Step | Hyp | Ref
| Expression |
1 | | cncmpmax.1 |
. . 3
⊢ 𝑇 = ∪
𝐽 |
2 | | cncmpmax.2 |
. . 3
⊢ 𝐾 = (topGen‘ran
(,)) |
3 | | cncmpmax.3 |
. . 3
⊢ (𝜑 → 𝐽 ∈ Comp) |
4 | | cncmpmax.4 |
. . 3
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
5 | | cncmpmax.5 |
. . 3
⊢ (𝜑 → 𝑇 ≠ ∅) |
6 | 1, 2, 3, 4, 5 | evth 22566 |
. 2
⊢ (𝜑 → ∃𝑥 ∈ 𝑇 ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) |
7 | | eqid 2610 |
. . . . . . . . 9
⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾) |
8 | 2, 1, 7, 4 | fcnre 38207 |
. . . . . . . 8
⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
9 | | frn 5966 |
. . . . . . . 8
⊢ (𝐹:𝑇⟶ℝ → ran 𝐹 ⊆ ℝ) |
10 | 8, 9 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
11 | 10 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → ran 𝐹 ⊆ ℝ) |
12 | | ffun 5961 |
. . . . . . . . . 10
⊢ (𝐹:𝑇⟶ℝ → Fun 𝐹) |
13 | 8, 12 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → Fun 𝐹) |
14 | 13 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → Fun 𝐹) |
15 | | simpr 476 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝑇) |
16 | 8 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝐹:𝑇⟶ℝ) |
17 | | fdm 5964 |
. . . . . . . . . 10
⊢ (𝐹:𝑇⟶ℝ → dom 𝐹 = 𝑇) |
18 | 16, 17 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → dom 𝐹 = 𝑇) |
19 | 15, 18 | eleqtrrd 2691 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ dom 𝐹) |
20 | | fvelrn 6260 |
. . . . . . . 8
⊢ ((Fun
𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) |
21 | 14, 19, 20 | syl2anc 691 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐹‘𝑥) ∈ ran 𝐹) |
22 | 21 | adantrr 749 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → (𝐹‘𝑥) ∈ ran 𝐹) |
23 | | ffn 5958 |
. . . . . . . . . . . . 13
⊢ (𝐹:𝑇⟶ℝ → 𝐹 Fn 𝑇) |
24 | | fvelrnb 6153 |
. . . . . . . . . . . . 13
⊢ (𝐹 Fn 𝑇 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑠 ∈ 𝑇 (𝐹‘𝑠) = 𝑦)) |
25 | 8, 23, 24 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑠 ∈ 𝑇 (𝐹‘𝑠) = 𝑦)) |
26 | 25 | biimpa 500 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑠 ∈ 𝑇 (𝐹‘𝑠) = 𝑦) |
27 | | df-rex 2902 |
. . . . . . . . . . 11
⊢
(∃𝑠 ∈
𝑇 (𝐹‘𝑠) = 𝑦 ↔ ∃𝑠(𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) |
28 | 26, 27 | sylib 207 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑠(𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) |
29 | 28 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑠(𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) |
30 | | simprr 792 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) ∧ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) → (𝐹‘𝑠) = 𝑦) |
31 | | simpllr 795 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) ∧ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) |
32 | | simprl 790 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) ∧ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) → 𝑠 ∈ 𝑇) |
33 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑠 → (𝐹‘𝑡) = (𝐹‘𝑠)) |
34 | 33 | breq1d 4593 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑠 → ((𝐹‘𝑡) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑠) ≤ (𝐹‘𝑥))) |
35 | 34 | rspccva 3281 |
. . . . . . . . . . 11
⊢
((∀𝑡 ∈
𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥) ∧ 𝑠 ∈ 𝑇) → (𝐹‘𝑠) ≤ (𝐹‘𝑥)) |
36 | 31, 32, 35 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) ∧ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) → (𝐹‘𝑠) ≤ (𝐹‘𝑥)) |
37 | 30, 36 | eqbrtrrd 4607 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) ∧ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) → 𝑦 ≤ (𝐹‘𝑥)) |
38 | 29, 37 | exlimddv 1850 |
. . . . . . . 8
⊢ (((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ≤ (𝐹‘𝑥)) |
39 | 38 | ralrimiva 2949 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) → ∀𝑦 ∈ ran 𝐹 𝑦 ≤ (𝐹‘𝑥)) |
40 | 39 | adantrl 748 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → ∀𝑦 ∈ ran 𝐹 𝑦 ≤ (𝐹‘𝑥)) |
41 | | ubelsupr 38202 |
. . . . . 6
⊢ ((ran
𝐹 ⊆ ℝ ∧
(𝐹‘𝑥) ∈ ran 𝐹 ∧ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ (𝐹‘𝑥)) → (𝐹‘𝑥) = sup(ran 𝐹, ℝ, < )) |
42 | 11, 22, 40, 41 | syl3anc 1318 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → (𝐹‘𝑥) = sup(ran 𝐹, ℝ, < )) |
43 | 42 | eqcomd 2616 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → sup(ran 𝐹, ℝ, < ) = (𝐹‘𝑥)) |
44 | 43, 22 | eqeltrd 2688 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹) |
45 | 11, 44 | sseldd 3569 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → sup(ran 𝐹, ℝ, < ) ∈
ℝ) |
46 | | simplrr 797 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) ∧ 𝑠 ∈ 𝑇) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) |
47 | 46, 35 | sylancom 698 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) ∧ 𝑠 ∈ 𝑇) → (𝐹‘𝑠) ≤ (𝐹‘𝑥)) |
48 | 43 | adantr 480 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) ∧ 𝑠 ∈ 𝑇) → sup(ran 𝐹, ℝ, < ) = (𝐹‘𝑥)) |
49 | 47, 48 | breqtrrd 4611 |
. . . . 5
⊢ (((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) ∧ 𝑠 ∈ 𝑇) → (𝐹‘𝑠) ≤ sup(ran 𝐹, ℝ, < )) |
50 | 49 | ralrimiva 2949 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → ∀𝑠 ∈ 𝑇 (𝐹‘𝑠) ≤ sup(ran 𝐹, ℝ, < )) |
51 | 33 | breq1d 4593 |
. . . . 5
⊢ (𝑡 = 𝑠 → ((𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < ) ↔ (𝐹‘𝑠) ≤ sup(ran 𝐹, ℝ, < ))) |
52 | 51 | cbvralv 3147 |
. . . 4
⊢
(∀𝑡 ∈
𝑇 (𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < ) ↔ ∀𝑠 ∈ 𝑇 (𝐹‘𝑠) ≤ sup(ran 𝐹, ℝ, < )) |
53 | 50, 52 | sylibr 223 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < )) |
54 | 44, 45, 53 | 3jca 1235 |
. 2
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < ))) |
55 | 6, 54 | rexlimddv 3017 |
1
⊢ (𝜑 → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < ))) |