| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | halfre 12480 | . . . . 5
⊢ (1 / 2)
∈ ℝ | 
| 2 |  | halfgt0 12482 | . . . . 5
⊢ 0 < (1
/ 2) | 
| 3 | 1, 2 | elrpii 13037 | . . . 4
⊢ (1 / 2)
∈ ℝ+ | 
| 4 | 3 | a1i 11 | . . 3
⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → (1 / 2) ∈
ℝ+) | 
| 5 |  | halflt1 12484 | . . . 4
⊢ (1 / 2)
< 1 | 
| 6 | 5 | a1i 11 | . . 3
⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → (1 / 2) <
1) | 
| 7 |  | nfcv 2905 | . . . . . . 7
⊢
Ⅎ𝑡𝑇 | 
| 8 |  | stoweidlem28.1 | . . . . . . 7
⊢
Ⅎ𝑡𝑈 | 
| 9 | 7, 8 | nfdif 4129 | . . . . . 6
⊢
Ⅎ𝑡(𝑇 ∖ 𝑈) | 
| 10 | 9 | nfeq1 2921 | . . . . 5
⊢
Ⅎ𝑡(𝑇 ∖ 𝑈) = ∅ | 
| 11 | 10 | rzalf 45022 | . . . 4
⊢ ((𝑇 ∖ 𝑈) = ∅ → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡)) | 
| 12 | 11 | adantl 481 | . . 3
⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡)) | 
| 13 |  | ovex 7464 | . . . 4
⊢ (1 / 2)
∈ V | 
| 14 |  | eleq1 2829 | . . . . 5
⊢ (𝑑 = (1 / 2) → (𝑑 ∈ ℝ+
↔ (1 / 2) ∈ ℝ+)) | 
| 15 |  | breq1 5146 | . . . . 5
⊢ (𝑑 = (1 / 2) → (𝑑 < 1 ↔ (1 / 2) <
1)) | 
| 16 |  | breq1 5146 | . . . . . 6
⊢ (𝑑 = (1 / 2) → (𝑑 ≤ (𝑃‘𝑡) ↔ (1 / 2) ≤ (𝑃‘𝑡))) | 
| 17 | 16 | ralbidv 3178 | . . . . 5
⊢ (𝑑 = (1 / 2) → (∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡))) | 
| 18 | 14, 15, 17 | 3anbi123d 1438 | . . . 4
⊢ (𝑑 = (1 / 2) → ((𝑑 ∈ ℝ+
∧ 𝑑 < 1 ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)) ↔ ((1 / 2) ∈ ℝ+
∧ (1 / 2) < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡)))) | 
| 19 | 13, 18 | spcev 3606 | . . 3
⊢ (((1 / 2)
∈ ℝ+ ∧ (1 / 2) < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡)) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡))) | 
| 20 | 4, 6, 12, 19 | syl3anc 1373 | . 2
⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡))) | 
| 21 |  | simplll 775 | . . . 4
⊢ ((((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → 𝜑) | 
| 22 |  | simplr 769 | . . . 4
⊢ ((((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → 𝑥 ∈ (𝑇 ∖ 𝑈)) | 
| 23 |  | simpr 484 | . . . 4
⊢ ((((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) | 
| 24 |  | stoweidlem28.3 | . . . . . . . . . . 11
⊢ 𝐾 = (topGen‘ran
(,)) | 
| 25 |  | stoweidlem28.4 | . . . . . . . . . . 11
⊢ 𝑇 = ∪
𝐽 | 
| 26 |  | eqid 2737 | . . . . . . . . . . 11
⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾) | 
| 27 |  | stoweidlem28.6 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾)) | 
| 28 | 24, 25, 26, 27 | fcnre 45030 | . . . . . . . . . 10
⊢ (𝜑 → 𝑃:𝑇⟶ℝ) | 
| 29 | 28 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 𝑃:𝑇⟶ℝ) | 
| 30 |  | eldifi 4131 | . . . . . . . . . 10
⊢ (𝑥 ∈ (𝑇 ∖ 𝑈) → 𝑥 ∈ 𝑇) | 
| 31 | 30 | adantl 481 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 𝑥 ∈ 𝑇) | 
| 32 | 29, 31 | ffvelcdmd 7105 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑥) ∈ ℝ) | 
| 33 |  | stoweidlem28.7 | . . . . . . . . 9
⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡)) | 
| 34 |  | nfcv 2905 | . . . . . . . . . . . 12
⊢
Ⅎ𝑥(𝑇 ∖ 𝑈) | 
| 35 |  | nfv 1914 | . . . . . . . . . . . 12
⊢
Ⅎ𝑥0 <
(𝑃‘𝑡) | 
| 36 |  | nfv 1914 | . . . . . . . . . . . 12
⊢
Ⅎ𝑡0 <
(𝑃‘𝑥) | 
| 37 |  | fveq2 6906 | . . . . . . . . . . . . 13
⊢ (𝑡 = 𝑥 → (𝑃‘𝑡) = (𝑃‘𝑥)) | 
| 38 | 37 | breq2d 5155 | . . . . . . . . . . . 12
⊢ (𝑡 = 𝑥 → (0 < (𝑃‘𝑡) ↔ 0 < (𝑃‘𝑥))) | 
| 39 | 9, 34, 35, 36, 38 | cbvralfw 3304 | . . . . . . . . . . 11
⊢
(∀𝑡 ∈
(𝑇 ∖ 𝑈)0 < (𝑃‘𝑡) ↔ ∀𝑥 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑥)) | 
| 40 | 39 | biimpi 216 | . . . . . . . . . 10
⊢
(∀𝑡 ∈
(𝑇 ∖ 𝑈)0 < (𝑃‘𝑡) → ∀𝑥 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑥)) | 
| 41 | 40 | r19.21bi 3251 | . . . . . . . . 9
⊢
((∀𝑡 ∈
(𝑇 ∖ 𝑈)0 < (𝑃‘𝑡) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 0 < (𝑃‘𝑥)) | 
| 42 | 33, 41 | sylan 580 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 0 < (𝑃‘𝑥)) | 
| 43 | 32, 42 | elrpd 13074 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑥) ∈
ℝ+) | 
| 44 | 43 | 3adant3 1133 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → (𝑃‘𝑥) ∈
ℝ+) | 
| 45 |  | stoweidlem28.2 | . . . . . . . 8
⊢
Ⅎ𝑡𝜑 | 
| 46 | 9 | nfcri 2897 | . . . . . . . 8
⊢
Ⅎ𝑡 𝑥 ∈ (𝑇 ∖ 𝑈) | 
| 47 |  | nfra1 3284 | . . . . . . . 8
⊢
Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) | 
| 48 | 45, 46, 47 | nf3an 1901 | . . . . . . 7
⊢
Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) | 
| 49 |  | rspa 3248 | . . . . . . . . . 10
⊢
((∀𝑡 ∈
(𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) | 
| 50 | 49 | 3ad2antl3 1188 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) | 
| 51 |  | simpl2 1193 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 𝑥 ∈ (𝑇 ∖ 𝑈)) | 
| 52 |  | fvres 6925 | . . . . . . . . . 10
⊢ (𝑥 ∈ (𝑇 ∖ 𝑈) → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) = (𝑃‘𝑥)) | 
| 53 | 51, 52 | syl 17 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) = (𝑃‘𝑥)) | 
| 54 |  | fvres 6925 | . . . . . . . . . 10
⊢ (𝑡 ∈ (𝑇 ∖ 𝑈) → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) = (𝑃‘𝑡)) | 
| 55 | 54 | adantl 481 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) = (𝑃‘𝑡)) | 
| 56 | 50, 53, 55 | 3brtr3d 5174 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑥) ≤ (𝑃‘𝑡)) | 
| 57 | 56 | ex 412 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → (𝑡 ∈ (𝑇 ∖ 𝑈) → (𝑃‘𝑥) ≤ (𝑃‘𝑡))) | 
| 58 | 48, 57 | ralrimi 3257 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑃‘𝑥) ≤ (𝑃‘𝑡)) | 
| 59 |  | eleq1 2829 | . . . . . . . . 9
⊢ (𝑐 = (𝑃‘𝑥) → (𝑐 ∈ ℝ+ ↔ (𝑃‘𝑥) ∈
ℝ+)) | 
| 60 |  | breq1 5146 | . . . . . . . . . 10
⊢ (𝑐 = (𝑃‘𝑥) → (𝑐 ≤ (𝑃‘𝑡) ↔ (𝑃‘𝑥) ≤ (𝑃‘𝑡))) | 
| 61 | 60 | ralbidv 3178 | . . . . . . . . 9
⊢ (𝑐 = (𝑃‘𝑥) → (∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑃‘𝑥) ≤ (𝑃‘𝑡))) | 
| 62 | 59, 61 | anbi12d 632 | . . . . . . . 8
⊢ (𝑐 = (𝑃‘𝑥) → ((𝑐 ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) ↔ ((𝑃‘𝑥) ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑃‘𝑥) ≤ (𝑃‘𝑡)))) | 
| 63 | 62 | spcegv 3597 | . . . . . . 7
⊢ ((𝑃‘𝑥) ∈ ℝ+ → (((𝑃‘𝑥) ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑃‘𝑥) ≤ (𝑃‘𝑡)) → ∃𝑐(𝑐 ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)))) | 
| 64 | 44, 63 | syl 17 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → (((𝑃‘𝑥) ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑃‘𝑥) ≤ (𝑃‘𝑡)) → ∃𝑐(𝑐 ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)))) | 
| 65 | 44, 58, 64 | mp2and 699 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → ∃𝑐(𝑐 ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))) | 
| 66 |  | simpl1 1192 | . . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ (𝑐 ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))) → 𝜑) | 
| 67 |  | simprl 771 | . . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ (𝑐 ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))) → 𝑐 ∈ ℝ+) | 
| 68 |  | simprr 773 | . . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ (𝑐 ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) | 
| 69 |  | nfv 1914 | . . . . . . . 8
⊢
Ⅎ𝑡 𝑐 ∈
ℝ+ | 
| 70 |  | nfra1 3284 | . . . . . . . 8
⊢
Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡) | 
| 71 | 45, 69, 70 | nf3an 1901 | . . . . . . 7
⊢
Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) | 
| 72 |  | eqid 2737 | . . . . . . 7
⊢ if(𝑐 ≤ (1 / 2), 𝑐, (1 / 2)) = if(𝑐 ≤ (1 / 2), 𝑐, (1 / 2)) | 
| 73 | 28 | 3ad2ant1 1134 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → 𝑃:𝑇⟶ℝ) | 
| 74 |  | difssd 4137 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → (𝑇 ∖ 𝑈) ⊆ 𝑇) | 
| 75 |  | simp2 1138 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → 𝑐 ∈ ℝ+) | 
| 76 |  | simp3 1139 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) | 
| 77 | 71, 72, 73, 74, 75, 76 | stoweidlem5 46020 | . . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡))) | 
| 78 | 66, 67, 68, 77 | syl3anc 1373 | . . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ (𝑐 ∈ ℝ+ ∧
∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡))) | 
| 79 | 65, 78 | exlimddv 1935 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡))) | 
| 80 | 21, 22, 23, 79 | syl3anc 1373 | . . 3
⊢ ((((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡))) | 
| 81 |  | eqid 2737 | . . . . . 6
⊢ ∪ (𝐽
↾t (𝑇
∖ 𝑈)) = ∪ (𝐽
↾t (𝑇
∖ 𝑈)) | 
| 82 |  | stoweidlem28.5 | . . . . . . . 8
⊢ (𝜑 → 𝐽 ∈ Comp) | 
| 83 |  | stoweidlem28.8 | . . . . . . . . 9
⊢ (𝜑 → 𝑈 ∈ 𝐽) | 
| 84 |  | cmptop 23403 | . . . . . . . . . . 11
⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top) | 
| 85 | 82, 84 | syl 17 | . . . . . . . . . 10
⊢ (𝜑 → 𝐽 ∈ Top) | 
| 86 |  | elssuni 4937 | . . . . . . . . . . . 12
⊢ (𝑈 ∈ 𝐽 → 𝑈 ⊆ ∪ 𝐽) | 
| 87 | 83, 86 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑈 ⊆ ∪ 𝐽) | 
| 88 | 87, 25 | sseqtrrdi 4025 | . . . . . . . . . 10
⊢ (𝜑 → 𝑈 ⊆ 𝑇) | 
| 89 | 25 | isopn2 23040 | . . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ 𝑈 ⊆ 𝑇) → (𝑈 ∈ 𝐽 ↔ (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽))) | 
| 90 | 85, 88, 89 | syl2anc 584 | . . . . . . . . 9
⊢ (𝜑 → (𝑈 ∈ 𝐽 ↔ (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽))) | 
| 91 | 83, 90 | mpbid 232 | . . . . . . . 8
⊢ (𝜑 → (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽)) | 
| 92 |  | cmpcld 23410 | . . . . . . . 8
⊢ ((𝐽 ∈ Comp ∧ (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽)) → (𝐽 ↾t (𝑇 ∖ 𝑈)) ∈ Comp) | 
| 93 | 82, 91, 92 | syl2anc 584 | . . . . . . 7
⊢ (𝜑 → (𝐽 ↾t (𝑇 ∖ 𝑈)) ∈ Comp) | 
| 94 | 93 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (𝐽 ↾t (𝑇 ∖ 𝑈)) ∈ Comp) | 
| 95 | 27 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → 𝑃 ∈ (𝐽 Cn 𝐾)) | 
| 96 |  | difssd 4137 | . . . . . . 7
⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (𝑇 ∖ 𝑈) ⊆ 𝑇) | 
| 97 | 25 | cnrest 23293 | . . . . . . 7
⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ (𝑇 ∖ 𝑈) ⊆ 𝑇) → (𝑃 ↾ (𝑇 ∖ 𝑈)) ∈ ((𝐽 ↾t (𝑇 ∖ 𝑈)) Cn 𝐾)) | 
| 98 | 95, 96, 97 | syl2anc 584 | . . . . . 6
⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (𝑃 ↾ (𝑇 ∖ 𝑈)) ∈ ((𝐽 ↾t (𝑇 ∖ 𝑈)) Cn 𝐾)) | 
| 99 |  | difssd 4137 | . . . . . . . . . 10
⊢ (𝜑 → (𝑇 ∖ 𝑈) ⊆ 𝑇) | 
| 100 | 25 | restuni 23170 | . . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ (𝑇 ∖ 𝑈) ⊆ 𝑇) → (𝑇 ∖ 𝑈) = ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))) | 
| 101 | 85, 99, 100 | syl2anc 584 | . . . . . . . . 9
⊢ (𝜑 → (𝑇 ∖ 𝑈) = ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))) | 
| 102 | 101 | neeq1d 3000 | . . . . . . . 8
⊢ (𝜑 → ((𝑇 ∖ 𝑈) ≠ ∅ ↔ ∪ (𝐽
↾t (𝑇
∖ 𝑈)) ≠
∅)) | 
| 103 |  | df-ne 2941 | . . . . . . . 8
⊢ ((𝑇 ∖ 𝑈) ≠ ∅ ↔ ¬ (𝑇 ∖ 𝑈) = ∅) | 
| 104 | 102, 103 | bitr3di 286 | . . . . . . 7
⊢ (𝜑 → (∪ (𝐽
↾t (𝑇
∖ 𝑈)) ≠ ∅
↔ ¬ (𝑇 ∖
𝑈) =
∅)) | 
| 105 | 104 | biimpar 477 | . . . . . 6
⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∪ (𝐽
↾t (𝑇
∖ 𝑈)) ≠
∅) | 
| 106 | 81, 24, 94, 98, 105 | evth2 24992 | . . . . 5
⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑥 ∈ ∪ (𝐽
↾t (𝑇
∖ 𝑈))∀𝑠 ∈ ∪ (𝐽
↾t (𝑇
∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠)) | 
| 107 |  | nfcv 2905 | . . . . . . 7
⊢
Ⅎ𝑠∪ (𝐽
↾t (𝑇
∖ 𝑈)) | 
| 108 |  | nfcv 2905 | . . . . . . . . 9
⊢
Ⅎ𝑡𝐽 | 
| 109 |  | nfcv 2905 | . . . . . . . . 9
⊢
Ⅎ𝑡
↾t | 
| 110 | 108, 109,
9 | nfov 7461 | . . . . . . . 8
⊢
Ⅎ𝑡(𝐽 ↾t (𝑇 ∖ 𝑈)) | 
| 111 | 110 | nfuni 4914 | . . . . . . 7
⊢
Ⅎ𝑡∪ (𝐽
↾t (𝑇
∖ 𝑈)) | 
| 112 |  | nfcv 2905 | . . . . . . . . . 10
⊢
Ⅎ𝑡𝑃 | 
| 113 | 112, 9 | nfres 5999 | . . . . . . . . 9
⊢
Ⅎ𝑡(𝑃 ↾ (𝑇 ∖ 𝑈)) | 
| 114 |  | nfcv 2905 | . . . . . . . . 9
⊢
Ⅎ𝑡𝑥 | 
| 115 | 113, 114 | nffv 6916 | . . . . . . . 8
⊢
Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) | 
| 116 |  | nfcv 2905 | . . . . . . . 8
⊢
Ⅎ𝑡
≤ | 
| 117 |  | nfcv 2905 | . . . . . . . . 9
⊢
Ⅎ𝑡𝑠 | 
| 118 | 113, 117 | nffv 6916 | . . . . . . . 8
⊢
Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠) | 
| 119 | 115, 116,
118 | nfbr 5190 | . . . . . . 7
⊢
Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠) | 
| 120 |  | nfv 1914 | . . . . . . 7
⊢
Ⅎ𝑠((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) | 
| 121 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑠 = 𝑡 → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠) = ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) | 
| 122 | 121 | breq2d 5155 | . . . . . . 7
⊢ (𝑠 = 𝑡 → (((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠) ↔ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))) | 
| 123 | 107, 111,
119, 120, 122 | cbvralfw 3304 | . . . . . 6
⊢
(∀𝑠 ∈
∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠) ↔ ∀𝑡 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) | 
| 124 | 123 | rexbii 3094 | . . . . 5
⊢
(∃𝑥 ∈
∪ (𝐽 ↾t (𝑇 ∖ 𝑈))∀𝑠 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠) ↔ ∃𝑥 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))∀𝑡 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) | 
| 125 | 106, 124 | sylib 218 | . . . 4
⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑥 ∈ ∪ (𝐽
↾t (𝑇
∖ 𝑈))∀𝑡 ∈ ∪ (𝐽
↾t (𝑇
∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) | 
| 126 | 9, 111 | raleqf 3353 | . . . . . . 7
⊢ ((𝑇 ∖ 𝑈) = ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)) → (∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) ↔ ∀𝑡 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))) | 
| 127 | 126 | rexeqbi1dv 3339 | . . . . . 6
⊢ ((𝑇 ∖ 𝑈) = ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)) → (∃𝑥 ∈ (𝑇 ∖ 𝑈)∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) ↔ ∃𝑥 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))∀𝑡 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))) | 
| 128 | 101, 127 | syl 17 | . . . . 5
⊢ (𝜑 → (∃𝑥 ∈ (𝑇 ∖ 𝑈)∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) ↔ ∃𝑥 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))∀𝑡 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))) | 
| 129 | 128 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (∃𝑥 ∈ (𝑇 ∖ 𝑈)∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) ↔ ∃𝑥 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))∀𝑡 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))) | 
| 130 | 125, 129 | mpbird 257 | . . 3
⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑥 ∈ (𝑇 ∖ 𝑈)∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) | 
| 131 | 80, 130 | r19.29a 3162 | . 2
⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡))) | 
| 132 | 20, 131 | pm2.61dan 813 | 1
⊢ (𝜑 → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡))) |