Step |
Hyp |
Ref |
Expression |
1 |
|
pstmval.1 |
⊢ ∼ = ( ~Met ‘ 𝐷 ) |
2 |
|
eqid |
⊢ ( 𝑥 ∈ ( 𝑋 / ∼ ) , 𝑦 ∈ ( 𝑋 / ∼ ) ↦ ∪ { 𝑧 ∣ ∃ 𝑎 ∈ 𝑥 ∃ 𝑏 ∈ 𝑦 𝑧 = ( 𝑎 𝐷 𝑏 ) } ) = ( 𝑥 ∈ ( 𝑋 / ∼ ) , 𝑦 ∈ ( 𝑋 / ∼ ) ↦ ∪ { 𝑧 ∣ ∃ 𝑎 ∈ 𝑥 ∃ 𝑏 ∈ 𝑦 𝑧 = ( 𝑎 𝐷 𝑏 ) } ) |
3 |
|
vex |
⊢ 𝑥 ∈ V |
4 |
|
vex |
⊢ 𝑦 ∈ V |
5 |
3 4
|
ab2rexex |
⊢ { 𝑧 ∣ ∃ 𝑎 ∈ 𝑥 ∃ 𝑏 ∈ 𝑦 𝑧 = ( 𝑎 𝐷 𝑏 ) } ∈ V |
6 |
5
|
uniex |
⊢ ∪ { 𝑧 ∣ ∃ 𝑎 ∈ 𝑥 ∃ 𝑏 ∈ 𝑦 𝑧 = ( 𝑎 𝐷 𝑏 ) } ∈ V |
7 |
2 6
|
fnmpoi |
⊢ ( 𝑥 ∈ ( 𝑋 / ∼ ) , 𝑦 ∈ ( 𝑋 / ∼ ) ↦ ∪ { 𝑧 ∣ ∃ 𝑎 ∈ 𝑥 ∃ 𝑏 ∈ 𝑦 𝑧 = ( 𝑎 𝐷 𝑏 ) } ) Fn ( ( 𝑋 / ∼ ) × ( 𝑋 / ∼ ) ) |
8 |
1
|
pstmval |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( pstoMet ‘ 𝐷 ) = ( 𝑥 ∈ ( 𝑋 / ∼ ) , 𝑦 ∈ ( 𝑋 / ∼ ) ↦ ∪ { 𝑧 ∣ ∃ 𝑎 ∈ 𝑥 ∃ 𝑏 ∈ 𝑦 𝑧 = ( 𝑎 𝐷 𝑏 ) } ) ) |
9 |
8
|
fneq1d |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( ( pstoMet ‘ 𝐷 ) Fn ( ( 𝑋 / ∼ ) × ( 𝑋 / ∼ ) ) ↔ ( 𝑥 ∈ ( 𝑋 / ∼ ) , 𝑦 ∈ ( 𝑋 / ∼ ) ↦ ∪ { 𝑧 ∣ ∃ 𝑎 ∈ 𝑥 ∃ 𝑏 ∈ 𝑦 𝑧 = ( 𝑎 𝐷 𝑏 ) } ) Fn ( ( 𝑋 / ∼ ) × ( 𝑋 / ∼ ) ) ) ) |
10 |
7 9
|
mpbiri |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( pstoMet ‘ 𝐷 ) Fn ( ( 𝑋 / ∼ ) × ( 𝑋 / ∼ ) ) ) |
11 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → 𝑥 = [ 𝑎 ] ∼ ) |
12 |
|
simpr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → 𝑦 = [ 𝑏 ] ∼ ) |
13 |
11 12
|
oveq12d |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) = ( [ 𝑎 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑏 ] ∼ ) ) |
14 |
|
simp-5l |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) |
15 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → 𝑎 ∈ 𝑋 ) |
16 |
|
simplr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → 𝑏 ∈ 𝑋 ) |
17 |
1
|
pstmfval |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) → ( [ 𝑎 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑏 ] ∼ ) = ( 𝑎 𝐷 𝑏 ) ) |
18 |
14 15 16 17
|
syl3anc |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → ( [ 𝑎 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑏 ] ∼ ) = ( 𝑎 𝐷 𝑏 ) ) |
19 |
13 18
|
eqtrd |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) = ( 𝑎 𝐷 𝑏 ) ) |
20 |
|
psmetf |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ) |
21 |
14 20
|
syl |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ) |
22 |
21 15 16
|
fovrnd |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → ( 𝑎 𝐷 𝑏 ) ∈ ℝ* ) |
23 |
19 22
|
eqeltrd |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ∈ ℝ* ) |
24 |
|
elqsi |
⊢ ( 𝑦 ∈ ( 𝑋 / ∼ ) → ∃ 𝑏 ∈ 𝑋 𝑦 = [ 𝑏 ] ∼ ) |
25 |
24
|
ad2antll |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) → ∃ 𝑏 ∈ 𝑋 𝑦 = [ 𝑏 ] ∼ ) |
26 |
25
|
ad2antrr |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) → ∃ 𝑏 ∈ 𝑋 𝑦 = [ 𝑏 ] ∼ ) |
27 |
23 26
|
r19.29a |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) → ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ∈ ℝ* ) |
28 |
|
elqsi |
⊢ ( 𝑥 ∈ ( 𝑋 / ∼ ) → ∃ 𝑎 ∈ 𝑋 𝑥 = [ 𝑎 ] ∼ ) |
29 |
28
|
ad2antrl |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) → ∃ 𝑎 ∈ 𝑋 𝑥 = [ 𝑎 ] ∼ ) |
30 |
27 29
|
r19.29a |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) → ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ∈ ℝ* ) |
31 |
30
|
ralrimivva |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ∀ 𝑥 ∈ ( 𝑋 / ∼ ) ∀ 𝑦 ∈ ( 𝑋 / ∼ ) ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ∈ ℝ* ) |
32 |
|
ffnov |
⊢ ( ( pstoMet ‘ 𝐷 ) : ( ( 𝑋 / ∼ ) × ( 𝑋 / ∼ ) ) ⟶ ℝ* ↔ ( ( pstoMet ‘ 𝐷 ) Fn ( ( 𝑋 / ∼ ) × ( 𝑋 / ∼ ) ) ∧ ∀ 𝑥 ∈ ( 𝑋 / ∼ ) ∀ 𝑦 ∈ ( 𝑋 / ∼ ) ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ∈ ℝ* ) ) |
33 |
10 31 32
|
sylanbrc |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( pstoMet ‘ 𝐷 ) : ( ( 𝑋 / ∼ ) × ( 𝑋 / ∼ ) ) ⟶ ℝ* ) |
34 |
17
|
3expa |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑏 ∈ 𝑋 ) → ( [ 𝑎 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑏 ] ∼ ) = ( 𝑎 𝐷 𝑏 ) ) |
35 |
34
|
eqeq1d |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑏 ∈ 𝑋 ) → ( ( [ 𝑎 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑏 ] ∼ ) = 0 ↔ ( 𝑎 𝐷 𝑏 ) = 0 ) ) |
36 |
1
|
breqi |
⊢ ( 𝑎 ∼ 𝑏 ↔ 𝑎 ( ~Met ‘ 𝐷 ) 𝑏 ) |
37 |
|
metidv |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → ( 𝑎 ( ~Met ‘ 𝐷 ) 𝑏 ↔ ( 𝑎 𝐷 𝑏 ) = 0 ) ) |
38 |
37
|
anassrs |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑏 ∈ 𝑋 ) → ( 𝑎 ( ~Met ‘ 𝐷 ) 𝑏 ↔ ( 𝑎 𝐷 𝑏 ) = 0 ) ) |
39 |
36 38
|
syl5bb |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑏 ∈ 𝑋 ) → ( 𝑎 ∼ 𝑏 ↔ ( 𝑎 𝐷 𝑏 ) = 0 ) ) |
40 |
|
metider |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( ~Met ‘ 𝐷 ) Er 𝑋 ) |
41 |
40
|
ad2antrr |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑏 ∈ 𝑋 ) → ( ~Met ‘ 𝐷 ) Er 𝑋 ) |
42 |
|
ereq1 |
⊢ ( ∼ = ( ~Met ‘ 𝐷 ) → ( ∼ Er 𝑋 ↔ ( ~Met ‘ 𝐷 ) Er 𝑋 ) ) |
43 |
1 42
|
ax-mp |
⊢ ( ∼ Er 𝑋 ↔ ( ~Met ‘ 𝐷 ) Er 𝑋 ) |
44 |
41 43
|
sylibr |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑏 ∈ 𝑋 ) → ∼ Er 𝑋 ) |
45 |
|
simplr |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑏 ∈ 𝑋 ) → 𝑎 ∈ 𝑋 ) |
46 |
44 45
|
erth |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑏 ∈ 𝑋 ) → ( 𝑎 ∼ 𝑏 ↔ [ 𝑎 ] ∼ = [ 𝑏 ] ∼ ) ) |
47 |
35 39 46
|
3bitr2d |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑏 ∈ 𝑋 ) → ( ( [ 𝑎 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑏 ] ∼ ) = 0 ↔ [ 𝑎 ] ∼ = [ 𝑏 ] ∼ ) ) |
48 |
47
|
adantllr |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑏 ∈ 𝑋 ) → ( ( [ 𝑎 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑏 ] ∼ ) = 0 ↔ [ 𝑎 ] ∼ = [ 𝑏 ] ∼ ) ) |
49 |
48
|
adantlr |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) → ( ( [ 𝑎 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑏 ] ∼ ) = 0 ↔ [ 𝑎 ] ∼ = [ 𝑏 ] ∼ ) ) |
50 |
49
|
adantr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → ( ( [ 𝑎 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑏 ] ∼ ) = 0 ↔ [ 𝑎 ] ∼ = [ 𝑏 ] ∼ ) ) |
51 |
13
|
eqeq1d |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → ( ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) = 0 ↔ ( [ 𝑎 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑏 ] ∼ ) = 0 ) ) |
52 |
11 12
|
eqeq12d |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → ( 𝑥 = 𝑦 ↔ [ 𝑎 ] ∼ = [ 𝑏 ] ∼ ) ) |
53 |
50 51 52
|
3bitr4d |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → ( ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ) |
54 |
53 26
|
r19.29a |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) → ( ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ) |
55 |
54 29
|
r19.29a |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) → ( ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ) |
56 |
|
simp-6l |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) |
57 |
|
simplr |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → 𝑐 ∈ 𝑋 ) |
58 |
|
simp-6r |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → 𝑎 ∈ 𝑋 ) |
59 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → 𝑏 ∈ 𝑋 ) |
60 |
|
psmettri2 |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → ( 𝑎 𝐷 𝑏 ) ≤ ( ( 𝑐 𝐷 𝑎 ) +𝑒 ( 𝑐 𝐷 𝑏 ) ) ) |
61 |
56 57 58 59 60
|
syl13anc |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → ( 𝑎 𝐷 𝑏 ) ≤ ( ( 𝑐 𝐷 𝑎 ) +𝑒 ( 𝑐 𝐷 𝑏 ) ) ) |
62 |
|
simp-5r |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → 𝑥 = [ 𝑎 ] ∼ ) |
63 |
|
simpllr |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → 𝑦 = [ 𝑏 ] ∼ ) |
64 |
62 63
|
oveq12d |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) = ( [ 𝑎 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑏 ] ∼ ) ) |
65 |
56 58 59 17
|
syl3anc |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → ( [ 𝑎 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑏 ] ∼ ) = ( 𝑎 𝐷 𝑏 ) ) |
66 |
64 65
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) = ( 𝑎 𝐷 𝑏 ) ) |
67 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → 𝑧 = [ 𝑐 ] ∼ ) |
68 |
67 62
|
oveq12d |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑥 ) = ( [ 𝑐 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑎 ] ∼ ) ) |
69 |
1
|
pstmfval |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ) → ( [ 𝑐 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑎 ] ∼ ) = ( 𝑐 𝐷 𝑎 ) ) |
70 |
56 57 58 69
|
syl3anc |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → ( [ 𝑐 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑎 ] ∼ ) = ( 𝑐 𝐷 𝑎 ) ) |
71 |
68 70
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑥 ) = ( 𝑐 𝐷 𝑎 ) ) |
72 |
67 63
|
oveq12d |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑦 ) = ( [ 𝑐 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑏 ] ∼ ) ) |
73 |
1
|
pstmfval |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑐 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) → ( [ 𝑐 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑏 ] ∼ ) = ( 𝑐 𝐷 𝑏 ) ) |
74 |
56 57 59 73
|
syl3anc |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → ( [ 𝑐 ] ∼ ( pstoMet ‘ 𝐷 ) [ 𝑏 ] ∼ ) = ( 𝑐 𝐷 𝑏 ) ) |
75 |
72 74
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑦 ) = ( 𝑐 𝐷 𝑏 ) ) |
76 |
71 75
|
oveq12d |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → ( ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑥 ) +𝑒 ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑦 ) ) = ( ( 𝑐 𝐷 𝑎 ) +𝑒 ( 𝑐 𝐷 𝑏 ) ) ) |
77 |
61 66 76
|
3brtr4d |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ≤ ( ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑥 ) +𝑒 ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑦 ) ) ) |
78 |
77
|
adantl6r |
⊢ ( ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) ∧ 𝑐 ∈ 𝑋 ) ∧ 𝑧 = [ 𝑐 ] ∼ ) → ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ≤ ( ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑥 ) +𝑒 ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑦 ) ) ) |
79 |
|
elqsi |
⊢ ( 𝑧 ∈ ( 𝑋 / ∼ ) → ∃ 𝑐 ∈ 𝑋 𝑧 = [ 𝑐 ] ∼ ) |
80 |
79
|
ad5antlr |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → ∃ 𝑐 ∈ 𝑋 𝑧 = [ 𝑐 ] ∼ ) |
81 |
78 80
|
r19.29a |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑧 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ≤ ( ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑥 ) +𝑒 ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑦 ) ) ) |
82 |
81
|
adantl5r |
⊢ ( ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑧 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) ∧ 𝑏 ∈ 𝑋 ) ∧ 𝑦 = [ 𝑏 ] ∼ ) → ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ≤ ( ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑥 ) +𝑒 ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑦 ) ) ) |
83 |
24
|
ad4antlr |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑧 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) → ∃ 𝑏 ∈ 𝑋 𝑦 = [ 𝑏 ] ∼ ) |
84 |
82 83
|
r19.29a |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑧 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) → ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ≤ ( ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑥 ) +𝑒 ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑦 ) ) ) |
85 |
84
|
adantl4r |
⊢ ( ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑥 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑧 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑎 ∈ 𝑋 ) ∧ 𝑥 = [ 𝑎 ] ∼ ) → ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ≤ ( ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑥 ) +𝑒 ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑦 ) ) ) |
86 |
28
|
ad3antlr |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑥 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑧 ∈ ( 𝑋 / ∼ ) ) → ∃ 𝑎 ∈ 𝑋 𝑥 = [ 𝑎 ] ∼ ) |
87 |
85 86
|
r19.29a |
⊢ ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑥 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑧 ∈ ( 𝑋 / ∼ ) ) → ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ≤ ( ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑥 ) +𝑒 ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑦 ) ) ) |
88 |
87
|
ralrimiva |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑥 ∈ ( 𝑋 / ∼ ) ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) → ∀ 𝑧 ∈ ( 𝑋 / ∼ ) ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ≤ ( ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑥 ) +𝑒 ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑦 ) ) ) |
89 |
88
|
anasss |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) → ∀ 𝑧 ∈ ( 𝑋 / ∼ ) ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ≤ ( ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑥 ) +𝑒 ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑦 ) ) ) |
90 |
55 89
|
jca |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( 𝑋 / ∼ ) ∧ 𝑦 ∈ ( 𝑋 / ∼ ) ) ) → ( ( ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ∧ ∀ 𝑧 ∈ ( 𝑋 / ∼ ) ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ≤ ( ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑥 ) +𝑒 ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑦 ) ) ) ) |
91 |
90
|
ralrimivva |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ∀ 𝑥 ∈ ( 𝑋 / ∼ ) ∀ 𝑦 ∈ ( 𝑋 / ∼ ) ( ( ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ∧ ∀ 𝑧 ∈ ( 𝑋 / ∼ ) ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ≤ ( ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑥 ) +𝑒 ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑦 ) ) ) ) |
92 |
|
elfvex |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → 𝑋 ∈ V ) |
93 |
|
qsexg |
⊢ ( 𝑋 ∈ V → ( 𝑋 / ∼ ) ∈ V ) |
94 |
|
isxmet |
⊢ ( ( 𝑋 / ∼ ) ∈ V → ( ( pstoMet ‘ 𝐷 ) ∈ ( ∞Met ‘ ( 𝑋 / ∼ ) ) ↔ ( ( pstoMet ‘ 𝐷 ) : ( ( 𝑋 / ∼ ) × ( 𝑋 / ∼ ) ) ⟶ ℝ* ∧ ∀ 𝑥 ∈ ( 𝑋 / ∼ ) ∀ 𝑦 ∈ ( 𝑋 / ∼ ) ( ( ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ∧ ∀ 𝑧 ∈ ( 𝑋 / ∼ ) ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ≤ ( ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑥 ) +𝑒 ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑦 ) ) ) ) ) ) |
95 |
92 93 94
|
3syl |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( ( pstoMet ‘ 𝐷 ) ∈ ( ∞Met ‘ ( 𝑋 / ∼ ) ) ↔ ( ( pstoMet ‘ 𝐷 ) : ( ( 𝑋 / ∼ ) × ( 𝑋 / ∼ ) ) ⟶ ℝ* ∧ ∀ 𝑥 ∈ ( 𝑋 / ∼ ) ∀ 𝑦 ∈ ( 𝑋 / ∼ ) ( ( ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ∧ ∀ 𝑧 ∈ ( 𝑋 / ∼ ) ( 𝑥 ( pstoMet ‘ 𝐷 ) 𝑦 ) ≤ ( ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑥 ) +𝑒 ( 𝑧 ( pstoMet ‘ 𝐷 ) 𝑦 ) ) ) ) ) ) |
96 |
33 91 95
|
mpbir2and |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( pstoMet ‘ 𝐷 ) ∈ ( ∞Met ‘ ( 𝑋 / ∼ ) ) ) |