| Step |
Hyp |
Ref |
Expression |
| 1 |
|
cnre2csqlem.1 |
⊢ ( 𝐺 ↾ ( ℝ × ℝ ) ) = ( 𝐻 ∘ 𝐹 ) |
| 2 |
|
cnre2csqlem.2 |
⊢ 𝐹 Fn ( ℝ × ℝ ) |
| 3 |
|
cnre2csqlem.3 |
⊢ 𝐺 Fn V |
| 4 |
|
cnre2csqlem.4 |
⊢ ( 𝑥 ∈ ( ℝ × ℝ ) → ( 𝐺 ‘ 𝑥 ) ∈ ℝ ) |
| 5 |
|
cnre2csqlem.5 |
⊢ ( ( 𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹 ) → ( 𝐻 ‘ ( 𝑥 − 𝑦 ) ) = ( ( 𝐻 ‘ 𝑥 ) − ( 𝐻 ‘ 𝑦 ) ) ) |
| 6 |
|
ssv |
⊢ ( ℝ × ℝ ) ⊆ V |
| 7 |
|
fnssres |
⊢ ( ( 𝐺 Fn V ∧ ( ℝ × ℝ ) ⊆ V ) → ( 𝐺 ↾ ( ℝ × ℝ ) ) Fn ( ℝ × ℝ ) ) |
| 8 |
3 6 7
|
mp2an |
⊢ ( 𝐺 ↾ ( ℝ × ℝ ) ) Fn ( ℝ × ℝ ) |
| 9 |
|
elpreima |
⊢ ( ( 𝐺 ↾ ( ℝ × ℝ ) ) Fn ( ℝ × ℝ ) → ( 𝑌 ∈ ( ◡ ( 𝐺 ↾ ( ℝ × ℝ ) ) “ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) ↔ ( 𝑌 ∈ ( ℝ × ℝ ) ∧ ( ( 𝐺 ↾ ( ℝ × ℝ ) ) ‘ 𝑌 ) ∈ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) ) ) |
| 10 |
8 9
|
mp1i |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( 𝑌 ∈ ( ◡ ( 𝐺 ↾ ( ℝ × ℝ ) ) “ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) ↔ ( 𝑌 ∈ ( ℝ × ℝ ) ∧ ( ( 𝐺 ↾ ( ℝ × ℝ ) ) ‘ 𝑌 ) ∈ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) ) ) |
| 11 |
10
|
simplbda |
⊢ ( ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) ∧ 𝑌 ∈ ( ◡ ( 𝐺 ↾ ( ℝ × ℝ ) ) “ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) ) → ( ( 𝐺 ↾ ( ℝ × ℝ ) ) ‘ 𝑌 ) ∈ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) |
| 12 |
11
|
ex |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( 𝑌 ∈ ( ◡ ( 𝐺 ↾ ( ℝ × ℝ ) ) “ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) → ( ( 𝐺 ↾ ( ℝ × ℝ ) ) ‘ 𝑌 ) ∈ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) ) |
| 13 |
|
simp2 |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → 𝑌 ∈ ( ℝ × ℝ ) ) |
| 14 |
|
fvres |
⊢ ( 𝑌 ∈ ( ℝ × ℝ ) → ( ( 𝐺 ↾ ( ℝ × ℝ ) ) ‘ 𝑌 ) = ( 𝐺 ‘ 𝑌 ) ) |
| 15 |
13 14
|
syl |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( ( 𝐺 ↾ ( ℝ × ℝ ) ) ‘ 𝑌 ) = ( 𝐺 ‘ 𝑌 ) ) |
| 16 |
15
|
eleq1d |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( ( ( 𝐺 ↾ ( ℝ × ℝ ) ) ‘ 𝑌 ) ∈ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ↔ ( 𝐺 ‘ 𝑌 ) ∈ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) ) |
| 17 |
|
simp1 |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → 𝑋 ∈ ( ℝ × ℝ ) ) |
| 18 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝐺 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑋 ) ) |
| 19 |
18
|
eleq1d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝐺 ‘ 𝑥 ) ∈ ℝ ↔ ( 𝐺 ‘ 𝑋 ) ∈ ℝ ) ) |
| 20 |
19 4
|
vtoclga |
⊢ ( 𝑋 ∈ ( ℝ × ℝ ) → ( 𝐺 ‘ 𝑋 ) ∈ ℝ ) |
| 21 |
17 20
|
syl |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( 𝐺 ‘ 𝑋 ) ∈ ℝ ) |
| 22 |
|
simp3 |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → 𝐷 ∈ ℝ+ ) |
| 23 |
22
|
rpred |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → 𝐷 ∈ ℝ ) |
| 24 |
21 23
|
resubcld |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) ∈ ℝ ) |
| 25 |
24
|
rexrd |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) ∈ ℝ* ) |
| 26 |
21 23
|
readdcld |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ∈ ℝ ) |
| 27 |
26
|
rexrd |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ∈ ℝ* ) |
| 28 |
|
elioo2 |
⊢ ( ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) ∈ ℝ* ∧ ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ∈ ℝ* ) → ( ( 𝐺 ‘ 𝑌 ) ∈ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ↔ ( ( 𝐺 ‘ 𝑌 ) ∈ ℝ ∧ ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) < ( 𝐺 ‘ 𝑌 ) ∧ ( 𝐺 ‘ 𝑌 ) < ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) ) |
| 29 |
25 27 28
|
syl2anc |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( ( 𝐺 ‘ 𝑌 ) ∈ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ↔ ( ( 𝐺 ‘ 𝑌 ) ∈ ℝ ∧ ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) < ( 𝐺 ‘ 𝑌 ) ∧ ( 𝐺 ‘ 𝑌 ) < ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) ) |
| 30 |
29
|
biimpa |
⊢ ( ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) ∧ ( 𝐺 ‘ 𝑌 ) ∈ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) → ( ( 𝐺 ‘ 𝑌 ) ∈ ℝ ∧ ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) < ( 𝐺 ‘ 𝑌 ) ∧ ( 𝐺 ‘ 𝑌 ) < ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) |
| 31 |
30
|
simp2d |
⊢ ( ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) ∧ ( 𝐺 ‘ 𝑌 ) ∈ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) → ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) < ( 𝐺 ‘ 𝑌 ) ) |
| 32 |
30
|
simp3d |
⊢ ( ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) ∧ ( 𝐺 ‘ 𝑌 ) ∈ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) → ( 𝐺 ‘ 𝑌 ) < ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) |
| 33 |
31 32
|
jca |
⊢ ( ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) ∧ ( 𝐺 ‘ 𝑌 ) ∈ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) → ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) < ( 𝐺 ‘ 𝑌 ) ∧ ( 𝐺 ‘ 𝑌 ) < ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) |
| 34 |
33
|
ex |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( ( 𝐺 ‘ 𝑌 ) ∈ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) → ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) < ( 𝐺 ‘ 𝑌 ) ∧ ( 𝐺 ‘ 𝑌 ) < ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) ) |
| 35 |
16 34
|
sylbid |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( ( ( 𝐺 ↾ ( ℝ × ℝ ) ) ‘ 𝑌 ) ∈ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) → ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) < ( 𝐺 ‘ 𝑌 ) ∧ ( 𝐺 ‘ 𝑌 ) < ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) ) |
| 36 |
|
fveq2 |
⊢ ( 𝑥 = 𝑌 → ( 𝐺 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) |
| 37 |
36
|
eleq1d |
⊢ ( 𝑥 = 𝑌 → ( ( 𝐺 ‘ 𝑥 ) ∈ ℝ ↔ ( 𝐺 ‘ 𝑌 ) ∈ ℝ ) ) |
| 38 |
37 4
|
vtoclga |
⊢ ( 𝑌 ∈ ( ℝ × ℝ ) → ( 𝐺 ‘ 𝑌 ) ∈ ℝ ) |
| 39 |
13 38
|
syl |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( 𝐺 ‘ 𝑌 ) ∈ ℝ ) |
| 40 |
|
absdiflt |
⊢ ( ( ( 𝐺 ‘ 𝑌 ) ∈ ℝ ∧ ( 𝐺 ‘ 𝑋 ) ∈ ℝ ∧ 𝐷 ∈ ℝ ) → ( ( abs ‘ ( ( 𝐺 ‘ 𝑌 ) − ( 𝐺 ‘ 𝑋 ) ) ) < 𝐷 ↔ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) < ( 𝐺 ‘ 𝑌 ) ∧ ( 𝐺 ‘ 𝑌 ) < ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) ) |
| 41 |
40
|
biimprd |
⊢ ( ( ( 𝐺 ‘ 𝑌 ) ∈ ℝ ∧ ( 𝐺 ‘ 𝑋 ) ∈ ℝ ∧ 𝐷 ∈ ℝ ) → ( ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) < ( 𝐺 ‘ 𝑌 ) ∧ ( 𝐺 ‘ 𝑌 ) < ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) → ( abs ‘ ( ( 𝐺 ‘ 𝑌 ) − ( 𝐺 ‘ 𝑋 ) ) ) < 𝐷 ) ) |
| 42 |
39 21 23 41
|
syl3anc |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) < ( 𝐺 ‘ 𝑌 ) ∧ ( 𝐺 ‘ 𝑌 ) < ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) → ( abs ‘ ( ( 𝐺 ‘ 𝑌 ) − ( 𝐺 ‘ 𝑋 ) ) ) < 𝐷 ) ) |
| 43 |
12 35 42
|
3syld |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( 𝑌 ∈ ( ◡ ( 𝐺 ↾ ( ℝ × ℝ ) ) “ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) → ( abs ‘ ( ( 𝐺 ‘ 𝑌 ) − ( 𝐺 ‘ 𝑋 ) ) ) < 𝐷 ) ) |
| 44 |
|
fnfvelrn |
⊢ ( ( 𝐹 Fn ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ) → ( 𝐹 ‘ 𝑌 ) ∈ ran 𝐹 ) |
| 45 |
2 13 44
|
sylancr |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( 𝐹 ‘ 𝑌 ) ∈ ran 𝐹 ) |
| 46 |
|
fnfvelrn |
⊢ ( ( 𝐹 Fn ( ℝ × ℝ ) ∧ 𝑋 ∈ ( ℝ × ℝ ) ) → ( 𝐹 ‘ 𝑋 ) ∈ ran 𝐹 ) |
| 47 |
2 17 46
|
sylancr |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( 𝐹 ‘ 𝑋 ) ∈ ran 𝐹 ) |
| 48 |
|
fvoveq1 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑌 ) → ( 𝐻 ‘ ( 𝑥 − 𝑦 ) ) = ( 𝐻 ‘ ( ( 𝐹 ‘ 𝑌 ) − 𝑦 ) ) ) |
| 49 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑌 ) → ( 𝐻 ‘ 𝑥 ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑌 ) ) ) |
| 50 |
49
|
oveq1d |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑌 ) → ( ( 𝐻 ‘ 𝑥 ) − ( 𝐻 ‘ 𝑦 ) ) = ( ( 𝐻 ‘ ( 𝐹 ‘ 𝑌 ) ) − ( 𝐻 ‘ 𝑦 ) ) ) |
| 51 |
48 50
|
eqeq12d |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑌 ) → ( ( 𝐻 ‘ ( 𝑥 − 𝑦 ) ) = ( ( 𝐻 ‘ 𝑥 ) − ( 𝐻 ‘ 𝑦 ) ) ↔ ( 𝐻 ‘ ( ( 𝐹 ‘ 𝑌 ) − 𝑦 ) ) = ( ( 𝐻 ‘ ( 𝐹 ‘ 𝑌 ) ) − ( 𝐻 ‘ 𝑦 ) ) ) ) |
| 52 |
|
oveq2 |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑋 ) → ( ( 𝐹 ‘ 𝑌 ) − 𝑦 ) = ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) |
| 53 |
52
|
fveq2d |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑋 ) → ( 𝐻 ‘ ( ( 𝐹 ‘ 𝑌 ) − 𝑦 ) ) = ( 𝐻 ‘ ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) ) |
| 54 |
|
fveq2 |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑋 ) → ( 𝐻 ‘ 𝑦 ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑋 ) ) ) |
| 55 |
54
|
oveq2d |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑋 ) → ( ( 𝐻 ‘ ( 𝐹 ‘ 𝑌 ) ) − ( 𝐻 ‘ 𝑦 ) ) = ( ( 𝐻 ‘ ( 𝐹 ‘ 𝑌 ) ) − ( 𝐻 ‘ ( 𝐹 ‘ 𝑋 ) ) ) ) |
| 56 |
53 55
|
eqeq12d |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑋 ) → ( ( 𝐻 ‘ ( ( 𝐹 ‘ 𝑌 ) − 𝑦 ) ) = ( ( 𝐻 ‘ ( 𝐹 ‘ 𝑌 ) ) − ( 𝐻 ‘ 𝑦 ) ) ↔ ( 𝐻 ‘ ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) = ( ( 𝐻 ‘ ( 𝐹 ‘ 𝑌 ) ) − ( 𝐻 ‘ ( 𝐹 ‘ 𝑋 ) ) ) ) ) |
| 57 |
51 56 5
|
vtocl2ga |
⊢ ( ( ( 𝐹 ‘ 𝑌 ) ∈ ran 𝐹 ∧ ( 𝐹 ‘ 𝑋 ) ∈ ran 𝐹 ) → ( 𝐻 ‘ ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) = ( ( 𝐻 ‘ ( 𝐹 ‘ 𝑌 ) ) − ( 𝐻 ‘ ( 𝐹 ‘ 𝑋 ) ) ) ) |
| 58 |
45 47 57
|
syl2anc |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( 𝐻 ‘ ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) = ( ( 𝐻 ‘ ( 𝐹 ‘ 𝑌 ) ) − ( 𝐻 ‘ ( 𝐹 ‘ 𝑋 ) ) ) ) |
| 59 |
1
|
fveq1i |
⊢ ( ( 𝐺 ↾ ( ℝ × ℝ ) ) ‘ 𝑌 ) = ( ( 𝐻 ∘ 𝐹 ) ‘ 𝑌 ) |
| 60 |
|
fvco2 |
⊢ ( ( 𝐹 Fn ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ) → ( ( 𝐻 ∘ 𝐹 ) ‘ 𝑌 ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑌 ) ) ) |
| 61 |
2 13 60
|
sylancr |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( ( 𝐻 ∘ 𝐹 ) ‘ 𝑌 ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑌 ) ) ) |
| 62 |
59 15 61
|
3eqtr3a |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( 𝐺 ‘ 𝑌 ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑌 ) ) ) |
| 63 |
1
|
fveq1i |
⊢ ( ( 𝐺 ↾ ( ℝ × ℝ ) ) ‘ 𝑋 ) = ( ( 𝐻 ∘ 𝐹 ) ‘ 𝑋 ) |
| 64 |
|
fvres |
⊢ ( 𝑋 ∈ ( ℝ × ℝ ) → ( ( 𝐺 ↾ ( ℝ × ℝ ) ) ‘ 𝑋 ) = ( 𝐺 ‘ 𝑋 ) ) |
| 65 |
17 64
|
syl |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( ( 𝐺 ↾ ( ℝ × ℝ ) ) ‘ 𝑋 ) = ( 𝐺 ‘ 𝑋 ) ) |
| 66 |
|
fvco2 |
⊢ ( ( 𝐹 Fn ( ℝ × ℝ ) ∧ 𝑋 ∈ ( ℝ × ℝ ) ) → ( ( 𝐻 ∘ 𝐹 ) ‘ 𝑋 ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑋 ) ) ) |
| 67 |
2 17 66
|
sylancr |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( ( 𝐻 ∘ 𝐹 ) ‘ 𝑋 ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑋 ) ) ) |
| 68 |
63 65 67
|
3eqtr3a |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( 𝐺 ‘ 𝑋 ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑋 ) ) ) |
| 69 |
62 68
|
oveq12d |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( ( 𝐺 ‘ 𝑌 ) − ( 𝐺 ‘ 𝑋 ) ) = ( ( 𝐻 ‘ ( 𝐹 ‘ 𝑌 ) ) − ( 𝐻 ‘ ( 𝐹 ‘ 𝑋 ) ) ) ) |
| 70 |
58 69
|
eqtr4d |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( 𝐻 ‘ ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) = ( ( 𝐺 ‘ 𝑌 ) − ( 𝐺 ‘ 𝑋 ) ) ) |
| 71 |
70
|
fveq2d |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( abs ‘ ( 𝐻 ‘ ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) ) = ( abs ‘ ( ( 𝐺 ‘ 𝑌 ) − ( 𝐺 ‘ 𝑋 ) ) ) ) |
| 72 |
71
|
breq1d |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( ( abs ‘ ( 𝐻 ‘ ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) ) < 𝐷 ↔ ( abs ‘ ( ( 𝐺 ‘ 𝑌 ) − ( 𝐺 ‘ 𝑋 ) ) ) < 𝐷 ) ) |
| 73 |
43 72
|
sylibrd |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( 𝑌 ∈ ( ◡ ( 𝐺 ↾ ( ℝ × ℝ ) ) “ ( ( ( 𝐺 ‘ 𝑋 ) − 𝐷 ) (,) ( ( 𝐺 ‘ 𝑋 ) + 𝐷 ) ) ) → ( abs ‘ ( 𝐻 ‘ ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) ) < 𝐷 ) ) |