| Step | Hyp | Ref
| Expression |
| 1 | | stoweidlem36.11 |
. . . . . 6
⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) / 𝑁)) |
| 2 | | stoweidlem36.5 |
. . . . . . 7
⊢
Ⅎ𝑡𝜑 |
| 3 | | stoweidlem36.6 |
. . . . . . . . . . . 12
⊢ 𝐾 = (topGen‘ran
(,)) |
| 4 | | stoweidlem36.8 |
. . . . . . . . . . . 12
⊢ 𝑇 = ∪
𝐽 |
| 5 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾) |
| 6 | | stoweidlem36.13 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾)) |
| 7 | | stoweidlem36.9 |
. . . . . . . . . . . . . 14
⊢ 𝐺 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐹‘𝑡))) |
| 8 | | stoweidlem36.18 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐹 ∈ 𝐴) |
| 9 | | stoweidlem36.3 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑡𝐹 |
| 10 | 9 | nfeq2 2923 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑡 𝑓 = 𝐹 |
| 11 | 9 | nfeq2 2923 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑡 𝑔 = 𝐹 |
| 12 | | stoweidlem36.14 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
| 13 | 10, 11, 12 | stoweidlem6 46021 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴) |
| 14 | 8, 8, 13 | mpd3an23 1465 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴) |
| 15 | 7, 14 | eqeltrid 2845 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐺 ∈ 𝐴) |
| 16 | 6, 15 | sseldd 3984 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) |
| 17 | 3, 4, 5, 16 | fcnre 45030 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺:𝑇⟶ℝ) |
| 18 | 17 | ffvelcdmda 7104 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ) |
| 19 | 18 | recnd 11289 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℂ) |
| 20 | | stoweidlem36.10 |
. . . . . . . . . . . 12
⊢ 𝑁 = sup(ran 𝐺, ℝ, < ) |
| 21 | | stoweidlem36.12 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐽 ∈ Comp) |
| 22 | | stoweidlem36.16 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑆 ∈ 𝑇) |
| 23 | 22 | ne0d 4342 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑇 ≠ ∅) |
| 24 | 4, 3, 21, 16, 23 | cncmpmax 45037 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (sup(ran 𝐺, ℝ, < ) ∈ ran 𝐺 ∧ sup(ran 𝐺, ℝ, < ) ∈ ℝ ∧
∀𝑠 ∈ 𝑇 (𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ))) |
| 25 | 24 | simp2d 1144 |
. . . . . . . . . . . 12
⊢ (𝜑 → sup(ran 𝐺, ℝ, < ) ∈
ℝ) |
| 26 | 20, 25 | eqeltrid 2845 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ ℝ) |
| 27 | 26 | recnd 11289 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 28 | 27 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈ ℂ) |
| 29 | | 0red 11264 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 ∈
ℝ) |
| 30 | 17, 22 | ffvelcdmd 7105 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐺‘𝑆) ∈ ℝ) |
| 31 | 6, 8 | sseldd 3984 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
| 32 | 3, 4, 5, 31 | fcnre 45030 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
| 33 | 32, 22 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐹‘𝑆) ∈ ℝ) |
| 34 | | stoweidlem36.19 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐹‘𝑆) ≠ (𝐹‘𝑍)) |
| 35 | | stoweidlem36.20 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐹‘𝑍) = 0) |
| 36 | 34, 35 | neeqtrd 3010 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐹‘𝑆) ≠ 0) |
| 37 | 33, 36 | msqgt0d 11830 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 < ((𝐹‘𝑆) · (𝐹‘𝑆))) |
| 38 | 33, 33 | remulcld 11291 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐹‘𝑆) · (𝐹‘𝑆)) ∈ ℝ) |
| 39 | | nfcv 2905 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑡𝑆 |
| 40 | 9, 39 | nffv 6916 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑡(𝐹‘𝑆) |
| 41 | | nfcv 2905 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑡
· |
| 42 | 40, 41, 40 | nfov 7461 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑡((𝐹‘𝑆) · (𝐹‘𝑆)) |
| 43 | | fveq2 6906 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑆 → (𝐹‘𝑡) = (𝐹‘𝑆)) |
| 44 | 43, 43 | oveq12d 7449 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑆 → ((𝐹‘𝑡) · (𝐹‘𝑡)) = ((𝐹‘𝑆) · (𝐹‘𝑆))) |
| 45 | 39, 42, 44, 7 | fvmptf 7037 |
. . . . . . . . . . . . . . 15
⊢ ((𝑆 ∈ 𝑇 ∧ ((𝐹‘𝑆) · (𝐹‘𝑆)) ∈ ℝ) → (𝐺‘𝑆) = ((𝐹‘𝑆) · (𝐹‘𝑆))) |
| 46 | 22, 38, 45 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐺‘𝑆) = ((𝐹‘𝑆) · (𝐹‘𝑆))) |
| 47 | 37, 46 | breqtrrd 5171 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 < (𝐺‘𝑆)) |
| 48 | 24 | simp3d 1145 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ∀𝑠 ∈ 𝑇 (𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < )) |
| 49 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 = 𝑆 → (𝐺‘𝑠) = (𝐺‘𝑆)) |
| 50 | 49 | breq1d 5153 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = 𝑆 → ((𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ) ↔ (𝐺‘𝑆) ≤ sup(ran 𝐺, ℝ, < ))) |
| 51 | 50 | rspccva 3621 |
. . . . . . . . . . . . . 14
⊢
((∀𝑠 ∈
𝑇 (𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ) ∧ 𝑆 ∈ 𝑇) → (𝐺‘𝑆) ≤ sup(ran 𝐺, ℝ, < )) |
| 52 | 48, 22, 51 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐺‘𝑆) ≤ sup(ran 𝐺, ℝ, < )) |
| 53 | 29, 30, 25, 47, 52 | ltletrd 11421 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 < sup(ran 𝐺, ℝ, <
)) |
| 54 | 53 | gt0ne0d 11827 |
. . . . . . . . . . 11
⊢ (𝜑 → sup(ran 𝐺, ℝ, < ) ≠ 0) |
| 55 | 20 | neeq1i 3005 |
. . . . . . . . . . 11
⊢ (𝑁 ≠ 0 ↔ sup(ran 𝐺, ℝ, < ) ≠
0) |
| 56 | 54, 55 | sylibr 234 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ≠ 0) |
| 57 | 56 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ≠ 0) |
| 58 | 19, 28, 57 | divrecd 12046 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 𝑁) = ((𝐺‘𝑡) · (1 / 𝑁))) |
| 59 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇) |
| 60 | 26, 56 | rereccld 12094 |
. . . . . . . . . . 11
⊢ (𝜑 → (1 / 𝑁) ∈ ℝ) |
| 61 | 60 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 / 𝑁) ∈ ℝ) |
| 62 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) = (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) |
| 63 | 62 | fvmpt2 7027 |
. . . . . . . . . 10
⊢ ((𝑡 ∈ 𝑇 ∧ (1 / 𝑁) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡) = (1 / 𝑁)) |
| 64 | 59, 61, 63 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡) = (1 / 𝑁)) |
| 65 | 64 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡)) = ((𝐺‘𝑡) · (1 / 𝑁))) |
| 66 | 58, 65 | eqtr4d 2780 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 𝑁) = ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡))) |
| 67 | 2, 66 | mpteq2da 5240 |
. . . . . 6
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) / 𝑁)) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡)))) |
| 68 | 1, 67 | eqtrid 2789 |
. . . . 5
⊢ (𝜑 → 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡)))) |
| 69 | | stoweidlem36.15 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) |
| 70 | 69 | stoweidlem4 46019 |
. . . . . . 7
⊢ ((𝜑 ∧ (1 / 𝑁) ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) ∈ 𝐴) |
| 71 | 60, 70 | mpdan 687 |
. . . . . 6
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) ∈ 𝐴) |
| 72 | | stoweidlem36.4 |
. . . . . . . 8
⊢
Ⅎ𝑡𝐺 |
| 73 | 72 | nfeq2 2923 |
. . . . . . 7
⊢
Ⅎ𝑡 𝑓 = 𝐺 |
| 74 | | nfmpt1 5250 |
. . . . . . . 8
⊢
Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) |
| 75 | 74 | nfeq2 2923 |
. . . . . . 7
⊢
Ⅎ𝑡 𝑔 = (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) |
| 76 | 73, 75, 12 | stoweidlem6 46021 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡))) ∈ 𝐴) |
| 77 | 15, 71, 76 | mpd3an23 1465 |
. . . . 5
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡))) ∈ 𝐴) |
| 78 | 68, 77 | eqeltrd 2841 |
. . . 4
⊢ (𝜑 → 𝐻 ∈ 𝐴) |
| 79 | | stoweidlem36.17 |
. . . . . . 7
⊢ (𝜑 → 𝑍 ∈ 𝑇) |
| 80 | 17, 79 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ (𝜑 → (𝐺‘𝑍) ∈ ℝ) |
| 81 | 80, 26, 56 | redivcld 12095 |
. . . . . . 7
⊢ (𝜑 → ((𝐺‘𝑍) / 𝑁) ∈ ℝ) |
| 82 | | nfcv 2905 |
. . . . . . . 8
⊢
Ⅎ𝑡𝑍 |
| 83 | 72, 82 | nffv 6916 |
. . . . . . . . 9
⊢
Ⅎ𝑡(𝐺‘𝑍) |
| 84 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑡
/ |
| 85 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑡𝑁 |
| 86 | 83, 84, 85 | nfov 7461 |
. . . . . . . 8
⊢
Ⅎ𝑡((𝐺‘𝑍) / 𝑁) |
| 87 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑡 = 𝑍 → (𝐺‘𝑡) = (𝐺‘𝑍)) |
| 88 | 87 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝑡 = 𝑍 → ((𝐺‘𝑡) / 𝑁) = ((𝐺‘𝑍) / 𝑁)) |
| 89 | 82, 86, 88, 1 | fvmptf 7037 |
. . . . . . 7
⊢ ((𝑍 ∈ 𝑇 ∧ ((𝐺‘𝑍) / 𝑁) ∈ ℝ) → (𝐻‘𝑍) = ((𝐺‘𝑍) / 𝑁)) |
| 90 | 79, 81, 89 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (𝐻‘𝑍) = ((𝐺‘𝑍) / 𝑁)) |
| 91 | | 0re 11263 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
| 92 | 35, 91 | eqeltrdi 2849 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹‘𝑍) ∈ ℝ) |
| 93 | 92, 92 | remulcld 11291 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐹‘𝑍) · (𝐹‘𝑍)) ∈ ℝ) |
| 94 | 9, 82 | nffv 6916 |
. . . . . . . . . . 11
⊢
Ⅎ𝑡(𝐹‘𝑍) |
| 95 | 94, 41, 94 | nfov 7461 |
. . . . . . . . . 10
⊢
Ⅎ𝑡((𝐹‘𝑍) · (𝐹‘𝑍)) |
| 96 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑍 → (𝐹‘𝑡) = (𝐹‘𝑍)) |
| 97 | 96, 96 | oveq12d 7449 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑍 → ((𝐹‘𝑡) · (𝐹‘𝑡)) = ((𝐹‘𝑍) · (𝐹‘𝑍))) |
| 98 | 82, 95, 97, 7 | fvmptf 7037 |
. . . . . . . . 9
⊢ ((𝑍 ∈ 𝑇 ∧ ((𝐹‘𝑍) · (𝐹‘𝑍)) ∈ ℝ) → (𝐺‘𝑍) = ((𝐹‘𝑍) · (𝐹‘𝑍))) |
| 99 | 79, 93, 98 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → (𝐺‘𝑍) = ((𝐹‘𝑍) · (𝐹‘𝑍))) |
| 100 | 35, 35 | oveq12d 7449 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐹‘𝑍) · (𝐹‘𝑍)) = (0 · 0)) |
| 101 | | 0cn 11253 |
. . . . . . . . . 10
⊢ 0 ∈
ℂ |
| 102 | 101 | mul02i 11450 |
. . . . . . . . 9
⊢ (0
· 0) = 0 |
| 103 | 100, 102 | eqtrdi 2793 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘𝑍) · (𝐹‘𝑍)) = 0) |
| 104 | 99, 103 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → (𝐺‘𝑍) = 0) |
| 105 | 104 | oveq1d 7446 |
. . . . . 6
⊢ (𝜑 → ((𝐺‘𝑍) / 𝑁) = (0 / 𝑁)) |
| 106 | 27, 56 | div0d 12042 |
. . . . . 6
⊢ (𝜑 → (0 / 𝑁) = 0) |
| 107 | 90, 105, 106 | 3eqtrd 2781 |
. . . . 5
⊢ (𝜑 → (𝐻‘𝑍) = 0) |
| 108 | 32 | ffvelcdmda 7104 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ) |
| 109 | 108 | msqge0d 11831 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝐹‘𝑡) · (𝐹‘𝑡))) |
| 110 | 108, 108 | remulcld 11291 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) · (𝐹‘𝑡)) ∈ ℝ) |
| 111 | 7 | fvmpt2 7027 |
. . . . . . . . . . . 12
⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐹‘𝑡) · (𝐹‘𝑡)) ∈ ℝ) → (𝐺‘𝑡) = ((𝐹‘𝑡) · (𝐹‘𝑡))) |
| 112 | 59, 110, 111 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) = ((𝐹‘𝑡) · (𝐹‘𝑡))) |
| 113 | 109, 112 | breqtrrd 5171 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐺‘𝑡)) |
| 114 | 26 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈ ℝ) |
| 115 | 53, 20 | breqtrrdi 5185 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 < 𝑁) |
| 116 | 115 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 < 𝑁) |
| 117 | | divge0 12137 |
. . . . . . . . . 10
⊢ ((((𝐺‘𝑡) ∈ ℝ ∧ 0 ≤ (𝐺‘𝑡)) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → 0 ≤ ((𝐺‘𝑡) / 𝑁)) |
| 118 | 18, 113, 114, 116, 117 | syl22anc 839 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝐺‘𝑡) / 𝑁)) |
| 119 | 18, 114, 57 | redivcld 12095 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 𝑁) ∈ ℝ) |
| 120 | 1 | fvmpt2 7027 |
. . . . . . . . . 10
⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐺‘𝑡) / 𝑁) ∈ ℝ) → (𝐻‘𝑡) = ((𝐺‘𝑡) / 𝑁)) |
| 121 | 59, 119, 120 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) = ((𝐺‘𝑡) / 𝑁)) |
| 122 | 118, 121 | breqtrrd 5171 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐻‘𝑡)) |
| 123 | 19 | div1d 12035 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 1) = (𝐺‘𝑡)) |
| 124 | | fveq2 6906 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = 𝑡 → (𝐺‘𝑠) = (𝐺‘𝑡)) |
| 125 | 124 | breq1d 5153 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑡 → ((𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ) ↔ (𝐺‘𝑡) ≤ sup(ran 𝐺, ℝ, < ))) |
| 126 | 125 | rspccva 3621 |
. . . . . . . . . . . . 13
⊢
((∀𝑠 ∈
𝑇 (𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ≤ sup(ran 𝐺, ℝ, < )) |
| 127 | 48, 126 | sylan 580 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ≤ sup(ran 𝐺, ℝ, < )) |
| 128 | 127, 20 | breqtrrdi 5185 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ≤ 𝑁) |
| 129 | 123, 128 | eqbrtrd 5165 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 1) ≤ 𝑁) |
| 130 | | 1red 11262 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ) |
| 131 | | 0lt1 11785 |
. . . . . . . . . . . 12
⊢ 0 <
1 |
| 132 | 131 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 < 1) |
| 133 | | lediv23 12160 |
. . . . . . . . . . 11
⊢ (((𝐺‘𝑡) ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁) ∧ (1 ∈ ℝ ∧
0 < 1)) → (((𝐺‘𝑡) / 𝑁) ≤ 1 ↔ ((𝐺‘𝑡) / 1) ≤ 𝑁)) |
| 134 | 18, 114, 116, 130, 132, 133 | syl122anc 1381 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((𝐺‘𝑡) / 𝑁) ≤ 1 ↔ ((𝐺‘𝑡) / 1) ≤ 𝑁)) |
| 135 | 129, 134 | mpbird 257 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 𝑁) ≤ 1) |
| 136 | 121, 135 | eqbrtrd 5165 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ≤ 1) |
| 137 | 122, 136 | jca 511 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)) |
| 138 | 137 | ex 412 |
. . . . . 6
⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))) |
| 139 | 2, 138 | ralrimi 3257 |
. . . . 5
⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)) |
| 140 | 107, 139 | jca 511 |
. . . 4
⊢ (𝜑 → ((𝐻‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))) |
| 141 | | fveq1 6905 |
. . . . . . 7
⊢ (ℎ = 𝐻 → (ℎ‘𝑍) = (𝐻‘𝑍)) |
| 142 | 141 | eqeq1d 2739 |
. . . . . 6
⊢ (ℎ = 𝐻 → ((ℎ‘𝑍) = 0 ↔ (𝐻‘𝑍) = 0)) |
| 143 | | stoweidlem36.2 |
. . . . . . . 8
⊢
Ⅎ𝑡𝐻 |
| 144 | 143 | nfeq2 2923 |
. . . . . . 7
⊢
Ⅎ𝑡 ℎ = 𝐻 |
| 145 | | fveq1 6905 |
. . . . . . . . 9
⊢ (ℎ = 𝐻 → (ℎ‘𝑡) = (𝐻‘𝑡)) |
| 146 | 145 | breq2d 5155 |
. . . . . . . 8
⊢ (ℎ = 𝐻 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝐻‘𝑡))) |
| 147 | 145 | breq1d 5153 |
. . . . . . . 8
⊢ (ℎ = 𝐻 → ((ℎ‘𝑡) ≤ 1 ↔ (𝐻‘𝑡) ≤ 1)) |
| 148 | 146, 147 | anbi12d 632 |
. . . . . . 7
⊢ (ℎ = 𝐻 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))) |
| 149 | 144, 148 | ralbid 3273 |
. . . . . 6
⊢ (ℎ = 𝐻 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))) |
| 150 | 142, 149 | anbi12d 632 |
. . . . 5
⊢ (ℎ = 𝐻 → (((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)) ↔ ((𝐻‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))) |
| 151 | 150 | elrab 3692 |
. . . 4
⊢ (𝐻 ∈ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} ↔ (𝐻 ∈ 𝐴 ∧ ((𝐻‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)))) |
| 152 | 78, 140, 151 | sylanbrc 583 |
. . 3
⊢ (𝜑 → 𝐻 ∈ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}) |
| 153 | | stoweidlem36.7 |
. . 3
⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} |
| 154 | 152, 153 | eleqtrrdi 2852 |
. 2
⊢ (𝜑 → 𝐻 ∈ 𝑄) |
| 155 | 30, 26, 47, 115 | divgt0d 12203 |
. . 3
⊢ (𝜑 → 0 < ((𝐺‘𝑆) / 𝑁)) |
| 156 | 30, 26, 56 | redivcld 12095 |
. . . 4
⊢ (𝜑 → ((𝐺‘𝑆) / 𝑁) ∈ ℝ) |
| 157 | 72, 39 | nffv 6916 |
. . . . . 6
⊢
Ⅎ𝑡(𝐺‘𝑆) |
| 158 | 157, 84, 85 | nfov 7461 |
. . . . 5
⊢
Ⅎ𝑡((𝐺‘𝑆) / 𝑁) |
| 159 | | fveq2 6906 |
. . . . . 6
⊢ (𝑡 = 𝑆 → (𝐺‘𝑡) = (𝐺‘𝑆)) |
| 160 | 159 | oveq1d 7446 |
. . . . 5
⊢ (𝑡 = 𝑆 → ((𝐺‘𝑡) / 𝑁) = ((𝐺‘𝑆) / 𝑁)) |
| 161 | 39, 158, 160, 1 | fvmptf 7037 |
. . . 4
⊢ ((𝑆 ∈ 𝑇 ∧ ((𝐺‘𝑆) / 𝑁) ∈ ℝ) → (𝐻‘𝑆) = ((𝐺‘𝑆) / 𝑁)) |
| 162 | 22, 156, 161 | syl2anc 584 |
. . 3
⊢ (𝜑 → (𝐻‘𝑆) = ((𝐺‘𝑆) / 𝑁)) |
| 163 | 155, 162 | breqtrrd 5171 |
. 2
⊢ (𝜑 → 0 < (𝐻‘𝑆)) |
| 164 | | nfcv 2905 |
. . . 4
⊢
Ⅎℎ𝐻 |
| 165 | | stoweidlem36.1 |
. . . . . 6
⊢
Ⅎℎ𝑄 |
| 166 | 165 | nfel2 2924 |
. . . . 5
⊢
Ⅎℎ 𝐻 ∈ 𝑄 |
| 167 | | nfv 1914 |
. . . . 5
⊢
Ⅎℎ0 < (𝐻‘𝑆) |
| 168 | 166, 167 | nfan 1899 |
. . . 4
⊢
Ⅎℎ(𝐻 ∈ 𝑄 ∧ 0 < (𝐻‘𝑆)) |
| 169 | | eleq1 2829 |
. . . . 5
⊢ (ℎ = 𝐻 → (ℎ ∈ 𝑄 ↔ 𝐻 ∈ 𝑄)) |
| 170 | | fveq1 6905 |
. . . . . 6
⊢ (ℎ = 𝐻 → (ℎ‘𝑆) = (𝐻‘𝑆)) |
| 171 | 170 | breq2d 5155 |
. . . . 5
⊢ (ℎ = 𝐻 → (0 < (ℎ‘𝑆) ↔ 0 < (𝐻‘𝑆))) |
| 172 | 169, 171 | anbi12d 632 |
. . . 4
⊢ (ℎ = 𝐻 → ((ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆)) ↔ (𝐻 ∈ 𝑄 ∧ 0 < (𝐻‘𝑆)))) |
| 173 | 164, 168,
172 | spcegf 3592 |
. . 3
⊢ (𝐻 ∈ 𝑄 → ((𝐻 ∈ 𝑄 ∧ 0 < (𝐻‘𝑆)) → ∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆)))) |
| 174 | 173 | anabsi5 669 |
. 2
⊢ ((𝐻 ∈ 𝑄 ∧ 0 < (𝐻‘𝑆)) → ∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆))) |
| 175 | 154, 163,
174 | syl2anc 584 |
1
⊢ (𝜑 → ∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆))) |