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 2622 |
. . . . . . . . . . . 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 2780 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑡 𝑓 = 𝐹 |
11 | 9 | nfeq2 2780 |
. . . . . . . . . . . . . . . 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 40223 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴) |
14 | 8, 8, 13 | mpd3an23 1426 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴) |
15 | 7, 14 | syl5eqel 2705 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐺 ∈ 𝐴) |
16 | 6, 15 | sseldd 3604 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) |
17 | 3, 4, 5, 16 | fcnre 39184 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺:𝑇⟶ℝ) |
18 | 17 | ffvelrnda 6359 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ) |
19 | 18 | recnd 10068 |
. . . . . . . . 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 | | ne0i 3921 |
. . . . . . . . . . . . . . 15
⊢ (𝑆 ∈ 𝑇 → 𝑇 ≠ ∅) |
24 | 22, 23 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑇 ≠ ∅) |
25 | 4, 3, 21, 16, 24 | cncmpmax 39191 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (sup(ran 𝐺, ℝ, < ) ∈ ran 𝐺 ∧ sup(ran 𝐺, ℝ, < ) ∈ ℝ ∧
∀𝑠 ∈ 𝑇 (𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ))) |
26 | 25 | simp2d 1074 |
. . . . . . . . . . . 12
⊢ (𝜑 → sup(ran 𝐺, ℝ, < ) ∈
ℝ) |
27 | 20, 26 | syl5eqel 2705 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ ℝ) |
28 | 27 | recnd 10068 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ ℂ) |
29 | 28 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈ ℂ) |
30 | | 0red 10041 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 ∈
ℝ) |
31 | 17, 22 | ffvelrnd 6360 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐺‘𝑆) ∈ ℝ) |
32 | 6, 8 | sseldd 3604 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
33 | 3, 4, 5, 32 | fcnre 39184 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
34 | 33, 22 | ffvelrnd 6360 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐹‘𝑆) ∈ ℝ) |
35 | | stoweidlem36.19 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐹‘𝑆) ≠ (𝐹‘𝑍)) |
36 | | stoweidlem36.20 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐹‘𝑍) = 0) |
37 | 35, 36 | neeqtrd 2863 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐹‘𝑆) ≠ 0) |
38 | 34, 37 | msqgt0d 10595 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 < ((𝐹‘𝑆) · (𝐹‘𝑆))) |
39 | 34, 34 | remulcld 10070 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐹‘𝑆) · (𝐹‘𝑆)) ∈ ℝ) |
40 | | nfcv 2764 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑡𝑆 |
41 | 9, 40 | nffv 6198 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑡(𝐹‘𝑆) |
42 | | nfcv 2764 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑡
· |
43 | 41, 42, 41 | nfov 6676 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑡((𝐹‘𝑆) · (𝐹‘𝑆)) |
44 | | fveq2 6191 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑆 → (𝐹‘𝑡) = (𝐹‘𝑆)) |
45 | 44, 44 | oveq12d 6668 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑆 → ((𝐹‘𝑡) · (𝐹‘𝑡)) = ((𝐹‘𝑆) · (𝐹‘𝑆))) |
46 | 40, 43, 45, 7 | fvmptf 6301 |
. . . . . . . . . . . . . . 15
⊢ ((𝑆 ∈ 𝑇 ∧ ((𝐹‘𝑆) · (𝐹‘𝑆)) ∈ ℝ) → (𝐺‘𝑆) = ((𝐹‘𝑆) · (𝐹‘𝑆))) |
47 | 22, 39, 46 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐺‘𝑆) = ((𝐹‘𝑆) · (𝐹‘𝑆))) |
48 | 38, 47 | breqtrrd 4681 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 < (𝐺‘𝑆)) |
49 | 25 | simp3d 1075 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ∀𝑠 ∈ 𝑇 (𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < )) |
50 | | fveq2 6191 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 = 𝑆 → (𝐺‘𝑠) = (𝐺‘𝑆)) |
51 | 50 | breq1d 4663 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = 𝑆 → ((𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ) ↔ (𝐺‘𝑆) ≤ sup(ran 𝐺, ℝ, < ))) |
52 | 51 | rspccva 3308 |
. . . . . . . . . . . . . 14
⊢
((∀𝑠 ∈
𝑇 (𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ) ∧ 𝑆 ∈ 𝑇) → (𝐺‘𝑆) ≤ sup(ran 𝐺, ℝ, < )) |
53 | 49, 22, 52 | syl2anc 693 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐺‘𝑆) ≤ sup(ran 𝐺, ℝ, < )) |
54 | 30, 31, 26, 48, 53 | ltletrd 10197 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 < sup(ran 𝐺, ℝ, <
)) |
55 | 54 | gt0ne0d 10592 |
. . . . . . . . . . 11
⊢ (𝜑 → sup(ran 𝐺, ℝ, < ) ≠ 0) |
56 | 20 | neeq1i 2858 |
. . . . . . . . . . 11
⊢ (𝑁 ≠ 0 ↔ sup(ran 𝐺, ℝ, < ) ≠
0) |
57 | 55, 56 | sylibr 224 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ≠ 0) |
58 | 57 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ≠ 0) |
59 | 19, 29, 58 | divrecd 10804 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 𝑁) = ((𝐺‘𝑡) · (1 / 𝑁))) |
60 | | simpr 477 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇) |
61 | 27, 57 | rereccld 10852 |
. . . . . . . . . . 11
⊢ (𝜑 → (1 / 𝑁) ∈ ℝ) |
62 | 61 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 / 𝑁) ∈ ℝ) |
63 | | eqid 2622 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) = (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) |
64 | 63 | fvmpt2 6291 |
. . . . . . . . . 10
⊢ ((𝑡 ∈ 𝑇 ∧ (1 / 𝑁) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡) = (1 / 𝑁)) |
65 | 60, 62, 64 | syl2anc 693 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡) = (1 / 𝑁)) |
66 | 65 | oveq2d 6666 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡)) = ((𝐺‘𝑡) · (1 / 𝑁))) |
67 | 59, 66 | eqtr4d 2659 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 𝑁) = ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡))) |
68 | 2, 67 | mpteq2da 4743 |
. . . . . 6
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) / 𝑁)) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡)))) |
69 | 1, 68 | syl5eq 2668 |
. . . . 5
⊢ (𝜑 → 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡)))) |
70 | | stoweidlem36.15 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) |
71 | 70 | stoweidlem4 40221 |
. . . . . . 7
⊢ ((𝜑 ∧ (1 / 𝑁) ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) ∈ 𝐴) |
72 | 61, 71 | mpdan 702 |
. . . . . 6
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) ∈ 𝐴) |
73 | | stoweidlem36.4 |
. . . . . . . 8
⊢
Ⅎ𝑡𝐺 |
74 | 73 | nfeq2 2780 |
. . . . . . 7
⊢
Ⅎ𝑡 𝑓 = 𝐺 |
75 | | nfmpt1 4747 |
. . . . . . . 8
⊢
Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) |
76 | 75 | nfeq2 2780 |
. . . . . . 7
⊢
Ⅎ𝑡 𝑔 = (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) |
77 | 74, 76, 12 | stoweidlem6 40223 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ (𝑡 ∈ 𝑇 ↦ (1 / 𝑁)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡))) ∈ 𝐴) |
78 | 15, 72, 77 | mpd3an23 1426 |
. . . . 5
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) · ((𝑡 ∈ 𝑇 ↦ (1 / 𝑁))‘𝑡))) ∈ 𝐴) |
79 | 69, 78 | eqeltrd 2701 |
. . . 4
⊢ (𝜑 → 𝐻 ∈ 𝐴) |
80 | | stoweidlem36.17 |
. . . . . . 7
⊢ (𝜑 → 𝑍 ∈ 𝑇) |
81 | 17, 80 | ffvelrnd 6360 |
. . . . . . . 8
⊢ (𝜑 → (𝐺‘𝑍) ∈ ℝ) |
82 | 81, 27, 57 | redivcld 10853 |
. . . . . . 7
⊢ (𝜑 → ((𝐺‘𝑍) / 𝑁) ∈ ℝ) |
83 | | nfcv 2764 |
. . . . . . . 8
⊢
Ⅎ𝑡𝑍 |
84 | 73, 83 | nffv 6198 |
. . . . . . . . 9
⊢
Ⅎ𝑡(𝐺‘𝑍) |
85 | | nfcv 2764 |
. . . . . . . . 9
⊢
Ⅎ𝑡
/ |
86 | | nfcv 2764 |
. . . . . . . . 9
⊢
Ⅎ𝑡𝑁 |
87 | 84, 85, 86 | nfov 6676 |
. . . . . . . 8
⊢
Ⅎ𝑡((𝐺‘𝑍) / 𝑁) |
88 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑡 = 𝑍 → (𝐺‘𝑡) = (𝐺‘𝑍)) |
89 | 88 | oveq1d 6665 |
. . . . . . . 8
⊢ (𝑡 = 𝑍 → ((𝐺‘𝑡) / 𝑁) = ((𝐺‘𝑍) / 𝑁)) |
90 | 83, 87, 89, 1 | fvmptf 6301 |
. . . . . . 7
⊢ ((𝑍 ∈ 𝑇 ∧ ((𝐺‘𝑍) / 𝑁) ∈ ℝ) → (𝐻‘𝑍) = ((𝐺‘𝑍) / 𝑁)) |
91 | 80, 82, 90 | syl2anc 693 |
. . . . . 6
⊢ (𝜑 → (𝐻‘𝑍) = ((𝐺‘𝑍) / 𝑁)) |
92 | | 0re 10040 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
93 | 36, 92 | syl6eqel 2709 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹‘𝑍) ∈ ℝ) |
94 | 93, 93 | remulcld 10070 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐹‘𝑍) · (𝐹‘𝑍)) ∈ ℝ) |
95 | 9, 83 | nffv 6198 |
. . . . . . . . . . 11
⊢
Ⅎ𝑡(𝐹‘𝑍) |
96 | 95, 42, 95 | nfov 6676 |
. . . . . . . . . 10
⊢
Ⅎ𝑡((𝐹‘𝑍) · (𝐹‘𝑍)) |
97 | | fveq2 6191 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑍 → (𝐹‘𝑡) = (𝐹‘𝑍)) |
98 | 97, 97 | oveq12d 6668 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑍 → ((𝐹‘𝑡) · (𝐹‘𝑡)) = ((𝐹‘𝑍) · (𝐹‘𝑍))) |
99 | 83, 96, 98, 7 | fvmptf 6301 |
. . . . . . . . 9
⊢ ((𝑍 ∈ 𝑇 ∧ ((𝐹‘𝑍) · (𝐹‘𝑍)) ∈ ℝ) → (𝐺‘𝑍) = ((𝐹‘𝑍) · (𝐹‘𝑍))) |
100 | 80, 94, 99 | syl2anc 693 |
. . . . . . . 8
⊢ (𝜑 → (𝐺‘𝑍) = ((𝐹‘𝑍) · (𝐹‘𝑍))) |
101 | 36, 36 | oveq12d 6668 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐹‘𝑍) · (𝐹‘𝑍)) = (0 · 0)) |
102 | | 0cn 10032 |
. . . . . . . . . 10
⊢ 0 ∈
ℂ |
103 | 102 | mul02i 10225 |
. . . . . . . . 9
⊢ (0
· 0) = 0 |
104 | 101, 103 | syl6eq 2672 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘𝑍) · (𝐹‘𝑍)) = 0) |
105 | 100, 104 | eqtrd 2656 |
. . . . . . 7
⊢ (𝜑 → (𝐺‘𝑍) = 0) |
106 | 105 | oveq1d 6665 |
. . . . . 6
⊢ (𝜑 → ((𝐺‘𝑍) / 𝑁) = (0 / 𝑁)) |
107 | 28, 57 | div0d 10800 |
. . . . . 6
⊢ (𝜑 → (0 / 𝑁) = 0) |
108 | 91, 106, 107 | 3eqtrd 2660 |
. . . . 5
⊢ (𝜑 → (𝐻‘𝑍) = 0) |
109 | 33 | ffvelrnda 6359 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ) |
110 | 109 | msqge0d 10596 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝐹‘𝑡) · (𝐹‘𝑡))) |
111 | 109, 109 | remulcld 10070 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) · (𝐹‘𝑡)) ∈ ℝ) |
112 | 7 | fvmpt2 6291 |
. . . . . . . . . . . 12
⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐹‘𝑡) · (𝐹‘𝑡)) ∈ ℝ) → (𝐺‘𝑡) = ((𝐹‘𝑡) · (𝐹‘𝑡))) |
113 | 60, 111, 112 | syl2anc 693 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) = ((𝐹‘𝑡) · (𝐹‘𝑡))) |
114 | 110, 113 | breqtrrd 4681 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐺‘𝑡)) |
115 | 27 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈ ℝ) |
116 | 54, 20 | syl6breqr 4695 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 < 𝑁) |
117 | 116 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 < 𝑁) |
118 | | divge0 10892 |
. . . . . . . . . 10
⊢ ((((𝐺‘𝑡) ∈ ℝ ∧ 0 ≤ (𝐺‘𝑡)) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → 0 ≤ ((𝐺‘𝑡) / 𝑁)) |
119 | 18, 114, 115, 117, 118 | syl22anc 1327 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝐺‘𝑡) / 𝑁)) |
120 | 18, 115, 58 | redivcld 10853 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 𝑁) ∈ ℝ) |
121 | 1 | fvmpt2 6291 |
. . . . . . . . . 10
⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐺‘𝑡) / 𝑁) ∈ ℝ) → (𝐻‘𝑡) = ((𝐺‘𝑡) / 𝑁)) |
122 | 60, 120, 121 | syl2anc 693 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) = ((𝐺‘𝑡) / 𝑁)) |
123 | 119, 122 | breqtrrd 4681 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐻‘𝑡)) |
124 | 19 | div1d 10793 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 1) = (𝐺‘𝑡)) |
125 | | fveq2 6191 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = 𝑡 → (𝐺‘𝑠) = (𝐺‘𝑡)) |
126 | 125 | breq1d 4663 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑡 → ((𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ) ↔ (𝐺‘𝑡) ≤ sup(ran 𝐺, ℝ, < ))) |
127 | 126 | rspccva 3308 |
. . . . . . . . . . . . 13
⊢
((∀𝑠 ∈
𝑇 (𝐺‘𝑠) ≤ sup(ran 𝐺, ℝ, < ) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ≤ sup(ran 𝐺, ℝ, < )) |
128 | 49, 127 | sylan 488 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ≤ sup(ran 𝐺, ℝ, < )) |
129 | 128, 20 | syl6breqr 4695 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ≤ 𝑁) |
130 | 124, 129 | eqbrtrd 4675 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 1) ≤ 𝑁) |
131 | | 1red 10055 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ) |
132 | | 0lt1 10550 |
. . . . . . . . . . . 12
⊢ 0 <
1 |
133 | 132 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 < 1) |
134 | | lediv23 10915 |
. . . . . . . . . . 11
⊢ (((𝐺‘𝑡) ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁) ∧ (1 ∈ ℝ ∧
0 < 1)) → (((𝐺‘𝑡) / 𝑁) ≤ 1 ↔ ((𝐺‘𝑡) / 1) ≤ 𝑁)) |
135 | 18, 115, 117, 131, 133, 134 | syl122anc 1335 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((𝐺‘𝑡) / 𝑁) ≤ 1 ↔ ((𝐺‘𝑡) / 1) ≤ 𝑁)) |
136 | 130, 135 | mpbird 247 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) / 𝑁) ≤ 1) |
137 | 122, 136 | eqbrtrd 4675 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ≤ 1) |
138 | 123, 137 | jca 554 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)) |
139 | 138 | ex 450 |
. . . . . 6
⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))) |
140 | 2, 139 | ralrimi 2957 |
. . . . 5
⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)) |
141 | 108, 140 | jca 554 |
. . . 4
⊢ (𝜑 → ((𝐻‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))) |
142 | | fveq1 6190 |
. . . . . . 7
⊢ (ℎ = 𝐻 → (ℎ‘𝑍) = (𝐻‘𝑍)) |
143 | 142 | eqeq1d 2624 |
. . . . . 6
⊢ (ℎ = 𝐻 → ((ℎ‘𝑍) = 0 ↔ (𝐻‘𝑍) = 0)) |
144 | | stoweidlem36.2 |
. . . . . . . 8
⊢
Ⅎ𝑡𝐻 |
145 | 144 | nfeq2 2780 |
. . . . . . 7
⊢
Ⅎ𝑡 ℎ = 𝐻 |
146 | | fveq1 6190 |
. . . . . . . . 9
⊢ (ℎ = 𝐻 → (ℎ‘𝑡) = (𝐻‘𝑡)) |
147 | 146 | breq2d 4665 |
. . . . . . . 8
⊢ (ℎ = 𝐻 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝐻‘𝑡))) |
148 | 146 | breq1d 4663 |
. . . . . . . 8
⊢ (ℎ = 𝐻 → ((ℎ‘𝑡) ≤ 1 ↔ (𝐻‘𝑡) ≤ 1)) |
149 | 147, 148 | anbi12d 747 |
. . . . . . 7
⊢ (ℎ = 𝐻 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))) |
150 | 145, 149 | ralbid 2983 |
. . . . . 6
⊢ (ℎ = 𝐻 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))) |
151 | 143, 150 | anbi12d 747 |
. . . . 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)))) |
152 | 151 | elrab 3363 |
. . . 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)))) |
153 | 79, 141, 152 | sylanbrc 698 |
. . 3
⊢ (𝜑 → 𝐻 ∈ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}) |
154 | | stoweidlem36.7 |
. . 3
⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} |
155 | 153, 154 | syl6eleqr 2712 |
. 2
⊢ (𝜑 → 𝐻 ∈ 𝑄) |
156 | 31, 27, 48, 116 | divgt0d 10959 |
. . 3
⊢ (𝜑 → 0 < ((𝐺‘𝑆) / 𝑁)) |
157 | 31, 27, 57 | redivcld 10853 |
. . . 4
⊢ (𝜑 → ((𝐺‘𝑆) / 𝑁) ∈ ℝ) |
158 | 73, 40 | nffv 6198 |
. . . . . 6
⊢
Ⅎ𝑡(𝐺‘𝑆) |
159 | 158, 85, 86 | nfov 6676 |
. . . . 5
⊢
Ⅎ𝑡((𝐺‘𝑆) / 𝑁) |
160 | | fveq2 6191 |
. . . . . 6
⊢ (𝑡 = 𝑆 → (𝐺‘𝑡) = (𝐺‘𝑆)) |
161 | 160 | oveq1d 6665 |
. . . . 5
⊢ (𝑡 = 𝑆 → ((𝐺‘𝑡) / 𝑁) = ((𝐺‘𝑆) / 𝑁)) |
162 | 40, 159, 161, 1 | fvmptf 6301 |
. . . 4
⊢ ((𝑆 ∈ 𝑇 ∧ ((𝐺‘𝑆) / 𝑁) ∈ ℝ) → (𝐻‘𝑆) = ((𝐺‘𝑆) / 𝑁)) |
163 | 22, 157, 162 | syl2anc 693 |
. . 3
⊢ (𝜑 → (𝐻‘𝑆) = ((𝐺‘𝑆) / 𝑁)) |
164 | 156, 163 | breqtrrd 4681 |
. 2
⊢ (𝜑 → 0 < (𝐻‘𝑆)) |
165 | | nfcv 2764 |
. . . 4
⊢
Ⅎℎ𝐻 |
166 | | stoweidlem36.1 |
. . . . . 6
⊢
Ⅎℎ𝑄 |
167 | 166 | nfel2 2781 |
. . . . 5
⊢
Ⅎℎ 𝐻 ∈ 𝑄 |
168 | | nfv 1843 |
. . . . 5
⊢
Ⅎℎ0 < (𝐻‘𝑆) |
169 | 167, 168 | nfan 1828 |
. . . 4
⊢
Ⅎℎ(𝐻 ∈ 𝑄 ∧ 0 < (𝐻‘𝑆)) |
170 | | eleq1 2689 |
. . . . 5
⊢ (ℎ = 𝐻 → (ℎ ∈ 𝑄 ↔ 𝐻 ∈ 𝑄)) |
171 | | fveq1 6190 |
. . . . . 6
⊢ (ℎ = 𝐻 → (ℎ‘𝑆) = (𝐻‘𝑆)) |
172 | 171 | breq2d 4665 |
. . . . 5
⊢ (ℎ = 𝐻 → (0 < (ℎ‘𝑆) ↔ 0 < (𝐻‘𝑆))) |
173 | 170, 172 | anbi12d 747 |
. . . 4
⊢ (ℎ = 𝐻 → ((ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆)) ↔ (𝐻 ∈ 𝑄 ∧ 0 < (𝐻‘𝑆)))) |
174 | 165, 169,
173 | spcegf 3289 |
. . 3
⊢ (𝐻 ∈ 𝑄 → ((𝐻 ∈ 𝑄 ∧ 0 < (𝐻‘𝑆)) → ∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆)))) |
175 | 174 | anabsi5 858 |
. 2
⊢ ((𝐻 ∈ 𝑄 ∧ 0 < (𝐻‘𝑆)) → ∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆))) |
176 | 155, 164,
175 | syl2anc 693 |
1
⊢ (𝜑 → ∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆))) |