝑅 ∈ 𝑋 ) → ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) = 𝐸 ) |
30 |
29
|
oveqd |
⊢ ( ( 𝑥 ∈ 𝐼 ∧ 𝑅 ∈ 𝑋 ) → ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) |
31 |
30
|
ralimiaa |
⊢ ( ∀ 𝑥 ∈ 𝐼 𝑅 ∈ 𝑋 → ∀ 𝑥 ∈ 𝐼 ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) |
32 |
5 31
|
syl |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) |
33 |
|
mpteq12 |
⊢ ( ( 𝐼 = 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) → ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) ) |
34 |
26 32 33
|
syl2anc |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) ) |
35 |
25 34
|
eqtrid |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑦 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑦 ) ) ( 𝐺 ‘ 𝑦 ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) ) |
36 |
35
|
rneqd |
⊢ ( 𝜑 → ran ( 𝑦 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑦 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑦 ) ) ( 𝐺 ‘ 𝑦 ) ) ) = ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) ) |
37 |
36
|
uneq1d |
⊢ ( 𝜑 → ( ran ( 𝑦 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑦 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑦 ) ) ( 𝐺 ‘ 𝑦 ) ) ) ∪ { 0 } ) = ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) ∪ { 0 } ) ) |
38 |
37
|
supeq1d |
⊢ ( 𝜑 → sup ( ( ran ( 𝑦 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑦 ) ( dist ‘ ( ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ‘ 𝑦 ) ) ( 𝐺 ‘ 𝑦 ) ) ) ∪ { 0 } ) , ℝ* , < ) = sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) |
39 |
13 38
|
eqtrd |
⊢ ( 𝜑 → ( 𝐹 𝐷 𝐺 ) = sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) 𝐸 ( 𝐺 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) |