Step | Hyp | Ref
| Expression |
1 | | evthicc.1 |
. . . 4
⊢ (𝜑 → 𝐴 ∈ ℝ) |
2 | | evthicc.2 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ ℝ) |
3 | | evthicc.3 |
. . . 4
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
4 | | evthicc.4 |
. . . 4
⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
5 | 1, 2, 3, 4 | evthicc 23035 |
. . 3
⊢ (𝜑 → (∃𝑎 ∈ (𝐴[,]𝐵)∀𝑧 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑎) ∧ ∃𝑏 ∈ (𝐴[,]𝐵)∀𝑧 ∈ (𝐴[,]𝐵)(𝐹‘𝑏) ≤ (𝐹‘𝑧))) |
6 | | reeanv 3086 |
. . 3
⊢
(∃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 223 |
. 2
⊢ (𝜑 → ∃𝑎 ∈ (𝐴[,]𝐵)∃𝑏 ∈ (𝐴[,]𝐵)(∀𝑧 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑎) ∧ ∀𝑧 ∈ (𝐴[,]𝐵)(𝐹‘𝑏) ≤ (𝐹‘𝑧))) |
8 | | r19.26 3046 |
. . . 4
⊢
(∀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 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
10 | | cncff 22504 |
. . . . . . . . 9
⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ) |
11 | 9, 10 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) → 𝐹:(𝐴[,]𝐵)⟶ℝ) |
12 | | simprr 792 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) → 𝑏 ∈ (𝐴[,]𝐵)) |
13 | 11, 12 | ffvelrnd 6268 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑏) ∈ ℝ) |
14 | 13 | adantr 480 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) ∧ ∀𝑧 ∈ (𝐴[,]𝐵)((𝐹‘𝑧) ≤ (𝐹‘𝑎) ∧ (𝐹‘𝑏) ≤ (𝐹‘𝑧))) → (𝐹‘𝑏) ∈ ℝ) |
15 | | simprl 790 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) → 𝑎 ∈ (𝐴[,]𝐵)) |
16 | 11, 15 | ffvelrnd 6268 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) → (𝐹‘𝑎) ∈ ℝ) |
17 | 16 | adantr 480 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) ∧ ∀𝑧 ∈ (𝐴[,]𝐵)((𝐹‘𝑧) ≤ (𝐹‘𝑎) ∧ (𝐹‘𝑏) ≤ (𝐹‘𝑧))) → (𝐹‘𝑎) ∈ ℝ) |
18 | 11 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) ∧ ∀𝑧 ∈ (𝐴[,]𝐵)((𝐹‘𝑧) ≤ (𝐹‘𝑎) ∧ (𝐹‘𝑏) ≤ (𝐹‘𝑧))) → 𝐹:(𝐴[,]𝐵)⟶ℝ) |
19 | | ffn 5958 |
. . . . . . . . . 10
⊢ (𝐹:(𝐴[,]𝐵)⟶ℝ → 𝐹 Fn (𝐴[,]𝐵)) |
20 | 18, 19 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) ∧ ∀𝑧 ∈ (𝐴[,]𝐵)((𝐹‘𝑧) ≤ (𝐹‘𝑎) ∧ (𝐹‘𝑏) ≤ (𝐹‘𝑧))) → 𝐹 Fn (𝐴[,]𝐵)) |
21 | 13 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑏) ∈ ℝ) |
22 | 16 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑎) ∈ ℝ) |
23 | | elicc2 12109 |
. . . . . . . . . . . . . 14
⊢ (((𝐹‘𝑏) ∈ ℝ ∧ (𝐹‘𝑎) ∈ ℝ) → ((𝐹‘𝑧) ∈ ((𝐹‘𝑏)[,](𝐹‘𝑎)) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑏) ≤ (𝐹‘𝑧) ∧ (𝐹‘𝑧) ≤ (𝐹‘𝑎)))) |
24 | 21, 22, 23 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ (((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 | | 3anass 1035 |
. . . . . . . . . . . . 13
⊢ (((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑏) ≤ (𝐹‘𝑧) ∧ (𝐹‘𝑧) ≤ (𝐹‘𝑎)) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ ((𝐹‘𝑏) ≤ (𝐹‘𝑧) ∧ (𝐹‘𝑧) ≤ (𝐹‘𝑎)))) |
26 | 24, 25 | syl6bb 275 |
. . . . . . . . . . . 12
⊢ (((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))))) |
27 | | ancom 465 |
. . . . . . . . . . . . 13
⊢ (((𝐹‘𝑧) ≤ (𝐹‘𝑎) ∧ (𝐹‘𝑏) ≤ (𝐹‘𝑧)) ↔ ((𝐹‘𝑏) ≤ (𝐹‘𝑧) ∧ (𝐹‘𝑧) ≤ (𝐹‘𝑎))) |
28 | 11 | ffvelrnda 6267 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑧) ∈ ℝ) |
29 | 28 | biantrurd 528 |
. . . . . . . . . . . . 13
⊢ (((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))))) |
30 | 27, 29 | syl5bb 271 |
. . . . . . . . . . . 12
⊢ (((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))))) |
31 | 26, 30 | bitr4d 270 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) ∧ 𝑧 ∈ (𝐴[,]𝐵)) → ((𝐹‘𝑧) ∈ ((𝐹‘𝑏)[,](𝐹‘𝑎)) ↔ ((𝐹‘𝑧) ≤ (𝐹‘𝑎) ∧ (𝐹‘𝑏) ≤ (𝐹‘𝑧)))) |
32 | 31 | ralbidva 2968 |
. . . . . . . . . 10
⊢ ((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)))) |
33 | 32 | biimpar 501 |
. . . . . . . . 9
⊢ (((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))) |
34 | | ffnfv 6295 |
. . . . . . . . 9
⊢ (𝐹:(𝐴[,]𝐵)⟶((𝐹‘𝑏)[,](𝐹‘𝑎)) ↔ (𝐹 Fn (𝐴[,]𝐵) ∧ ∀𝑧 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ∈ ((𝐹‘𝑏)[,](𝐹‘𝑎)))) |
35 | 20, 33, 34 | sylanbrc 695 |
. . . . . . . 8
⊢ (((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))) |
36 | | frn 5966 |
. . . . . . . 8
⊢ (𝐹:(𝐴[,]𝐵)⟶((𝐹‘𝑏)[,](𝐹‘𝑎)) → ran 𝐹 ⊆ ((𝐹‘𝑏)[,](𝐹‘𝑎))) |
37 | 35, 36 | syl 17 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) ∧ ∀𝑧 ∈ (𝐴[,]𝐵)((𝐹‘𝑧) ≤ (𝐹‘𝑎) ∧ (𝐹‘𝑏) ≤ (𝐹‘𝑧))) → ran 𝐹 ⊆ ((𝐹‘𝑏)[,](𝐹‘𝑎))) |
38 | 1 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) → 𝐴 ∈ ℝ) |
39 | 2 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) → 𝐵 ∈ ℝ) |
40 | | ssid 3587 |
. . . . . . . . . 10
⊢ (𝐴[,]𝐵) ⊆ (𝐴[,]𝐵) |
41 | 40 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐵)) |
42 | | ax-resscn 9872 |
. . . . . . . . . . 11
⊢ ℝ
⊆ ℂ |
43 | | ssid 3587 |
. . . . . . . . . . 11
⊢ ℂ
⊆ ℂ |
44 | | cncfss 22510 |
. . . . . . . . . . 11
⊢ ((ℝ
⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)) |
45 | 42, 43, 44 | mp2an 704 |
. . . . . . . . . 10
⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ) |
46 | 45, 9 | sseldi 3566 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
47 | 11 | ffvelrnda 6267 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) |
48 | 38, 39, 12, 15, 41, 46, 47 | ivthicc 23034 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) → ((𝐹‘𝑏)[,](𝐹‘𝑎)) ⊆ ran 𝐹) |
49 | 48 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) ∧ ∀𝑧 ∈ (𝐴[,]𝐵)((𝐹‘𝑧) ≤ (𝐹‘𝑎) ∧ (𝐹‘𝑏) ≤ (𝐹‘𝑧))) → ((𝐹‘𝑏)[,](𝐹‘𝑎)) ⊆ ran 𝐹) |
50 | 37, 49 | eqssd 3585 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑏 ∈ (𝐴[,]𝐵))) ∧ ∀𝑧 ∈ (𝐴[,]𝐵)((𝐹‘𝑧) ≤ (𝐹‘𝑎) ∧ (𝐹‘𝑏) ≤ (𝐹‘𝑧))) → ran 𝐹 = ((𝐹‘𝑏)[,](𝐹‘𝑎))) |
51 | | rspceov 6590 |
. . . . . 6
⊢ (((𝐹‘𝑏) ∈ ℝ ∧ (𝐹‘𝑎) ∈ ℝ ∧ ran 𝐹 = ((𝐹‘𝑏)[,](𝐹‘𝑎))) → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran 𝐹 = (𝑥[,]𝑦)) |
52 | 14, 17, 50, 51 | syl3anc 1318 |
. . . . 5
⊢ (((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)) |
53 | 52 | ex 449 |
. . . 4
⊢ ((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))) |
54 | 8, 53 | syl5bir 232 |
. . 3
⊢ ((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))) |
55 | 54 | rexlimdvva 3020 |
. 2
⊢ (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))) |
56 | 7, 55 | mpd 15 |
1
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ran 𝐹 = (𝑥[,]𝑦)) |