Proof of Theorem stoweidlem38
Step | Hyp | Ref
| Expression |
1 | | stoweidlem38.3 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℕ) |
2 | 1 | nnrecred 12022 |
. . . . 5
⊢ (𝜑 → (1 / 𝑀) ∈ ℝ) |
3 | 2 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (1 / 𝑀) ∈ ℝ) |
4 | | fzfid 13689 |
. . . . 5
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (1...𝑀) ∈ Fin) |
5 | | stoweidlem38.1 |
. . . . . . . 8
⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} |
6 | | stoweidlem38.4 |
. . . . . . . 8
⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝑄) |
7 | | stoweidlem38.5 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) |
8 | 5, 6, 7 | stoweidlem15 43525 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) → (((𝐺‘𝑖)‘𝑆) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝑖)‘𝑆) ∧ ((𝐺‘𝑖)‘𝑆) ≤ 1)) |
9 | 8 | simp1d 1141 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) → ((𝐺‘𝑖)‘𝑆) ∈ ℝ) |
10 | 9 | an32s 649 |
. . . . 5
⊢ (((𝜑 ∧ 𝑆 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺‘𝑖)‘𝑆) ∈ ℝ) |
11 | 4, 10 | fsumrecl 15442 |
. . . 4
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆) ∈ ℝ) |
12 | | 1red 10975 |
. . . . . 6
⊢ (𝜑 → 1 ∈
ℝ) |
13 | | 0le1 11496 |
. . . . . . 7
⊢ 0 ≤
1 |
14 | 13 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 0 ≤ 1) |
15 | 1 | nnred 11986 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℝ) |
16 | 1 | nngt0d 12020 |
. . . . . 6
⊢ (𝜑 → 0 < 𝑀) |
17 | | divge0 11842 |
. . . . . 6
⊢ (((1
∈ ℝ ∧ 0 ≤ 1) ∧ (𝑀 ∈ ℝ ∧ 0 < 𝑀)) → 0 ≤ (1 / 𝑀)) |
18 | 12, 14, 15, 16, 17 | syl22anc 836 |
. . . . 5
⊢ (𝜑 → 0 ≤ (1 / 𝑀)) |
19 | 18 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → 0 ≤ (1 / 𝑀)) |
20 | 8 | simp2d 1142 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) → 0 ≤ ((𝐺‘𝑖)‘𝑆)) |
21 | 20 | an32s 649 |
. . . . 5
⊢ (((𝜑 ∧ 𝑆 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ ((𝐺‘𝑖)‘𝑆)) |
22 | 4, 10, 21 | fsumge0 15503 |
. . . 4
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → 0 ≤ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆)) |
23 | 3, 11, 19, 22 | mulge0d 11550 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → 0 ≤ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆))) |
24 | | stoweidlem38.2 |
. . . 4
⊢ 𝑃 = (𝑡 ∈ 𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))) |
25 | 5, 24, 1, 6, 7 | stoweidlem30 43540 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (𝑃‘𝑆) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆))) |
26 | 23, 25 | breqtrrd 5107 |
. 2
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → 0 ≤ (𝑃‘𝑆)) |
27 | | 1red 10975 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑆 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 1 ∈ ℝ) |
28 | 8 | simp3d 1143 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) → ((𝐺‘𝑖)‘𝑆) ≤ 1) |
29 | 28 | an32s 649 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑆 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺‘𝑖)‘𝑆) ≤ 1) |
30 | 4, 10, 27, 29 | fsumle 15507 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆) ≤ Σ𝑖 ∈ (1...𝑀)1) |
31 | | fzfid 13689 |
. . . . . . . . 9
⊢ (𝜑 → (1...𝑀) ∈ Fin) |
32 | | ax-1cn 10928 |
. . . . . . . . 9
⊢ 1 ∈
ℂ |
33 | | fsumconst 15498 |
. . . . . . . . 9
⊢
(((1...𝑀) ∈ Fin
∧ 1 ∈ ℂ) → Σ𝑖 ∈ (1...𝑀)1 = ((♯‘(1...𝑀)) · 1)) |
34 | 31, 32, 33 | sylancl 586 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑖 ∈ (1...𝑀)1 = ((♯‘(1...𝑀)) · 1)) |
35 | 1 | nnnn0d 12291 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
36 | | hashfz1 14056 |
. . . . . . . . . 10
⊢ (𝑀 ∈ ℕ0
→ (♯‘(1...𝑀)) = 𝑀) |
37 | 35, 36 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀) |
38 | 37 | oveq1d 7284 |
. . . . . . . 8
⊢ (𝜑 → ((♯‘(1...𝑀)) · 1) = (𝑀 · 1)) |
39 | 1 | nncnd 11987 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ ℂ) |
40 | 39 | mulid1d 10991 |
. . . . . . . 8
⊢ (𝜑 → (𝑀 · 1) = 𝑀) |
41 | 34, 38, 40 | 3eqtrd 2784 |
. . . . . . 7
⊢ (𝜑 → Σ𝑖 ∈ (1...𝑀)1 = 𝑀) |
42 | 41 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → Σ𝑖 ∈ (1...𝑀)1 = 𝑀) |
43 | 30, 42 | breqtrd 5105 |
. . . . 5
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆) ≤ 𝑀) |
44 | 15 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → 𝑀 ∈ ℝ) |
45 | | 1red 10975 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → 1 ∈ ℝ) |
46 | | 0lt1 11495 |
. . . . . . . 8
⊢ 0 <
1 |
47 | 46 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → 0 < 1) |
48 | 15, 16 | jca 512 |
. . . . . . . 8
⊢ (𝜑 → (𝑀 ∈ ℝ ∧ 0 < 𝑀)) |
49 | 48 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (𝑀 ∈ ℝ ∧ 0 < 𝑀)) |
50 | | divgt0 11841 |
. . . . . . 7
⊢ (((1
∈ ℝ ∧ 0 < 1) ∧ (𝑀 ∈ ℝ ∧ 0 < 𝑀)) → 0 < (1 / 𝑀)) |
51 | 45, 47, 49, 50 | syl21anc 835 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → 0 < (1 / 𝑀)) |
52 | | lemul2 11826 |
. . . . . 6
⊢
((Σ𝑖 ∈
(1...𝑀)((𝐺‘𝑖)‘𝑆) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ((1 / 𝑀) ∈ ℝ ∧ 0 < (1
/ 𝑀))) → (Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆) ≤ 𝑀 ↔ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆)) ≤ ((1 / 𝑀) · 𝑀))) |
53 | 11, 44, 3, 51, 52 | syl112anc 1373 |
. . . . 5
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆) ≤ 𝑀 ↔ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆)) ≤ ((1 / 𝑀) · 𝑀))) |
54 | 43, 53 | mpbid 231 |
. . . 4
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆)) ≤ ((1 / 𝑀) · 𝑀)) |
55 | 25, 54 | eqbrtrd 5101 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (𝑃‘𝑆) ≤ ((1 / 𝑀) · 𝑀)) |
56 | 32 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 1 ∈
ℂ) |
57 | 1 | nnne0d 12021 |
. . . . . 6
⊢ (𝜑 → 𝑀 ≠ 0) |
58 | 56, 39, 57 | 3jca 1127 |
. . . . 5
⊢ (𝜑 → (1 ∈ ℂ ∧
𝑀 ∈ ℂ ∧
𝑀 ≠ 0)) |
59 | 58 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (1 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑀 ≠ 0)) |
60 | | divcan1 11640 |
. . . 4
⊢ ((1
∈ ℂ ∧ 𝑀
∈ ℂ ∧ 𝑀 ≠
0) → ((1 / 𝑀) ·
𝑀) = 1) |
61 | 59, 60 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → ((1 / 𝑀) · 𝑀) = 1) |
62 | 55, 61 | breqtrd 5105 |
. 2
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (𝑃‘𝑆) ≤ 1) |
63 | 26, 62 | jca 512 |
1
⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (0 ≤ (𝑃‘𝑆) ∧ (𝑃‘𝑆) ≤ 1)) |