| Step |
Hyp |
Ref |
Expression |
| 1 |
|
stdbdmet.1 |
⊢ 𝐷 = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ if ( ( 𝑥 𝐶 𝑦 ) ≤ 𝑅 , ( 𝑥 𝐶 𝑦 ) , 𝑅 ) ) |
| 2 |
|
simp1 |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 3 |
|
xmetcl |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → ( 𝑥 𝐶 𝑦 ) ∈ ℝ* ) |
| 4 |
|
xmetge0 |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → 0 ≤ ( 𝑥 𝐶 𝑦 ) ) |
| 5 |
|
elxrge0 |
⊢ ( ( 𝑥 𝐶 𝑦 ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝑥 𝐶 𝑦 ) ∈ ℝ* ∧ 0 ≤ ( 𝑥 𝐶 𝑦 ) ) ) |
| 6 |
3 4 5
|
sylanbrc |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → ( 𝑥 𝐶 𝑦 ) ∈ ( 0 [,] +∞ ) ) |
| 7 |
6
|
3expb |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) ) → ( 𝑥 𝐶 𝑦 ) ∈ ( 0 [,] +∞ ) ) |
| 8 |
2 7
|
sylan |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) ) → ( 𝑥 𝐶 𝑦 ) ∈ ( 0 [,] +∞ ) ) |
| 9 |
|
xmetf |
⊢ ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) → 𝐶 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ) |
| 10 |
9
|
3ad2ant1 |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → 𝐶 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ) |
| 11 |
10
|
ffnd |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → 𝐶 Fn ( 𝑋 × 𝑋 ) ) |
| 12 |
|
fnov |
⊢ ( 𝐶 Fn ( 𝑋 × 𝑋 ) ↔ 𝐶 = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝑥 𝐶 𝑦 ) ) ) |
| 13 |
11 12
|
sylib |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → 𝐶 = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝑥 𝐶 𝑦 ) ) ) |
| 14 |
|
eqidd |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) = ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ) |
| 15 |
|
breq1 |
⊢ ( 𝑧 = ( 𝑥 𝐶 𝑦 ) → ( 𝑧 ≤ 𝑅 ↔ ( 𝑥 𝐶 𝑦 ) ≤ 𝑅 ) ) |
| 16 |
|
id |
⊢ ( 𝑧 = ( 𝑥 𝐶 𝑦 ) → 𝑧 = ( 𝑥 𝐶 𝑦 ) ) |
| 17 |
15 16
|
ifbieq1d |
⊢ ( 𝑧 = ( 𝑥 𝐶 𝑦 ) → if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) = if ( ( 𝑥 𝐶 𝑦 ) ≤ 𝑅 , ( 𝑥 𝐶 𝑦 ) , 𝑅 ) ) |
| 18 |
8 13 14 17
|
fmpoco |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ∘ 𝐶 ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ if ( ( 𝑥 𝐶 𝑦 ) ≤ 𝑅 , ( 𝑥 𝐶 𝑦 ) , 𝑅 ) ) ) |
| 19 |
18 1
|
eqtr4di |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ∘ 𝐶 ) = 𝐷 ) |
| 20 |
|
eliccxr |
⊢ ( 𝑧 ∈ ( 0 [,] +∞ ) → 𝑧 ∈ ℝ* ) |
| 21 |
|
simp2 |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → 𝑅 ∈ ℝ* ) |
| 22 |
|
ifcl |
⊢ ( ( 𝑧 ∈ ℝ* ∧ 𝑅 ∈ ℝ* ) → if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ∈ ℝ* ) |
| 23 |
20 21 22
|
syl2anr |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ 𝑧 ∈ ( 0 [,] +∞ ) ) → if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ∈ ℝ* ) |
| 24 |
23
|
fmpttd |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) : ( 0 [,] +∞ ) ⟶ ℝ* ) |
| 25 |
|
id |
⊢ ( 𝑎 ∈ ( 0 [,] +∞ ) → 𝑎 ∈ ( 0 [,] +∞ ) ) |
| 26 |
|
vex |
⊢ 𝑎 ∈ V |
| 27 |
|
ifexg |
⊢ ( ( 𝑎 ∈ V ∧ 𝑅 ∈ ℝ* ) → if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ∈ V ) |
| 28 |
26 21 27
|
sylancr |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ∈ V ) |
| 29 |
|
breq1 |
⊢ ( 𝑧 = 𝑎 → ( 𝑧 ≤ 𝑅 ↔ 𝑎 ≤ 𝑅 ) ) |
| 30 |
|
id |
⊢ ( 𝑧 = 𝑎 → 𝑧 = 𝑎 ) |
| 31 |
29 30
|
ifbieq1d |
⊢ ( 𝑧 = 𝑎 → if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) = if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ) |
| 32 |
|
eqid |
⊢ ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) = ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) |
| 33 |
31 32
|
fvmptg |
⊢ ( ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ∈ V ) → ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ 𝑎 ) = if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ) |
| 34 |
25 28 33
|
syl2anr |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ 𝑎 ∈ ( 0 [,] +∞ ) ) → ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ 𝑎 ) = if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ) |
| 35 |
34
|
eqeq1d |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ 𝑎 ∈ ( 0 [,] +∞ ) ) → ( ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ 𝑎 ) = 0 ↔ if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) = 0 ) ) |
| 36 |
|
eqeq1 |
⊢ ( 𝑎 = if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) → ( 𝑎 = 0 ↔ if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) = 0 ) ) |
| 37 |
36
|
bibi1d |
⊢ ( 𝑎 = if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) → ( ( 𝑎 = 0 ↔ 𝑎 = 0 ) ↔ ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) = 0 ↔ 𝑎 = 0 ) ) ) |
| 38 |
|
eqeq1 |
⊢ ( 𝑅 = if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) → ( 𝑅 = 0 ↔ if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) = 0 ) ) |
| 39 |
38
|
bibi1d |
⊢ ( 𝑅 = if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) → ( ( 𝑅 = 0 ↔ 𝑎 = 0 ) ↔ ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) = 0 ↔ 𝑎 = 0 ) ) ) |
| 40 |
|
biidd |
⊢ ( ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ 𝑎 ∈ ( 0 [,] +∞ ) ) ∧ 𝑎 ≤ 𝑅 ) → ( 𝑎 = 0 ↔ 𝑎 = 0 ) ) |
| 41 |
|
simp3 |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → 0 < 𝑅 ) |
| 42 |
41
|
gt0ne0d |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → 𝑅 ≠ 0 ) |
| 43 |
42
|
neneqd |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → ¬ 𝑅 = 0 ) |
| 44 |
43
|
ad2antrr |
⊢ ( ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ 𝑎 ∈ ( 0 [,] +∞ ) ) ∧ ¬ 𝑎 ≤ 𝑅 ) → ¬ 𝑅 = 0 ) |
| 45 |
|
0xr |
⊢ 0 ∈ ℝ* |
| 46 |
|
xrltle |
⊢ ( ( 0 ∈ ℝ* ∧ 𝑅 ∈ ℝ* ) → ( 0 < 𝑅 → 0 ≤ 𝑅 ) ) |
| 47 |
45 21 46
|
sylancr |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → ( 0 < 𝑅 → 0 ≤ 𝑅 ) ) |
| 48 |
41 47
|
mpd |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → 0 ≤ 𝑅 ) |
| 49 |
48
|
adantr |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ 𝑎 ∈ ( 0 [,] +∞ ) ) → 0 ≤ 𝑅 ) |
| 50 |
|
breq1 |
⊢ ( 𝑎 = 0 → ( 𝑎 ≤ 𝑅 ↔ 0 ≤ 𝑅 ) ) |
| 51 |
49 50
|
syl5ibrcom |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ 𝑎 ∈ ( 0 [,] +∞ ) ) → ( 𝑎 = 0 → 𝑎 ≤ 𝑅 ) ) |
| 52 |
51
|
con3dimp |
⊢ ( ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ 𝑎 ∈ ( 0 [,] +∞ ) ) ∧ ¬ 𝑎 ≤ 𝑅 ) → ¬ 𝑎 = 0 ) |
| 53 |
44 52
|
2falsed |
⊢ ( ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ 𝑎 ∈ ( 0 [,] +∞ ) ) ∧ ¬ 𝑎 ≤ 𝑅 ) → ( 𝑅 = 0 ↔ 𝑎 = 0 ) ) |
| 54 |
37 39 40 53
|
ifbothda |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ 𝑎 ∈ ( 0 [,] +∞ ) ) → ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) = 0 ↔ 𝑎 = 0 ) ) |
| 55 |
35 54
|
bitrd |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ 𝑎 ∈ ( 0 [,] +∞ ) ) → ( ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ 𝑎 ) = 0 ↔ 𝑎 = 0 ) ) |
| 56 |
|
eliccxr |
⊢ ( 𝑎 ∈ ( 0 [,] +∞ ) → 𝑎 ∈ ℝ* ) |
| 57 |
56
|
ad2antrl |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → 𝑎 ∈ ℝ* ) |
| 58 |
21
|
adantr |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → 𝑅 ∈ ℝ* ) |
| 59 |
|
xrmin1 |
⊢ ( ( 𝑎 ∈ ℝ* ∧ 𝑅 ∈ ℝ* ) → if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ 𝑎 ) |
| 60 |
57 58 59
|
syl2anc |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ 𝑎 ) |
| 61 |
57 58
|
ifcld |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ∈ ℝ* ) |
| 62 |
|
eliccxr |
⊢ ( 𝑏 ∈ ( 0 [,] +∞ ) → 𝑏 ∈ ℝ* ) |
| 63 |
62
|
ad2antll |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → 𝑏 ∈ ℝ* ) |
| 64 |
|
xrletr |
⊢ ( ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ∈ ℝ* ∧ 𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) → ( ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ 𝑎 ∧ 𝑎 ≤ 𝑏 ) → if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ 𝑏 ) ) |
| 65 |
61 57 63 64
|
syl3anc |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ 𝑎 ∧ 𝑎 ≤ 𝑏 ) → if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ 𝑏 ) ) |
| 66 |
60 65
|
mpand |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( 𝑎 ≤ 𝑏 → if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ 𝑏 ) ) |
| 67 |
|
xrmin2 |
⊢ ( ( 𝑎 ∈ ℝ* ∧ 𝑅 ∈ ℝ* ) → if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ 𝑅 ) |
| 68 |
57 58 67
|
syl2anc |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ 𝑅 ) |
| 69 |
66 68
|
jctird |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( 𝑎 ≤ 𝑏 → ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ 𝑏 ∧ if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ 𝑅 ) ) ) |
| 70 |
|
xrlemin |
⊢ ( ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ∈ ℝ* ∧ 𝑏 ∈ ℝ* ∧ 𝑅 ∈ ℝ* ) → ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ↔ ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ 𝑏 ∧ if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ 𝑅 ) ) ) |
| 71 |
61 63 58 70
|
syl3anc |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ↔ ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ 𝑏 ∧ if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ 𝑅 ) ) ) |
| 72 |
69 71
|
sylibrd |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( 𝑎 ≤ 𝑏 → if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 73 |
34
|
adantrr |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ 𝑎 ) = if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ) |
| 74 |
|
simpr |
⊢ ( ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) → 𝑏 ∈ ( 0 [,] +∞ ) ) |
| 75 |
|
vex |
⊢ 𝑏 ∈ V |
| 76 |
|
ifexg |
⊢ ( ( 𝑏 ∈ V ∧ 𝑅 ∈ ℝ* ) → if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ∈ V ) |
| 77 |
75 21 76
|
sylancr |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ∈ V ) |
| 78 |
|
breq1 |
⊢ ( 𝑧 = 𝑏 → ( 𝑧 ≤ 𝑅 ↔ 𝑏 ≤ 𝑅 ) ) |
| 79 |
|
id |
⊢ ( 𝑧 = 𝑏 → 𝑧 = 𝑏 ) |
| 80 |
78 79
|
ifbieq1d |
⊢ ( 𝑧 = 𝑏 → if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) = if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) |
| 81 |
80 32
|
fvmptg |
⊢ ( ( 𝑏 ∈ ( 0 [,] +∞ ) ∧ if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ∈ V ) → ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ 𝑏 ) = if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) |
| 82 |
74 77 81
|
syl2anr |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ 𝑏 ) = if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) |
| 83 |
73 82
|
breq12d |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ 𝑎 ) ≤ ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ 𝑏 ) ↔ if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) ≤ if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 84 |
72 83
|
sylibrd |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( 𝑎 ≤ 𝑏 → ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ 𝑎 ) ≤ ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ 𝑏 ) ) ) |
| 85 |
57 63
|
xaddcld |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( 𝑎 +e 𝑏 ) ∈ ℝ* ) |
| 86 |
|
xrmin1 |
⊢ ( ( ( 𝑎 +e 𝑏 ) ∈ ℝ* ∧ 𝑅 ∈ ℝ* ) → if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( 𝑎 +e 𝑏 ) ) |
| 87 |
85 58 86
|
syl2anc |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( 𝑎 +e 𝑏 ) ) |
| 88 |
85 58
|
ifcld |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ∈ ℝ* ) |
| 89 |
57 58
|
xaddcld |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( 𝑎 +e 𝑅 ) ∈ ℝ* ) |
| 90 |
|
xrmin2 |
⊢ ( ( ( 𝑎 +e 𝑏 ) ∈ ℝ* ∧ 𝑅 ∈ ℝ* ) → if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ 𝑅 ) |
| 91 |
85 58 90
|
syl2anc |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ 𝑅 ) |
| 92 |
|
xaddlid |
⊢ ( 𝑅 ∈ ℝ* → ( 0 +e 𝑅 ) = 𝑅 ) |
| 93 |
58 92
|
syl |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( 0 +e 𝑅 ) = 𝑅 ) |
| 94 |
45
|
a1i |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → 0 ∈ ℝ* ) |
| 95 |
|
elxrge0 |
⊢ ( 𝑎 ∈ ( 0 [,] +∞ ) ↔ ( 𝑎 ∈ ℝ* ∧ 0 ≤ 𝑎 ) ) |
| 96 |
95
|
simprbi |
⊢ ( 𝑎 ∈ ( 0 [,] +∞ ) → 0 ≤ 𝑎 ) |
| 97 |
96
|
ad2antrl |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → 0 ≤ 𝑎 ) |
| 98 |
|
xleadd1a |
⊢ ( ( ( 0 ∈ ℝ* ∧ 𝑎 ∈ ℝ* ∧ 𝑅 ∈ ℝ* ) ∧ 0 ≤ 𝑎 ) → ( 0 +e 𝑅 ) ≤ ( 𝑎 +e 𝑅 ) ) |
| 99 |
94 57 58 97 98
|
syl31anc |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( 0 +e 𝑅 ) ≤ ( 𝑎 +e 𝑅 ) ) |
| 100 |
93 99
|
eqbrtrrd |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → 𝑅 ≤ ( 𝑎 +e 𝑅 ) ) |
| 101 |
88 58 89 91 100
|
xrletrd |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( 𝑎 +e 𝑅 ) ) |
| 102 |
|
oveq2 |
⊢ ( 𝑏 = if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) → ( 𝑎 +e 𝑏 ) = ( 𝑎 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 103 |
102
|
breq2d |
⊢ ( 𝑏 = if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) → ( if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( 𝑎 +e 𝑏 ) ↔ if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( 𝑎 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) ) |
| 104 |
|
oveq2 |
⊢ ( 𝑅 = if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) → ( 𝑎 +e 𝑅 ) = ( 𝑎 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 105 |
104
|
breq2d |
⊢ ( 𝑅 = if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) → ( if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( 𝑎 +e 𝑅 ) ↔ if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( 𝑎 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) ) |
| 106 |
103 105
|
ifboth |
⊢ ( ( if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( 𝑎 +e 𝑏 ) ∧ if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( 𝑎 +e 𝑅 ) ) → if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( 𝑎 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 107 |
87 101 106
|
syl2anc |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( 𝑎 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 108 |
63 58
|
ifcld |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ∈ ℝ* ) |
| 109 |
58 108
|
xaddcld |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( 𝑅 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ∈ ℝ* ) |
| 110 |
58
|
xaddridd |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( 𝑅 +e 0 ) = 𝑅 ) |
| 111 |
|
elxrge0 |
⊢ ( 𝑏 ∈ ( 0 [,] +∞ ) ↔ ( 𝑏 ∈ ℝ* ∧ 0 ≤ 𝑏 ) ) |
| 112 |
111
|
simprbi |
⊢ ( 𝑏 ∈ ( 0 [,] +∞ ) → 0 ≤ 𝑏 ) |
| 113 |
112
|
ad2antll |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → 0 ≤ 𝑏 ) |
| 114 |
48
|
adantr |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → 0 ≤ 𝑅 ) |
| 115 |
|
breq2 |
⊢ ( 𝑏 = if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) → ( 0 ≤ 𝑏 ↔ 0 ≤ if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 116 |
|
breq2 |
⊢ ( 𝑅 = if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) → ( 0 ≤ 𝑅 ↔ 0 ≤ if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 117 |
115 116
|
ifboth |
⊢ ( ( 0 ≤ 𝑏 ∧ 0 ≤ 𝑅 ) → 0 ≤ if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) |
| 118 |
113 114 117
|
syl2anc |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → 0 ≤ if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) |
| 119 |
|
xleadd2a |
⊢ ( ( ( 0 ∈ ℝ* ∧ if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ∈ ℝ* ∧ 𝑅 ∈ ℝ* ) ∧ 0 ≤ if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) → ( 𝑅 +e 0 ) ≤ ( 𝑅 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 120 |
94 108 58 118 119
|
syl31anc |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( 𝑅 +e 0 ) ≤ ( 𝑅 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 121 |
110 120
|
eqbrtrrd |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → 𝑅 ≤ ( 𝑅 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 122 |
88 58 109 91 121
|
xrletrd |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( 𝑅 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 123 |
|
oveq1 |
⊢ ( 𝑎 = if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) → ( 𝑎 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) = ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 124 |
123
|
breq2d |
⊢ ( 𝑎 = if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) → ( if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( 𝑎 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ↔ if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) ) |
| 125 |
|
oveq1 |
⊢ ( 𝑅 = if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) → ( 𝑅 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) = ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 126 |
125
|
breq2d |
⊢ ( 𝑅 = if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) → ( if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( 𝑅 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ↔ if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) ) |
| 127 |
124 126
|
ifboth |
⊢ ( ( if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( 𝑎 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ∧ if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( 𝑅 +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) → if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 128 |
107 122 127
|
syl2anc |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ≤ ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 129 |
|
ge0xaddcl |
⊢ ( ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) → ( 𝑎 +e 𝑏 ) ∈ ( 0 [,] +∞ ) ) |
| 130 |
|
ovex |
⊢ ( 𝑎 +e 𝑏 ) ∈ V |
| 131 |
|
ifexg |
⊢ ( ( ( 𝑎 +e 𝑏 ) ∈ V ∧ 𝑅 ∈ ℝ* ) → if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ∈ V ) |
| 132 |
130 21 131
|
sylancr |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ∈ V ) |
| 133 |
|
breq1 |
⊢ ( 𝑧 = ( 𝑎 +e 𝑏 ) → ( 𝑧 ≤ 𝑅 ↔ ( 𝑎 +e 𝑏 ) ≤ 𝑅 ) ) |
| 134 |
|
id |
⊢ ( 𝑧 = ( 𝑎 +e 𝑏 ) → 𝑧 = ( 𝑎 +e 𝑏 ) ) |
| 135 |
133 134
|
ifbieq1d |
⊢ ( 𝑧 = ( 𝑎 +e 𝑏 ) → if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) = if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ) |
| 136 |
135 32
|
fvmptg |
⊢ ( ( ( 𝑎 +e 𝑏 ) ∈ ( 0 [,] +∞ ) ∧ if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ∈ V ) → ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ ( 𝑎 +e 𝑏 ) ) = if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ) |
| 137 |
129 132 136
|
syl2anr |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ ( 𝑎 +e 𝑏 ) ) = if ( ( 𝑎 +e 𝑏 ) ≤ 𝑅 , ( 𝑎 +e 𝑏 ) , 𝑅 ) ) |
| 138 |
73 82
|
oveq12d |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ 𝑎 ) +e ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ 𝑏 ) ) = ( if ( 𝑎 ≤ 𝑅 , 𝑎 , 𝑅 ) +e if ( 𝑏 ≤ 𝑅 , 𝑏 , 𝑅 ) ) ) |
| 139 |
128 137 138
|
3brtr4d |
⊢ ( ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) ∧ ( 𝑎 ∈ ( 0 [,] +∞ ) ∧ 𝑏 ∈ ( 0 [,] +∞ ) ) ) → ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ ( 𝑎 +e 𝑏 ) ) ≤ ( ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ 𝑎 ) +e ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ‘ 𝑏 ) ) ) |
| 140 |
2 24 55 84 139
|
comet |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → ( ( 𝑧 ∈ ( 0 [,] +∞ ) ↦ if ( 𝑧 ≤ 𝑅 , 𝑧 , 𝑅 ) ) ∘ 𝐶 ) ∈ ( ∞Met ‘ 𝑋 ) ) |
| 141 |
19 140
|
eqeltrrd |
⊢ ( ( 𝐶 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |