| Step |
Hyp |
Ref |
Expression |
| 1 |
|
simpl |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) |
| 2 |
|
metidss |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( ~Met ‘ 𝐷 ) ⊆ ( 𝑋 × 𝑋 ) ) |
| 3 |
|
dmss |
⊢ ( ( ~Met ‘ 𝐷 ) ⊆ ( 𝑋 × 𝑋 ) → dom ( ~Met ‘ 𝐷 ) ⊆ dom ( 𝑋 × 𝑋 ) ) |
| 4 |
2 3
|
syl |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → dom ( ~Met ‘ 𝐷 ) ⊆ dom ( 𝑋 × 𝑋 ) ) |
| 5 |
|
dmxpid |
⊢ dom ( 𝑋 × 𝑋 ) = 𝑋 |
| 6 |
4 5
|
sseqtrdi |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → dom ( ~Met ‘ 𝐷 ) ⊆ 𝑋 ) |
| 7 |
1 6
|
syl |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → dom ( ~Met ‘ 𝐷 ) ⊆ 𝑋 ) |
| 8 |
|
xpss |
⊢ ( 𝑋 × 𝑋 ) ⊆ ( V × V ) |
| 9 |
2 8
|
sstrdi |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( ~Met ‘ 𝐷 ) ⊆ ( V × V ) ) |
| 10 |
|
df-rel |
⊢ ( Rel ( ~Met ‘ 𝐷 ) ↔ ( ~Met ‘ 𝐷 ) ⊆ ( V × V ) ) |
| 11 |
9 10
|
sylibr |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → Rel ( ~Met ‘ 𝐷 ) ) |
| 12 |
1 11
|
syl |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → Rel ( ~Met ‘ 𝐷 ) ) |
| 13 |
|
simprl |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ) |
| 14 |
|
releldm |
⊢ ( ( Rel ( ~Met ‘ 𝐷 ) ∧ 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ) → 𝐴 ∈ dom ( ~Met ‘ 𝐷 ) ) |
| 15 |
12 13 14
|
syl2anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → 𝐴 ∈ dom ( ~Met ‘ 𝐷 ) ) |
| 16 |
7 15
|
sseldd |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → 𝐴 ∈ 𝑋 ) |
| 17 |
|
simprr |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) |
| 18 |
|
releldm |
⊢ ( ( Rel ( ~Met ‘ 𝐷 ) ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) → 𝐸 ∈ dom ( ~Met ‘ 𝐷 ) ) |
| 19 |
12 17 18
|
syl2anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → 𝐸 ∈ dom ( ~Met ‘ 𝐷 ) ) |
| 20 |
7 19
|
sseldd |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → 𝐸 ∈ 𝑋 ) |
| 21 |
|
psmetsym |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐸 ) = ( 𝐸 𝐷 𝐴 ) ) |
| 22 |
1 16 20 21
|
syl3anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐴 𝐷 𝐸 ) = ( 𝐸 𝐷 𝐴 ) ) |
| 23 |
|
psmetf |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ) |
| 24 |
23
|
fovrnda |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐸 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( 𝐸 𝐷 𝐴 ) ∈ ℝ* ) |
| 25 |
1 20 16 24
|
syl12anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐸 𝐷 𝐴 ) ∈ ℝ* ) |
| 26 |
22 25
|
eqeltrd |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐴 𝐷 𝐸 ) ∈ ℝ* ) |
| 27 |
|
rnss |
⊢ ( ( ~Met ‘ 𝐷 ) ⊆ ( 𝑋 × 𝑋 ) → ran ( ~Met ‘ 𝐷 ) ⊆ ran ( 𝑋 × 𝑋 ) ) |
| 28 |
2 27
|
syl |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ran ( ~Met ‘ 𝐷 ) ⊆ ran ( 𝑋 × 𝑋 ) ) |
| 29 |
|
rnxpid |
⊢ ran ( 𝑋 × 𝑋 ) = 𝑋 |
| 30 |
28 29
|
sseqtrdi |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ran ( ~Met ‘ 𝐷 ) ⊆ 𝑋 ) |
| 31 |
1 30
|
syl |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ran ( ~Met ‘ 𝐷 ) ⊆ 𝑋 ) |
| 32 |
|
relelrn |
⊢ ( ( Rel ( ~Met ‘ 𝐷 ) ∧ 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ) → 𝐵 ∈ ran ( ~Met ‘ 𝐷 ) ) |
| 33 |
12 13 32
|
syl2anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → 𝐵 ∈ ran ( ~Met ‘ 𝐷 ) ) |
| 34 |
31 33
|
sseldd |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → 𝐵 ∈ 𝑋 ) |
| 35 |
23
|
fovrnda |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐵 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋 ) ) → ( 𝐵 𝐷 𝐸 ) ∈ ℝ* ) |
| 36 |
1 34 20 35
|
syl12anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐵 𝐷 𝐸 ) ∈ ℝ* ) |
| 37 |
|
relelrn |
⊢ ( ( Rel ( ~Met ‘ 𝐷 ) ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) → 𝐹 ∈ ran ( ~Met ‘ 𝐷 ) ) |
| 38 |
12 17 37
|
syl2anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → 𝐹 ∈ ran ( ~Met ‘ 𝐷 ) ) |
| 39 |
31 38
|
sseldd |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → 𝐹 ∈ 𝑋 ) |
| 40 |
|
psmetsym |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐹 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐹 𝐷 𝐵 ) = ( 𝐵 𝐷 𝐹 ) ) |
| 41 |
1 39 34 40
|
syl3anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐹 𝐷 𝐵 ) = ( 𝐵 𝐷 𝐹 ) ) |
| 42 |
23
|
fovrnda |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐹 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐹 𝐷 𝐵 ) ∈ ℝ* ) |
| 43 |
1 39 34 42
|
syl12anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐹 𝐷 𝐵 ) ∈ ℝ* ) |
| 44 |
41 43
|
eqeltrrd |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐵 𝐷 𝐹 ) ∈ ℝ* ) |
| 45 |
|
psmettri2 |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋 ) ) → ( 𝐴 𝐷 𝐸 ) ≤ ( ( 𝐵 𝐷 𝐴 ) +𝑒 ( 𝐵 𝐷 𝐸 ) ) ) |
| 46 |
1 34 16 20 45
|
syl13anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐴 𝐷 𝐸 ) ≤ ( ( 𝐵 𝐷 𝐴 ) +𝑒 ( 𝐵 𝐷 𝐸 ) ) ) |
| 47 |
|
psmetsym |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝐵 𝐷 𝐴 ) ) |
| 48 |
1 16 34 47
|
syl3anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐴 𝐷 𝐵 ) = ( 𝐵 𝐷 𝐴 ) ) |
| 49 |
16 34
|
jca |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) |
| 50 |
|
metidv |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ↔ ( 𝐴 𝐷 𝐵 ) = 0 ) ) |
| 51 |
50
|
biimpa |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) ∧ 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ) → ( 𝐴 𝐷 𝐵 ) = 0 ) |
| 52 |
1 49 13 51
|
syl21anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐴 𝐷 𝐵 ) = 0 ) |
| 53 |
48 52
|
eqtr3d |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐵 𝐷 𝐴 ) = 0 ) |
| 54 |
53
|
oveq1d |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( ( 𝐵 𝐷 𝐴 ) +𝑒 ( 𝐵 𝐷 𝐸 ) ) = ( 0 +𝑒 ( 𝐵 𝐷 𝐸 ) ) ) |
| 55 |
|
xaddid2 |
⊢ ( ( 𝐵 𝐷 𝐸 ) ∈ ℝ* → ( 0 +𝑒 ( 𝐵 𝐷 𝐸 ) ) = ( 𝐵 𝐷 𝐸 ) ) |
| 56 |
36 55
|
syl |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 0 +𝑒 ( 𝐵 𝐷 𝐸 ) ) = ( 𝐵 𝐷 𝐸 ) ) |
| 57 |
54 56
|
eqtrd |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( ( 𝐵 𝐷 𝐴 ) +𝑒 ( 𝐵 𝐷 𝐸 ) ) = ( 𝐵 𝐷 𝐸 ) ) |
| 58 |
46 57
|
breqtrd |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐴 𝐷 𝐸 ) ≤ ( 𝐵 𝐷 𝐸 ) ) |
| 59 |
|
psmettri2 |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐹 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋 ) ) → ( 𝐵 𝐷 𝐸 ) ≤ ( ( 𝐹 𝐷 𝐵 ) +𝑒 ( 𝐹 𝐷 𝐸 ) ) ) |
| 60 |
1 39 34 20 59
|
syl13anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐵 𝐷 𝐸 ) ≤ ( ( 𝐹 𝐷 𝐵 ) +𝑒 ( 𝐹 𝐷 𝐸 ) ) ) |
| 61 |
|
psmetsym |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐹 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋 ) → ( 𝐹 𝐷 𝐸 ) = ( 𝐸 𝐷 𝐹 ) ) |
| 62 |
1 39 20 61
|
syl3anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐹 𝐷 𝐸 ) = ( 𝐸 𝐷 𝐹 ) ) |
| 63 |
20 39
|
jca |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋 ) ) |
| 64 |
|
metidv |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋 ) ) → ( 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ↔ ( 𝐸 𝐷 𝐹 ) = 0 ) ) |
| 65 |
64
|
biimpa |
⊢ ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐸 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋 ) ) ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) → ( 𝐸 𝐷 𝐹 ) = 0 ) |
| 66 |
1 63 17 65
|
syl21anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐸 𝐷 𝐹 ) = 0 ) |
| 67 |
62 66
|
eqtrd |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐹 𝐷 𝐸 ) = 0 ) |
| 68 |
67
|
oveq2d |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( ( 𝐹 𝐷 𝐵 ) +𝑒 ( 𝐹 𝐷 𝐸 ) ) = ( ( 𝐹 𝐷 𝐵 ) +𝑒 0 ) ) |
| 69 |
|
xaddid1 |
⊢ ( ( 𝐹 𝐷 𝐵 ) ∈ ℝ* → ( ( 𝐹 𝐷 𝐵 ) +𝑒 0 ) = ( 𝐹 𝐷 𝐵 ) ) |
| 70 |
43 69
|
syl |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( ( 𝐹 𝐷 𝐵 ) +𝑒 0 ) = ( 𝐹 𝐷 𝐵 ) ) |
| 71 |
68 70 41
|
3eqtrd |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( ( 𝐹 𝐷 𝐵 ) +𝑒 ( 𝐹 𝐷 𝐸 ) ) = ( 𝐵 𝐷 𝐹 ) ) |
| 72 |
60 71
|
breqtrd |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐵 𝐷 𝐸 ) ≤ ( 𝐵 𝐷 𝐹 ) ) |
| 73 |
26 36 44 58 72
|
xrletrd |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐴 𝐷 𝐸 ) ≤ ( 𝐵 𝐷 𝐹 ) ) |
| 74 |
23
|
fovrnda |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋 ) ) → ( 𝐴 𝐷 𝐹 ) ∈ ℝ* ) |
| 75 |
1 16 39 74
|
syl12anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐴 𝐷 𝐹 ) ∈ ℝ* ) |
| 76 |
|
psmettri2 |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋 ) ) → ( 𝐵 𝐷 𝐹 ) ≤ ( ( 𝐴 𝐷 𝐵 ) +𝑒 ( 𝐴 𝐷 𝐹 ) ) ) |
| 77 |
1 16 34 39 76
|
syl13anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐵 𝐷 𝐹 ) ≤ ( ( 𝐴 𝐷 𝐵 ) +𝑒 ( 𝐴 𝐷 𝐹 ) ) ) |
| 78 |
52
|
oveq1d |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( ( 𝐴 𝐷 𝐵 ) +𝑒 ( 𝐴 𝐷 𝐹 ) ) = ( 0 +𝑒 ( 𝐴 𝐷 𝐹 ) ) ) |
| 79 |
|
xaddid2 |
⊢ ( ( 𝐴 𝐷 𝐹 ) ∈ ℝ* → ( 0 +𝑒 ( 𝐴 𝐷 𝐹 ) ) = ( 𝐴 𝐷 𝐹 ) ) |
| 80 |
75 79
|
syl |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 0 +𝑒 ( 𝐴 𝐷 𝐹 ) ) = ( 𝐴 𝐷 𝐹 ) ) |
| 81 |
78 80
|
eqtrd |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( ( 𝐴 𝐷 𝐵 ) +𝑒 ( 𝐴 𝐷 𝐹 ) ) = ( 𝐴 𝐷 𝐹 ) ) |
| 82 |
77 81
|
breqtrd |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐵 𝐷 𝐹 ) ≤ ( 𝐴 𝐷 𝐹 ) ) |
| 83 |
|
psmettri2 |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐸 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋 ) ) → ( 𝐴 𝐷 𝐹 ) ≤ ( ( 𝐸 𝐷 𝐴 ) +𝑒 ( 𝐸 𝐷 𝐹 ) ) ) |
| 84 |
1 20 16 39 83
|
syl13anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐴 𝐷 𝐹 ) ≤ ( ( 𝐸 𝐷 𝐴 ) +𝑒 ( 𝐸 𝐷 𝐹 ) ) ) |
| 85 |
|
xaddid1 |
⊢ ( ( 𝐸 𝐷 𝐴 ) ∈ ℝ* → ( ( 𝐸 𝐷 𝐴 ) +𝑒 0 ) = ( 𝐸 𝐷 𝐴 ) ) |
| 86 |
25 85
|
syl |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( ( 𝐸 𝐷 𝐴 ) +𝑒 0 ) = ( 𝐸 𝐷 𝐴 ) ) |
| 87 |
66
|
oveq2d |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( ( 𝐸 𝐷 𝐴 ) +𝑒 ( 𝐸 𝐷 𝐹 ) ) = ( ( 𝐸 𝐷 𝐴 ) +𝑒 0 ) ) |
| 88 |
86 87 22
|
3eqtr4d |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( ( 𝐸 𝐷 𝐴 ) +𝑒 ( 𝐸 𝐷 𝐹 ) ) = ( 𝐴 𝐷 𝐸 ) ) |
| 89 |
84 88
|
breqtrd |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐴 𝐷 𝐹 ) ≤ ( 𝐴 𝐷 𝐸 ) ) |
| 90 |
44 75 26 82 89
|
xrletrd |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐵 𝐷 𝐹 ) ≤ ( 𝐴 𝐷 𝐸 ) ) |
| 91 |
|
xrletri3 |
⊢ ( ( ( 𝐴 𝐷 𝐸 ) ∈ ℝ* ∧ ( 𝐵 𝐷 𝐹 ) ∈ ℝ* ) → ( ( 𝐴 𝐷 𝐸 ) = ( 𝐵 𝐷 𝐹 ) ↔ ( ( 𝐴 𝐷 𝐸 ) ≤ ( 𝐵 𝐷 𝐹 ) ∧ ( 𝐵 𝐷 𝐹 ) ≤ ( 𝐴 𝐷 𝐸 ) ) ) ) |
| 92 |
26 44 91
|
syl2anc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( ( 𝐴 𝐷 𝐸 ) = ( 𝐵 𝐷 𝐹 ) ↔ ( ( 𝐴 𝐷 𝐸 ) ≤ ( 𝐵 𝐷 𝐹 ) ∧ ( 𝐵 𝐷 𝐹 ) ≤ ( 𝐴 𝐷 𝐸 ) ) ) ) |
| 93 |
73 90 92
|
mpbir2and |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝐴 ( ~Met ‘ 𝐷 ) 𝐵 ∧ 𝐸 ( ~Met ‘ 𝐷 ) 𝐹 ) ) → ( 𝐴 𝐷 𝐸 ) = ( 𝐵 𝐷 𝐹 ) ) |