Step |
Hyp |
Ref |
Expression |
1 |
|
evthicc.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
evthicc.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
evthicc.3 |
⊢ ( 𝜑 → 𝐴 ≤ 𝐵 ) |
4 |
|
evthicc.4 |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) ) |
5 |
1 2 3 4
|
evthicc |
⊢ ( 𝜑 → ( ∃ 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ∃ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) |
6 |
|
reeanv |
⊢ ( ∃ 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∃ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ( ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ↔ ( ∃ 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ∃ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) |
7 |
5 6
|
sylibr |
⊢ ( 𝜑 → ∃ 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∃ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ( ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) |
8 |
|
r19.26 |
⊢ ( ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ↔ ( ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) |
9 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) → 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) ) |
10 |
|
cncff |
⊢ ( 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℝ ) |
11 |
9 10
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℝ ) |
12 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) → 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) |
13 |
11 12
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐹 ‘ 𝑏 ) ∈ ℝ ) |
14 |
13
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) → ( 𝐹 ‘ 𝑏 ) ∈ ℝ ) |
15 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) → 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ) |
16 |
11 15
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐹 ‘ 𝑎 ) ∈ ℝ ) |
17 |
16
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) → ( 𝐹 ‘ 𝑎 ) ∈ ℝ ) |
18 |
11
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℝ ) |
19 |
18
|
ffnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) ) |
20 |
16
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑎 ) ∈ ℝ ) |
21 |
|
elicc2 |
⊢ ( ( ( 𝐹 ‘ 𝑏 ) ∈ ℝ ∧ ( 𝐹 ‘ 𝑎 ) ∈ ℝ ) → ( ( 𝐹 ‘ 𝑧 ) ∈ ( ( 𝐹 ‘ 𝑏 ) [,] ( 𝐹 ‘ 𝑎 ) ) ↔ ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ) ) ) |
22 |
13 20 21
|
syl2an2r |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( 𝐹 ‘ 𝑧 ) ∈ ( ( 𝐹 ‘ 𝑏 ) [,] ( 𝐹 ‘ 𝑎 ) ) ↔ ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ) ) ) |
23 |
|
3anass |
⊢ ( ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ) ↔ ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ) ) ) |
24 |
22 23
|
bitrdi |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( 𝐹 ‘ 𝑧 ) ∈ ( ( 𝐹 ‘ 𝑏 ) [,] ( 𝐹 ‘ 𝑎 ) ) ↔ ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ) ) ) ) |
25 |
|
ancom |
⊢ ( ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ↔ ( ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ) ) |
26 |
11
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ ℝ ) |
27 |
26
|
biantrurd |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ) ↔ ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ) ) ) ) |
28 |
25 27
|
syl5bb |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ↔ ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ ( ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ) ) ) ) |
29 |
24 28
|
bitr4d |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( 𝐹 ‘ 𝑧 ) ∈ ( ( 𝐹 ‘ 𝑏 ) [,] ( 𝐹 ‘ 𝑎 ) ) ↔ ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) ) |
30 |
29
|
ralbidva |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) → ( ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑧 ) ∈ ( ( 𝐹 ‘ 𝑏 ) [,] ( 𝐹 ‘ 𝑎 ) ) ↔ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) ) |
31 |
30
|
biimpar |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) → ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑧 ) ∈ ( ( 𝐹 ‘ 𝑏 ) [,] ( 𝐹 ‘ 𝑎 ) ) ) |
32 |
|
ffnfv |
⊢ ( 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ( ( 𝐹 ‘ 𝑏 ) [,] ( 𝐹 ‘ 𝑎 ) ) ↔ ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) ∧ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑧 ) ∈ ( ( 𝐹 ‘ 𝑏 ) [,] ( 𝐹 ‘ 𝑎 ) ) ) ) |
33 |
19 31 32
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ( ( 𝐹 ‘ 𝑏 ) [,] ( 𝐹 ‘ 𝑎 ) ) ) |
34 |
33
|
frnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) → ran 𝐹 ⊆ ( ( 𝐹 ‘ 𝑏 ) [,] ( 𝐹 ‘ 𝑎 ) ) ) |
35 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) → 𝐴 ∈ ℝ ) |
36 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) → 𝐵 ∈ ℝ ) |
37 |
|
ssidd |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐴 [,] 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
38 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
39 |
|
ssid |
⊢ ℂ ⊆ ℂ |
40 |
|
cncfss |
⊢ ( ( ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) ⊆ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
41 |
38 39 40
|
mp2an |
⊢ ( ( 𝐴 [,] 𝐵 ) –cn→ ℝ ) ⊆ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) |
42 |
41 9
|
sselid |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) → 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
43 |
11
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
44 |
35 36 12 15 37 42 43
|
ivthicc |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) → ( ( 𝐹 ‘ 𝑏 ) [,] ( 𝐹 ‘ 𝑎 ) ) ⊆ ran 𝐹 ) |
45 |
44
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) → ( ( 𝐹 ‘ 𝑏 ) [,] ( 𝐹 ‘ 𝑎 ) ) ⊆ ran 𝐹 ) |
46 |
34 45
|
eqssd |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) → ran 𝐹 = ( ( 𝐹 ‘ 𝑏 ) [,] ( 𝐹 ‘ 𝑎 ) ) ) |
47 |
|
rspceov |
⊢ ( ( ( 𝐹 ‘ 𝑏 ) ∈ ℝ ∧ ( 𝐹 ‘ 𝑎 ) ∈ ℝ ∧ ran 𝐹 = ( ( 𝐹 ‘ 𝑏 ) [,] ( 𝐹 ‘ 𝑎 ) ) ) → ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ ran 𝐹 = ( 𝑥 [,] 𝑦 ) ) |
48 |
14 17 46 47
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) ∧ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) ) → ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ ran 𝐹 = ( 𝑥 [,] 𝑦 ) ) |
49 |
48
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) → ( ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) → ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ ran 𝐹 = ( 𝑥 [,] 𝑦 ) ) ) |
50 |
8 49
|
syl5bir |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ) ) → ( ( ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) → ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ ran 𝐹 = ( 𝑥 [,] 𝑦 ) ) ) |
51 |
50
|
rexlimdvva |
⊢ ( 𝜑 → ( ∃ 𝑎 ∈ ( 𝐴 [,] 𝐵 ) ∃ 𝑏 ∈ ( 𝐴 [,] 𝐵 ) ( ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐹 ‘ 𝑎 ) ∧ ∀ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ( 𝐹 ‘ 𝑏 ) ≤ ( 𝐹 ‘ 𝑧 ) ) → ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ ran 𝐹 = ( 𝑥 [,] 𝑦 ) ) ) |
52 |
7 51
|
mpd |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ ran 𝐹 = ( 𝑥 [,] 𝑦 ) ) |