Step |
Hyp |
Ref |
Expression |
1 |
|
prdsdsf.y |
⊢ 𝑌 = ( 𝑆 Xs ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ) |
2 |
|
prdsdsf.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
3 |
|
prdsdsf.v |
⊢ 𝑉 = ( Base ‘ 𝑅 ) |
4 |
|
prdsdsf.e |
⊢ 𝐸 = ( ( dist ‘ 𝑅 ) ↾ ( 𝑉 × 𝑉 ) ) |
5 |
|
prdsdsf.d |
⊢ 𝐷 = ( dist ‘ 𝑌 ) |
6 |
|
prdsdsf.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑊 ) |
7 |
|
prdsdsf.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑋 ) |
8 |
|
prdsdsf.r |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝑅 ∈ 𝑍 ) |
9 |
|
prdsdsf.m |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐸 ∈ ( ∞Met ‘ 𝑉 ) ) |
10 |
|
nfcv |
⊢ Ⅎ 𝑦 𝑅 |
11 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝑅 |
12 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑦 → 𝑅 = ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) |
13 |
10 11 12
|
cbvmpt |
⊢ ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) = ( 𝑦 ∈ 𝐼 ↦ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) |
14 |
13
|
oveq2i |
⊢ ( 𝑆 Xs ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) ) = ( 𝑆 Xs ( 𝑦 ∈ 𝐼 ↦ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) |
15 |
1 14
|
eqtri |
⊢ 𝑌 = ( 𝑆 Xs ( 𝑦 ∈ 𝐼 ↦ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) |
16 |
|
eqid |
⊢ ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) = ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) |
17 |
|
eqid |
⊢ ( ( dist ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ↾ ( ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) × ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) ) = ( ( dist ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ↾ ( ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) × ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) ) |
18 |
8
|
elexd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝑅 ∈ V ) |
19 |
18
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 𝑅 ∈ V ) |
20 |
11
|
nfel1 |
⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ∈ V |
21 |
12
|
eleq1d |
⊢ ( 𝑥 = 𝑦 → ( 𝑅 ∈ V ↔ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ∈ V ) ) |
22 |
20 21
|
rspc |
⊢ ( 𝑦 ∈ 𝐼 → ( ∀ 𝑥 ∈ 𝐼 𝑅 ∈ V → ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ∈ V ) ) |
23 |
19 22
|
mpan9 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ∈ V ) |
24 |
9
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 𝐸 ∈ ( ∞Met ‘ 𝑉 ) ) |
25 |
|
nfcv |
⊢ Ⅎ 𝑥 dist |
26 |
25 11
|
nffv |
⊢ Ⅎ 𝑥 ( dist ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) |
27 |
|
nfcv |
⊢ Ⅎ 𝑥 Base |
28 |
27 11
|
nffv |
⊢ Ⅎ 𝑥 ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) |
29 |
28 28
|
nfxp |
⊢ Ⅎ 𝑥 ( ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) × ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) |
30 |
26 29
|
nfres |
⊢ Ⅎ 𝑥 ( ( dist ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ↾ ( ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) × ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) ) |
31 |
|
nfcv |
⊢ Ⅎ 𝑥 ∞Met |
32 |
31 28
|
nffv |
⊢ Ⅎ 𝑥 ( ∞Met ‘ ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) |
33 |
30 32
|
nfel |
⊢ Ⅎ 𝑥 ( ( dist ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ↾ ( ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) × ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) |
34 |
12
|
fveq2d |
⊢ ( 𝑥 = 𝑦 → ( dist ‘ 𝑅 ) = ( dist ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) |
35 |
12
|
fveq2d |
⊢ ( 𝑥 = 𝑦 → ( Base ‘ 𝑅 ) = ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) |
36 |
3 35
|
eqtrid |
⊢ ( 𝑥 = 𝑦 → 𝑉 = ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) |
37 |
36
|
sqxpeqd |
⊢ ( 𝑥 = 𝑦 → ( 𝑉 × 𝑉 ) = ( ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) × ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) ) |
38 |
34 37
|
reseq12d |
⊢ ( 𝑥 = 𝑦 → ( ( dist ‘ 𝑅 ) ↾ ( 𝑉 × 𝑉 ) ) = ( ( dist ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ↾ ( ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) × ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) ) ) |
39 |
4 38
|
eqtrid |
⊢ ( 𝑥 = 𝑦 → 𝐸 = ( ( dist ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ↾ ( ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) × ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) ) ) |
40 |
36
|
fveq2d |
⊢ ( 𝑥 = 𝑦 → ( ∞Met ‘ 𝑉 ) = ( ∞Met ‘ ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) ) |
41 |
39 40
|
eleq12d |
⊢ ( 𝑥 = 𝑦 → ( 𝐸 ∈ ( ∞Met ‘ 𝑉 ) ↔ ( ( dist ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ↾ ( ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) × ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) ) ) |
42 |
33 41
|
rspc |
⊢ ( 𝑦 ∈ 𝐼 → ( ∀ 𝑥 ∈ 𝐼 𝐸 ∈ ( ∞Met ‘ 𝑉 ) → ( ( dist ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ↾ ( ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) × ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) ) ) |
43 |
24 42
|
mpan9 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( ( dist ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ↾ ( ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) × ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ ⦋ 𝑦 / 𝑥 ⦌ 𝑅 ) ) ) |
44 |
15 2 16 17 5 6 7 23 43
|
prdsxmetlem |
⊢ ( 𝜑 → 𝐷 ∈ ( ∞Met ‘ 𝐵 ) ) |