Step |
Hyp |
Ref |
Expression |
1 |
|
prdsbasmpt.y |
⊢ 𝑌 = ( 𝑆 Xs 𝑅 ) |
2 |
|
prdsbasmpt.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
3 |
|
prdsbasmpt.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑉 ) |
4 |
|
prdsbasmpt.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
5 |
|
prdsbasmpt.r |
⊢ ( 𝜑 → 𝑅 Fn 𝐼 ) |
6 |
|
prdsplusgval.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
7 |
|
prdsplusgval.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐵 ) |
8 |
|
prdsdsval.d |
⊢ 𝐷 = ( dist ‘ 𝑌 ) |
9 |
|
fnex |
⊢ ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑊 ) → 𝑅 ∈ V ) |
10 |
5 4 9
|
syl2anc |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
11 |
|
fndm |
⊢ ( 𝑅 Fn 𝐼 → dom 𝑅 = 𝐼 ) |
12 |
5 11
|
syl |
⊢ ( 𝜑 → dom 𝑅 = 𝐼 ) |
13 |
1 3 10 2 12 8
|
prdsds |
⊢ ( 𝜑 → 𝐷 = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) ) |
14 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
15 |
|
fveq1 |
⊢ ( 𝑔 = 𝐺 → ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
16 |
14 15
|
oveqan12d |
⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) |
17 |
16
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) → ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) |
18 |
17
|
mpteq2dv |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) → ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ) |
19 |
18
|
rneqd |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) → ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) = ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ) |
20 |
19
|
uneq1d |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) → ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∪ { 0 } ) = ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ∪ { 0 } ) ) |
21 |
20
|
supeq1d |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) → sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) = sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) |
22 |
|
xrltso |
⊢ < Or ℝ* |
23 |
22
|
supex |
⊢ sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ∈ V |
24 |
23
|
a1i |
⊢ ( 𝜑 → sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ∈ V ) |
25 |
13 21 6 7 24
|
ovmpod |
⊢ ( 𝜑 → ( 𝐹 𝐷 𝐺 ) = sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) |