Step |
Hyp |
Ref |
Expression |
1 |
|
isbnd3 |
⊢ ( 𝑀 ∈ ( Bnd ‘ 𝑋 ) ↔ ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ ∃ 𝑥 ∈ ℝ 𝑀 : ( 𝑋 × 𝑋 ) ⟶ ( 0 [,] 𝑥 ) ) ) |
2 |
|
metf |
⊢ ( 𝑀 ∈ ( Met ‘ 𝑋 ) → 𝑀 : ( 𝑋 × 𝑋 ) ⟶ ℝ ) |
3 |
2
|
adantr |
⊢ ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑥 ∈ ℝ ) → 𝑀 : ( 𝑋 × 𝑋 ) ⟶ ℝ ) |
4 |
|
ffn |
⊢ ( 𝑀 : ( 𝑋 × 𝑋 ) ⟶ ℝ → 𝑀 Fn ( 𝑋 × 𝑋 ) ) |
5 |
|
ffnov |
⊢ ( 𝑀 : ( 𝑋 × 𝑋 ) ⟶ ( 0 [,] 𝑥 ) ↔ ( 𝑀 Fn ( 𝑋 × 𝑋 ) ∧ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑀 𝑧 ) ∈ ( 0 [,] 𝑥 ) ) ) |
6 |
5
|
baib |
⊢ ( 𝑀 Fn ( 𝑋 × 𝑋 ) → ( 𝑀 : ( 𝑋 × 𝑋 ) ⟶ ( 0 [,] 𝑥 ) ↔ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑀 𝑧 ) ∈ ( 0 [,] 𝑥 ) ) ) |
7 |
3 4 6
|
3syl |
⊢ ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑥 ∈ ℝ ) → ( 𝑀 : ( 𝑋 × 𝑋 ) ⟶ ( 0 [,] 𝑥 ) ↔ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑀 𝑧 ) ∈ ( 0 [,] 𝑥 ) ) ) |
8 |
|
0red |
⊢ ( ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ) ) → 0 ∈ ℝ ) |
9 |
|
simplr |
⊢ ( ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ) ) → 𝑥 ∈ ℝ ) |
10 |
|
metcl |
⊢ ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ) → ( 𝑦 𝑀 𝑧 ) ∈ ℝ ) |
11 |
10
|
3expb |
⊢ ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ) ) → ( 𝑦 𝑀 𝑧 ) ∈ ℝ ) |
12 |
11
|
adantlr |
⊢ ( ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ) ) → ( 𝑦 𝑀 𝑧 ) ∈ ℝ ) |
13 |
|
metge0 |
⊢ ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ) → 0 ≤ ( 𝑦 𝑀 𝑧 ) ) |
14 |
13
|
3expb |
⊢ ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ) ) → 0 ≤ ( 𝑦 𝑀 𝑧 ) ) |
15 |
14
|
adantlr |
⊢ ( ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ) ) → 0 ≤ ( 𝑦 𝑀 𝑧 ) ) |
16 |
|
elicc2 |
⊢ ( ( 0 ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( ( 𝑦 𝑀 𝑧 ) ∈ ( 0 [,] 𝑥 ) ↔ ( ( 𝑦 𝑀 𝑧 ) ∈ ℝ ∧ 0 ≤ ( 𝑦 𝑀 𝑧 ) ∧ ( 𝑦 𝑀 𝑧 ) ≤ 𝑥 ) ) ) |
17 |
|
df-3an |
⊢ ( ( ( 𝑦 𝑀 𝑧 ) ∈ ℝ ∧ 0 ≤ ( 𝑦 𝑀 𝑧 ) ∧ ( 𝑦 𝑀 𝑧 ) ≤ 𝑥 ) ↔ ( ( ( 𝑦 𝑀 𝑧 ) ∈ ℝ ∧ 0 ≤ ( 𝑦 𝑀 𝑧 ) ) ∧ ( 𝑦 𝑀 𝑧 ) ≤ 𝑥 ) ) |
18 |
16 17
|
syl6bb |
⊢ ( ( 0 ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( ( 𝑦 𝑀 𝑧 ) ∈ ( 0 [,] 𝑥 ) ↔ ( ( ( 𝑦 𝑀 𝑧 ) ∈ ℝ ∧ 0 ≤ ( 𝑦 𝑀 𝑧 ) ) ∧ ( 𝑦 𝑀 𝑧 ) ≤ 𝑥 ) ) ) |
19 |
18
|
baibd |
⊢ ( ( ( 0 ∈ ℝ ∧ 𝑥 ∈ ℝ ) ∧ ( ( 𝑦 𝑀 𝑧 ) ∈ ℝ ∧ 0 ≤ ( 𝑦 𝑀 𝑧 ) ) ) → ( ( 𝑦 𝑀 𝑧 ) ∈ ( 0 [,] 𝑥 ) ↔ ( 𝑦 𝑀 𝑧 ) ≤ 𝑥 ) ) |
20 |
8 9 12 15 19
|
syl22anc |
⊢ ( ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ) ) → ( ( 𝑦 𝑀 𝑧 ) ∈ ( 0 [,] 𝑥 ) ↔ ( 𝑦 𝑀 𝑧 ) ≤ 𝑥 ) ) |
21 |
20
|
2ralbidva |
⊢ ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑥 ∈ ℝ ) → ( ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑀 𝑧 ) ∈ ( 0 [,] 𝑥 ) ↔ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑀 𝑧 ) ≤ 𝑥 ) ) |
22 |
7 21
|
bitrd |
⊢ ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ 𝑥 ∈ ℝ ) → ( 𝑀 : ( 𝑋 × 𝑋 ) ⟶ ( 0 [,] 𝑥 ) ↔ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑀 𝑧 ) ≤ 𝑥 ) ) |
23 |
22
|
rexbidva |
⊢ ( 𝑀 ∈ ( Met ‘ 𝑋 ) → ( ∃ 𝑥 ∈ ℝ 𝑀 : ( 𝑋 × 𝑋 ) ⟶ ( 0 [,] 𝑥 ) ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑀 𝑧 ) ≤ 𝑥 ) ) |
24 |
23
|
pm5.32i |
⊢ ( ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ ∃ 𝑥 ∈ ℝ 𝑀 : ( 𝑋 × 𝑋 ) ⟶ ( 0 [,] 𝑥 ) ) ↔ ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑀 𝑧 ) ≤ 𝑥 ) ) |
25 |
1 24
|
bitri |
⊢ ( 𝑀 ∈ ( Bnd ‘ 𝑋 ) ↔ ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑀 𝑧 ) ≤ 𝑥 ) ) |