Step |
Hyp |
Ref |
Expression |
1 |
|
dvcnvre.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 –cn→ ℝ ) ) |
2 |
|
dvcnvre.d |
⊢ ( 𝜑 → dom ( ℝ D 𝐹 ) = 𝑋 ) |
3 |
|
dvcnvre.z |
⊢ ( 𝜑 → ¬ 0 ∈ ran ( ℝ D 𝐹 ) ) |
4 |
|
dvcnvre.1 |
⊢ ( 𝜑 → 𝐹 : 𝑋 –1-1-onto→ 𝑌 ) |
5 |
|
dvcnvre.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑋 ) |
6 |
|
dvcnvre.r |
⊢ ( 𝜑 → 𝑅 ∈ ℝ+ ) |
7 |
|
dvcnvre.s |
⊢ ( 𝜑 → ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ⊆ 𝑋 ) |
8 |
|
dvbsss |
⊢ dom ( ℝ D 𝐹 ) ⊆ ℝ |
9 |
2 8
|
eqsstrrdi |
⊢ ( 𝜑 → 𝑋 ⊆ ℝ ) |
10 |
9 5
|
sseldd |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
11 |
6
|
rpred |
⊢ ( 𝜑 → 𝑅 ∈ ℝ ) |
12 |
10 11
|
resubcld |
⊢ ( 𝜑 → ( 𝐶 − 𝑅 ) ∈ ℝ ) |
13 |
10 11
|
readdcld |
⊢ ( 𝜑 → ( 𝐶 + 𝑅 ) ∈ ℝ ) |
14 |
10 6
|
ltsubrpd |
⊢ ( 𝜑 → ( 𝐶 − 𝑅 ) < 𝐶 ) |
15 |
10 6
|
ltaddrpd |
⊢ ( 𝜑 → 𝐶 < ( 𝐶 + 𝑅 ) ) |
16 |
12 10 13 14 15
|
lttrd |
⊢ ( 𝜑 → ( 𝐶 − 𝑅 ) < ( 𝐶 + 𝑅 ) ) |
17 |
12 13 16
|
ltled |
⊢ ( 𝜑 → ( 𝐶 − 𝑅 ) ≤ ( 𝐶 + 𝑅 ) ) |
18 |
|
rescncf |
⊢ ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ⊆ 𝑋 → ( 𝐹 ∈ ( 𝑋 –cn→ ℝ ) → ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ∈ ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) –cn→ ℝ ) ) ) |
19 |
7 1 18
|
sylc |
⊢ ( 𝜑 → ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ∈ ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) –cn→ ℝ ) ) |
20 |
12 13 17 19
|
evthicc2 |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) |
21 |
|
cncff |
⊢ ( 𝐹 ∈ ( 𝑋 –cn→ ℝ ) → 𝐹 : 𝑋 ⟶ ℝ ) |
22 |
1 21
|
syl |
⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ℝ ) |
23 |
22 5
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐶 ) ∈ ℝ ) |
24 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( 𝐹 ‘ 𝐶 ) ∈ ℝ ) |
25 |
12
|
rexrd |
⊢ ( 𝜑 → ( 𝐶 − 𝑅 ) ∈ ℝ* ) |
26 |
13
|
rexrd |
⊢ ( 𝜑 → ( 𝐶 + 𝑅 ) ∈ ℝ* ) |
27 |
|
lbicc2 |
⊢ ( ( ( 𝐶 − 𝑅 ) ∈ ℝ* ∧ ( 𝐶 + 𝑅 ) ∈ ℝ* ∧ ( 𝐶 − 𝑅 ) ≤ ( 𝐶 + 𝑅 ) ) → ( 𝐶 − 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) |
28 |
25 26 17 27
|
syl3anc |
⊢ ( 𝜑 → ( 𝐶 − 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) |
29 |
28
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( 𝐶 − 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) |
30 |
12 10 14
|
ltled |
⊢ ( 𝜑 → ( 𝐶 − 𝑅 ) ≤ 𝐶 ) |
31 |
10 13 15
|
ltled |
⊢ ( 𝜑 → 𝐶 ≤ ( 𝐶 + 𝑅 ) ) |
32 |
|
elicc2 |
⊢ ( ( ( 𝐶 − 𝑅 ) ∈ ℝ ∧ ( 𝐶 + 𝑅 ) ∈ ℝ ) → ( 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ↔ ( 𝐶 ∈ ℝ ∧ ( 𝐶 − 𝑅 ) ≤ 𝐶 ∧ 𝐶 ≤ ( 𝐶 + 𝑅 ) ) ) ) |
33 |
12 13 32
|
syl2anc |
⊢ ( 𝜑 → ( 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ↔ ( 𝐶 ∈ ℝ ∧ ( 𝐶 − 𝑅 ) ≤ 𝐶 ∧ 𝐶 ≤ ( 𝐶 + 𝑅 ) ) ) ) |
34 |
10 30 31 33
|
mpbir3and |
⊢ ( 𝜑 → 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) |
35 |
34
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) |
36 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( 𝐶 − 𝑅 ) < 𝐶 ) |
37 |
|
isorel |
⊢ ( ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ∧ ( ( 𝐶 − 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ∧ 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( ( 𝐶 − 𝑅 ) < 𝐶 ↔ ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ) ) |
38 |
37
|
biimpd |
⊢ ( ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ∧ ( ( 𝐶 − 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ∧ 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( ( 𝐶 − 𝑅 ) < 𝐶 → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ) ) |
39 |
38
|
exp32 |
⊢ ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( ( 𝐶 − 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( ( 𝐶 − 𝑅 ) < 𝐶 → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ) ) ) ) |
40 |
39
|
com4l |
⊢ ( ( 𝐶 − 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( ( 𝐶 − 𝑅 ) < 𝐶 → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ) ) ) ) |
41 |
29 35 36 40
|
syl3c |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ) ) |
42 |
29
|
fvresd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) = ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ) |
43 |
35
|
fvresd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) = ( 𝐹 ‘ 𝐶 ) ) |
44 |
42 43
|
breq12d |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ↔ ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) < ( 𝐹 ‘ 𝐶 ) ) ) |
45 |
41 44
|
sylibd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) < ( 𝐹 ‘ 𝐶 ) ) ) |
46 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → 𝐹 : 𝑋 ⟶ ℝ ) |
47 |
46
|
ffund |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → Fun 𝐹 ) |
48 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ⊆ 𝑋 ) |
49 |
46
|
fdmd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → dom 𝐹 = 𝑋 ) |
50 |
48 49
|
sseqtrrd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ⊆ dom 𝐹 ) |
51 |
|
funfvima2 |
⊢ ( ( Fun 𝐹 ∧ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ⊆ dom 𝐹 ) → ( ( 𝐶 − 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∈ ( 𝐹 “ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ) |
52 |
47 50 51
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐶 − 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∈ ( 𝐹 “ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ) |
53 |
29 52
|
mpd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∈ ( 𝐹 “ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) |
54 |
|
df-ima |
⊢ ( 𝐹 “ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) |
55 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) |
56 |
54 55
|
eqtrid |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( 𝐹 “ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) |
57 |
53 56
|
eleqtrd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∈ ( 𝑥 [,] 𝑦 ) ) |
58 |
|
elicc2 |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∈ ( 𝑥 [,] 𝑦 ) ↔ ( ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∈ ℝ ∧ 𝑥 ≤ ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ≤ 𝑦 ) ) ) |
59 |
58
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∈ ( 𝑥 [,] 𝑦 ) ↔ ( ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∈ ℝ ∧ 𝑥 ≤ ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ≤ 𝑦 ) ) ) |
60 |
57 59
|
mpbid |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∈ ℝ ∧ 𝑥 ≤ ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ≤ 𝑦 ) ) |
61 |
60
|
simp2d |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → 𝑥 ≤ ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ) |
62 |
|
simprll |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → 𝑥 ∈ ℝ ) |
63 |
7 28
|
sseldd |
⊢ ( 𝜑 → ( 𝐶 − 𝑅 ) ∈ 𝑋 ) |
64 |
22 63
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∈ ℝ ) |
65 |
64
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∈ ℝ ) |
66 |
|
lelttr |
⊢ ( ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∈ ℝ ∧ ( 𝐹 ‘ 𝐶 ) ∈ ℝ ) → ( ( 𝑥 ≤ ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) < ( 𝐹 ‘ 𝐶 ) ) → 𝑥 < ( 𝐹 ‘ 𝐶 ) ) ) |
67 |
62 65 24 66
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝑥 ≤ ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) < ( 𝐹 ‘ 𝐶 ) ) → 𝑥 < ( 𝐹 ‘ 𝐶 ) ) ) |
68 |
61 67
|
mpand |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) < ( 𝐹 ‘ 𝐶 ) → 𝑥 < ( 𝐹 ‘ 𝐶 ) ) ) |
69 |
45 68
|
syld |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → 𝑥 < ( 𝐹 ‘ 𝐶 ) ) ) |
70 |
|
ubicc2 |
⊢ ( ( ( 𝐶 − 𝑅 ) ∈ ℝ* ∧ ( 𝐶 + 𝑅 ) ∈ ℝ* ∧ ( 𝐶 − 𝑅 ) ≤ ( 𝐶 + 𝑅 ) ) → ( 𝐶 + 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) |
71 |
25 26 17 70
|
syl3anc |
⊢ ( 𝜑 → ( 𝐶 + 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) |
72 |
71
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( 𝐶 + 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) |
73 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → 𝐶 < ( 𝐶 + 𝑅 ) ) |
74 |
|
isorel |
⊢ ( ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , ◡ < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ∧ ( 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ∧ ( 𝐶 + 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( 𝐶 < ( 𝐶 + 𝑅 ) ↔ ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ◡ < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) ) ) |
75 |
74
|
biimpd |
⊢ ( ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , ◡ < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ∧ ( 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ∧ ( 𝐶 + 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( 𝐶 < ( 𝐶 + 𝑅 ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ◡ < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) ) ) |
76 |
75
|
exp32 |
⊢ ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , ◡ < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( ( 𝐶 + 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( 𝐶 < ( 𝐶 + 𝑅 ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ◡ < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) ) ) ) ) |
77 |
76
|
com4l |
⊢ ( 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( ( 𝐶 + 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( 𝐶 < ( 𝐶 + 𝑅 ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , ◡ < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ◡ < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) ) ) ) ) |
78 |
35 72 73 77
|
syl3c |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , ◡ < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ◡ < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) ) ) |
79 |
|
fvex |
⊢ ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ∈ V |
80 |
|
fvex |
⊢ ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) ∈ V |
81 |
79 80
|
brcnv |
⊢ ( ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ◡ < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) ↔ ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ) |
82 |
72
|
fvresd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) = ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ) |
83 |
82 43
|
breq12d |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ↔ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) < ( 𝐹 ‘ 𝐶 ) ) ) |
84 |
81 83
|
syl5bb |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ◡ < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) ↔ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) < ( 𝐹 ‘ 𝐶 ) ) ) |
85 |
78 84
|
sylibd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , ◡ < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) < ( 𝐹 ‘ 𝐶 ) ) ) |
86 |
|
funfvima2 |
⊢ ( ( Fun 𝐹 ∧ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ⊆ dom 𝐹 ) → ( ( 𝐶 + 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∈ ( 𝐹 “ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ) |
87 |
47 50 86
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐶 + 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∈ ( 𝐹 “ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ) |
88 |
72 87
|
mpd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∈ ( 𝐹 “ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) |
89 |
88 56
|
eleqtrd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∈ ( 𝑥 [,] 𝑦 ) ) |
90 |
|
elicc2 |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∈ ( 𝑥 [,] 𝑦 ) ↔ ( ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∈ ℝ ∧ 𝑥 ≤ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ≤ 𝑦 ) ) ) |
91 |
90
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∈ ( 𝑥 [,] 𝑦 ) ↔ ( ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∈ ℝ ∧ 𝑥 ≤ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ≤ 𝑦 ) ) ) |
92 |
89 91
|
mpbid |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∈ ℝ ∧ 𝑥 ≤ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ≤ 𝑦 ) ) |
93 |
92
|
simp2d |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → 𝑥 ≤ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ) |
94 |
7 71
|
sseldd |
⊢ ( 𝜑 → ( 𝐶 + 𝑅 ) ∈ 𝑋 ) |
95 |
22 94
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∈ ℝ ) |
96 |
95
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∈ ℝ ) |
97 |
|
lelttr |
⊢ ( ( 𝑥 ∈ ℝ ∧ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∈ ℝ ∧ ( 𝐹 ‘ 𝐶 ) ∈ ℝ ) → ( ( 𝑥 ≤ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) < ( 𝐹 ‘ 𝐶 ) ) → 𝑥 < ( 𝐹 ‘ 𝐶 ) ) ) |
98 |
62 96 24 97
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝑥 ≤ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) < ( 𝐹 ‘ 𝐶 ) ) → 𝑥 < ( 𝐹 ‘ 𝐶 ) ) ) |
99 |
93 98
|
mpand |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) < ( 𝐹 ‘ 𝐶 ) → 𝑥 < ( 𝐹 ‘ 𝐶 ) ) ) |
100 |
85 99
|
syld |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , ◡ < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → 𝑥 < ( 𝐹 ‘ 𝐶 ) ) ) |
101 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
102 |
101
|
a1i |
⊢ ( 𝜑 → ℝ ⊆ ℂ ) |
103 |
|
fss |
⊢ ( ( 𝐹 : 𝑋 ⟶ ℝ ∧ ℝ ⊆ ℂ ) → 𝐹 : 𝑋 ⟶ ℂ ) |
104 |
22 101 103
|
sylancl |
⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ℂ ) |
105 |
7 9
|
sstrd |
⊢ ( 𝜑 → ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ⊆ ℝ ) |
106 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
107 |
106
|
tgioo2 |
⊢ ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) |
108 |
106 107
|
dvres |
⊢ ( ( ( ℝ ⊆ ℂ ∧ 𝐹 : 𝑋 ⟶ ℂ ) ∧ ( 𝑋 ⊆ ℝ ∧ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ⊆ ℝ ) ) → ( ℝ D ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ) |
109 |
102 104 9 105 108
|
syl22anc |
⊢ ( 𝜑 → ( ℝ D ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ) |
110 |
|
iccntr |
⊢ ( ( ( 𝐶 − 𝑅 ) ∈ ℝ ∧ ( 𝐶 + 𝑅 ) ∈ ℝ ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ) |
111 |
12 13 110
|
syl2anc |
⊢ ( 𝜑 → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ) |
112 |
111
|
reseq2d |
⊢ ( 𝜑 → ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ) ) |
113 |
109 112
|
eqtrd |
⊢ ( 𝜑 → ( ℝ D ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ) ) |
114 |
113
|
dmeqd |
⊢ ( 𝜑 → dom ( ℝ D ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) = dom ( ( ℝ D 𝐹 ) ↾ ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ) ) |
115 |
|
dmres |
⊢ dom ( ( ℝ D 𝐹 ) ↾ ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ) = ( ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ∩ dom ( ℝ D 𝐹 ) ) |
116 |
|
ioossicc |
⊢ ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ⊆ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) |
117 |
116 7
|
sstrid |
⊢ ( 𝜑 → ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ⊆ 𝑋 ) |
118 |
117 2
|
sseqtrrd |
⊢ ( 𝜑 → ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ⊆ dom ( ℝ D 𝐹 ) ) |
119 |
|
df-ss |
⊢ ( ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ⊆ dom ( ℝ D 𝐹 ) ↔ ( ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ∩ dom ( ℝ D 𝐹 ) ) = ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ) |
120 |
118 119
|
sylib |
⊢ ( 𝜑 → ( ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ∩ dom ( ℝ D 𝐹 ) ) = ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ) |
121 |
115 120
|
eqtrid |
⊢ ( 𝜑 → dom ( ( ℝ D 𝐹 ) ↾ ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ) = ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ) |
122 |
114 121
|
eqtrd |
⊢ ( 𝜑 → dom ( ℝ D ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) = ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ) |
123 |
|
resss |
⊢ ( ( ℝ D 𝐹 ) ↾ ( ( 𝐶 − 𝑅 ) (,) ( 𝐶 + 𝑅 ) ) ) ⊆ ( ℝ D 𝐹 ) |
124 |
113 123
|
eqsstrdi |
⊢ ( 𝜑 → ( ℝ D ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ⊆ ( ℝ D 𝐹 ) ) |
125 |
|
rnss |
⊢ ( ( ℝ D ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ⊆ ( ℝ D 𝐹 ) → ran ( ℝ D ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ⊆ ran ( ℝ D 𝐹 ) ) |
126 |
124 125
|
syl |
⊢ ( 𝜑 → ran ( ℝ D ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ⊆ ran ( ℝ D 𝐹 ) ) |
127 |
126 3
|
ssneldd |
⊢ ( 𝜑 → ¬ 0 ∈ ran ( ℝ D ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ) |
128 |
12 13 19 122 127
|
dvne0 |
⊢ ( 𝜑 → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ∨ ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , ◡ < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ) ) |
129 |
128
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ∨ ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , ◡ < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ) ) |
130 |
69 100 129
|
mpjaod |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → 𝑥 < ( 𝐹 ‘ 𝐶 ) ) |
131 |
|
isorel |
⊢ ( ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ∧ ( 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ∧ ( 𝐶 + 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( 𝐶 < ( 𝐶 + 𝑅 ) ↔ ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) ) ) |
132 |
131
|
biimpd |
⊢ ( ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ∧ ( 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ∧ ( 𝐶 + 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( 𝐶 < ( 𝐶 + 𝑅 ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) ) ) |
133 |
132
|
exp32 |
⊢ ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( ( 𝐶 + 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( 𝐶 < ( 𝐶 + 𝑅 ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) ) ) ) ) |
134 |
133
|
com4l |
⊢ ( 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( ( 𝐶 + 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( 𝐶 < ( 𝐶 + 𝑅 ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) ) ) ) ) |
135 |
35 72 73 134
|
syl3c |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) ) ) |
136 |
43 82
|
breq12d |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 + 𝑅 ) ) ↔ ( 𝐹 ‘ 𝐶 ) < ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ) ) |
137 |
135 136
|
sylibd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝐶 ) < ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ) ) |
138 |
92
|
simp3d |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ≤ 𝑦 ) |
139 |
|
simprlr |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → 𝑦 ∈ ℝ ) |
140 |
|
ltletr |
⊢ ( ( ( 𝐹 ‘ 𝐶 ) ∈ ℝ ∧ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ( ( 𝐹 ‘ 𝐶 ) < ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ≤ 𝑦 ) → ( 𝐹 ‘ 𝐶 ) < 𝑦 ) ) |
141 |
24 96 139 140
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( ( 𝐹 ‘ 𝐶 ) < ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) ≤ 𝑦 ) → ( 𝐹 ‘ 𝐶 ) < 𝑦 ) ) |
142 |
138 141
|
mpan2d |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ‘ 𝐶 ) < ( 𝐹 ‘ ( 𝐶 + 𝑅 ) ) → ( 𝐹 ‘ 𝐶 ) < 𝑦 ) ) |
143 |
137 142
|
syld |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝐶 ) < 𝑦 ) ) |
144 |
|
isorel |
⊢ ( ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , ◡ < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ∧ ( ( 𝐶 − 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ∧ 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( ( 𝐶 − 𝑅 ) < 𝐶 ↔ ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) ◡ < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ) ) |
145 |
144
|
biimpd |
⊢ ( ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , ◡ < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ∧ ( ( 𝐶 − 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ∧ 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( ( 𝐶 − 𝑅 ) < 𝐶 → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) ◡ < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ) ) |
146 |
145
|
exp32 |
⊢ ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , ◡ < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( ( 𝐶 − 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( ( 𝐶 − 𝑅 ) < 𝐶 → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) ◡ < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ) ) ) ) |
147 |
146
|
com4l |
⊢ ( ( 𝐶 − 𝑅 ) ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( 𝐶 ∈ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) → ( ( 𝐶 − 𝑅 ) < 𝐶 → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , ◡ < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) ◡ < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ) ) ) ) |
148 |
29 35 36 147
|
syl3c |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , ◡ < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) ◡ < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ) ) |
149 |
|
fvex |
⊢ ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) ∈ V |
150 |
149 79
|
brcnv |
⊢ ( ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) ◡ < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ↔ ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) ) |
151 |
43 42
|
breq12d |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) ↔ ( 𝐹 ‘ 𝐶 ) < ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ) ) |
152 |
150 151
|
syl5bb |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ ( 𝐶 − 𝑅 ) ) ◡ < ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ‘ 𝐶 ) ↔ ( 𝐹 ‘ 𝐶 ) < ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ) ) |
153 |
148 152
|
sylibd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , ◡ < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝐶 ) < ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ) ) |
154 |
60
|
simp3d |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ≤ 𝑦 ) |
155 |
|
ltletr |
⊢ ( ( ( 𝐹 ‘ 𝐶 ) ∈ ℝ ∧ ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ( ( 𝐹 ‘ 𝐶 ) < ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ≤ 𝑦 ) → ( 𝐹 ‘ 𝐶 ) < 𝑦 ) ) |
156 |
24 65 139 155
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( ( 𝐹 ‘ 𝐶 ) < ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) ≤ 𝑦 ) → ( 𝐹 ‘ 𝐶 ) < 𝑦 ) ) |
157 |
154 156
|
mpan2d |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ‘ 𝐶 ) < ( 𝐹 ‘ ( 𝐶 − 𝑅 ) ) → ( 𝐹 ‘ 𝐶 ) < 𝑦 ) ) |
158 |
153 157
|
syld |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) Isom < , ◡ < ( ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) , ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝐶 ) < 𝑦 ) ) |
159 |
143 158 129
|
mpjaod |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( 𝐹 ‘ 𝐶 ) < 𝑦 ) |
160 |
62
|
rexrd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → 𝑥 ∈ ℝ* ) |
161 |
139
|
rexrd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → 𝑦 ∈ ℝ* ) |
162 |
|
elioo2 |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( ( 𝐹 ‘ 𝐶 ) ∈ ( 𝑥 (,) 𝑦 ) ↔ ( ( 𝐹 ‘ 𝐶 ) ∈ ℝ ∧ 𝑥 < ( 𝐹 ‘ 𝐶 ) ∧ ( 𝐹 ‘ 𝐶 ) < 𝑦 ) ) ) |
163 |
160 161 162
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( 𝐹 ‘ 𝐶 ) ∈ ( 𝑥 (,) 𝑦 ) ↔ ( ( 𝐹 ‘ 𝐶 ) ∈ ℝ ∧ 𝑥 < ( 𝐹 ‘ 𝐶 ) ∧ ( 𝐹 ‘ 𝐶 ) < 𝑦 ) ) ) |
164 |
24 130 159 163
|
mpbir3and |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( 𝐹 ‘ 𝐶 ) ∈ ( 𝑥 (,) 𝑦 ) ) |
165 |
56
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐹 “ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) = ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝑥 [,] 𝑦 ) ) ) |
166 |
|
iccntr |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝑥 [,] 𝑦 ) ) = ( 𝑥 (,) 𝑦 ) ) |
167 |
166
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝑥 [,] 𝑦 ) ) = ( 𝑥 (,) 𝑦 ) ) |
168 |
165 167
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐹 “ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) = ( 𝑥 (,) 𝑦 ) ) |
169 |
164 168
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) ) ) → ( 𝐹 ‘ 𝐶 ) ∈ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐹 “ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ) |
170 |
169
|
expr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → ( ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) → ( 𝐹 ‘ 𝐶 ) ∈ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐹 “ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ) ) |
171 |
170
|
rexlimdvva |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ ran ( 𝐹 ↾ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) = ( 𝑥 [,] 𝑦 ) → ( 𝐹 ‘ 𝐶 ) ∈ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐹 “ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ) ) |
172 |
20 171
|
mpd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐶 ) ∈ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐹 “ ( ( 𝐶 − 𝑅 ) [,] ( 𝐶 + 𝑅 ) ) ) ) ) |