Proof of Theorem stoweidlem11
Step | Hyp | Ref
| Expression |
1 | | stoweidlem11.2 |
. . 3
⊢ (𝜑 → 𝑡 ∈ 𝑇) |
2 | | sumex 14418 |
. . 3
⊢
Σ𝑖 ∈
(0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ V |
3 | | eqid 2622 |
. . . 4
⊢ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) |
4 | 3 | fvmpt2 6291 |
. . 3
⊢ ((𝑡 ∈ 𝑇 ∧ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ V) → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) = Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) |
5 | 1, 2, 4 | sylancl 694 |
. 2
⊢ (𝜑 → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) = Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) |
6 | | fzfid 12772 |
. . . 4
⊢ (𝜑 → (0...𝑁) ∈ Fin) |
7 | | stoweidlem11.7 |
. . . . . . 7
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
8 | 7 | rpred 11872 |
. . . . . 6
⊢ (𝜑 → 𝐸 ∈ ℝ) |
9 | 8 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → 𝐸 ∈ ℝ) |
10 | | stoweidlem11.4 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → (𝑋‘𝑖):𝑇⟶ℝ) |
11 | 1 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → 𝑡 ∈ 𝑇) |
12 | 10, 11 | ffvelrnd 6360 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ) |
13 | 9, 12 | remulcld 10070 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ) |
14 | 6, 13 | fsumrecl 14465 |
. . 3
⊢ (𝜑 → Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ) |
15 | | stoweidlem11.3 |
. . . . . . . . 9
⊢ (𝜑 → 𝑗 ∈ (1...𝑁)) |
16 | | elfzuz 12338 |
. . . . . . . . 9
⊢ (𝑗 ∈ (1...𝑁) → 𝑗 ∈
(ℤ≥‘1)) |
17 | 15, 16 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑗 ∈
(ℤ≥‘1)) |
18 | | eluz2 11693 |
. . . . . . . 8
⊢ (𝑗 ∈
(ℤ≥‘1) ↔ (1 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 1 ≤
𝑗)) |
19 | 17, 18 | sylib 208 |
. . . . . . 7
⊢ (𝜑 → (1 ∈ ℤ ∧
𝑗 ∈ ℤ ∧ 1
≤ 𝑗)) |
20 | 19 | simp2d 1074 |
. . . . . 6
⊢ (𝜑 → 𝑗 ∈ ℤ) |
21 | 20 | zred 11482 |
. . . . 5
⊢ (𝜑 → 𝑗 ∈ ℝ) |
22 | 8, 21 | remulcld 10070 |
. . . 4
⊢ (𝜑 → (𝐸 · 𝑗) ∈ ℝ) |
23 | | stoweidlem11.1 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℕ) |
24 | 23 | nnred 11035 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℝ) |
25 | 24, 21 | resubcld 10458 |
. . . . . 6
⊢ (𝜑 → (𝑁 − 𝑗) ∈ ℝ) |
26 | | 1red 10055 |
. . . . . 6
⊢ (𝜑 → 1 ∈
ℝ) |
27 | 25, 26 | readdcld 10069 |
. . . . 5
⊢ (𝜑 → ((𝑁 − 𝑗) + 1) ∈ ℝ) |
28 | 8, 23 | nndivred 11069 |
. . . . . 6
⊢ (𝜑 → (𝐸 / 𝑁) ∈ ℝ) |
29 | 8, 28 | remulcld 10070 |
. . . . 5
⊢ (𝜑 → (𝐸 · (𝐸 / 𝑁)) ∈ ℝ) |
30 | 27, 29 | remulcld 10070 |
. . . 4
⊢ (𝜑 → (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁))) ∈ ℝ) |
31 | 22, 30 | readdcld 10069 |
. . 3
⊢ (𝜑 → ((𝐸 · 𝑗) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))) ∈ ℝ) |
32 | | 3re 11094 |
. . . . . . 7
⊢ 3 ∈
ℝ |
33 | 32 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 3 ∈
ℝ) |
34 | | 3ne0 11115 |
. . . . . . 7
⊢ 3 ≠
0 |
35 | 34 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 3 ≠ 0) |
36 | 33, 35 | rereccld 10852 |
. . . . 5
⊢ (𝜑 → (1 / 3) ∈
ℝ) |
37 | 21, 36 | readdcld 10069 |
. . . 4
⊢ (𝜑 → (𝑗 + (1 / 3)) ∈ ℝ) |
38 | 37, 8 | remulcld 10070 |
. . 3
⊢ (𝜑 → ((𝑗 + (1 / 3)) · 𝐸) ∈ ℝ) |
39 | | fzfid 12772 |
. . . . . 6
⊢ (𝜑 → (0...(𝑗 − 1)) ∈ Fin) |
40 | 8 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → 𝐸 ∈ ℝ) |
41 | | elfzelz 12342 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ (1...𝑁) → 𝑗 ∈ ℤ) |
42 | | peano2zm 11420 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ ℤ → (𝑗 − 1) ∈
ℤ) |
43 | 15, 41, 42 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑗 − 1) ∈ ℤ) |
44 | 23 | nnzd 11481 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ ℤ) |
45 | 21, 26 | resubcld 10458 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑗 − 1) ∈ ℝ) |
46 | 21 | lem1d 10957 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑗 − 1) ≤ 𝑗) |
47 | | elfzuz3 12339 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ (1...𝑁) → 𝑁 ∈ (ℤ≥‘𝑗)) |
48 | | eluzle 11700 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈
(ℤ≥‘𝑗) → 𝑗 ≤ 𝑁) |
49 | 15, 47, 48 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑗 ≤ 𝑁) |
50 | 45, 21, 24, 46, 49 | letrd 10194 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑗 − 1) ≤ 𝑁) |
51 | | eluz2 11693 |
. . . . . . . . . . 11
⊢ (𝑁 ∈
(ℤ≥‘(𝑗 − 1)) ↔ ((𝑗 − 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑗 − 1) ≤ 𝑁)) |
52 | 43, 44, 50, 51 | syl3anbrc 1246 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑗 − 1))) |
53 | | fzss2 12381 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘(𝑗 − 1)) → (0...(𝑗 − 1)) ⊆ (0...𝑁)) |
54 | 52, 53 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (0...(𝑗 − 1)) ⊆ (0...𝑁)) |
55 | 54 | sselda 3603 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → 𝑖 ∈ (0...𝑁)) |
56 | 55, 12 | syldan 487 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ) |
57 | 40, 56 | remulcld 10070 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ) |
58 | 39, 57 | fsumrecl 14465 |
. . . . 5
⊢ (𝜑 → Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ) |
59 | 58, 30 | readdcld 10069 |
. . . 4
⊢ (𝜑 → (Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))) ∈ ℝ) |
60 | 21 | ltm1d 10956 |
. . . . . . 7
⊢ (𝜑 → (𝑗 − 1) < 𝑗) |
61 | | fzdisj 12368 |
. . . . . . 7
⊢ ((𝑗 − 1) < 𝑗 → ((0...(𝑗 − 1)) ∩ (𝑗...𝑁)) = ∅) |
62 | 60, 61 | syl 17 |
. . . . . 6
⊢ (𝜑 → ((0...(𝑗 − 1)) ∩ (𝑗...𝑁)) = ∅) |
63 | | fzssp1 12384 |
. . . . . . . . . 10
⊢
(0...(𝑁 − 1))
⊆ (0...((𝑁 − 1)
+ 1)) |
64 | 23 | nncnd 11036 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ ℂ) |
65 | | 1cnd 10056 |
. . . . . . . . . . . 12
⊢ (𝜑 → 1 ∈
ℂ) |
66 | 64, 65 | npcand 10396 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
67 | 66 | oveq2d 6666 |
. . . . . . . . . 10
⊢ (𝜑 → (0...((𝑁 − 1) + 1)) = (0...𝑁)) |
68 | 63, 67 | syl5sseq 3653 |
. . . . . . . . 9
⊢ (𝜑 → (0...(𝑁 − 1)) ⊆ (0...𝑁)) |
69 | | 1zzd 11408 |
. . . . . . . . . . . 12
⊢ (𝜑 → 1 ∈
ℤ) |
70 | | fzsubel 12377 |
. . . . . . . . . . . 12
⊢ (((1
∈ ℤ ∧ 𝑁
∈ ℤ) ∧ (𝑗
∈ ℤ ∧ 1 ∈ ℤ)) → (𝑗 ∈ (1...𝑁) ↔ (𝑗 − 1) ∈ ((1 − 1)...(𝑁 − 1)))) |
71 | 69, 44, 20, 69, 70 | syl22anc 1327 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑗 ∈ (1...𝑁) ↔ (𝑗 − 1) ∈ ((1 − 1)...(𝑁 − 1)))) |
72 | 15, 71 | mpbid 222 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑗 − 1) ∈ ((1 − 1)...(𝑁 − 1))) |
73 | | 1m1e0 11089 |
. . . . . . . . . . 11
⊢ (1
− 1) = 0 |
74 | 73 | oveq1i 6660 |
. . . . . . . . . 10
⊢ ((1
− 1)...(𝑁 − 1))
= (0...(𝑁 −
1)) |
75 | 72, 74 | syl6eleq 2711 |
. . . . . . . . 9
⊢ (𝜑 → (𝑗 − 1) ∈ (0...(𝑁 − 1))) |
76 | 68, 75 | sseldd 3604 |
. . . . . . . 8
⊢ (𝜑 → (𝑗 − 1) ∈ (0...𝑁)) |
77 | | fzsplit 12367 |
. . . . . . . 8
⊢ ((𝑗 − 1) ∈ (0...𝑁) → (0...𝑁) = ((0...(𝑗 − 1)) ∪ (((𝑗 − 1) + 1)...𝑁))) |
78 | 76, 77 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (0...𝑁) = ((0...(𝑗 − 1)) ∪ (((𝑗 − 1) + 1)...𝑁))) |
79 | 20 | zcnd 11483 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑗 ∈ ℂ) |
80 | 79, 65 | npcand 10396 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑗 − 1) + 1) = 𝑗) |
81 | 80 | oveq1d 6665 |
. . . . . . . 8
⊢ (𝜑 → (((𝑗 − 1) + 1)...𝑁) = (𝑗...𝑁)) |
82 | 81 | uneq2d 3767 |
. . . . . . 7
⊢ (𝜑 → ((0...(𝑗 − 1)) ∪ (((𝑗 − 1) + 1)...𝑁)) = ((0...(𝑗 − 1)) ∪ (𝑗...𝑁))) |
83 | 78, 82 | eqtrd 2656 |
. . . . . 6
⊢ (𝜑 → (0...𝑁) = ((0...(𝑗 − 1)) ∪ (𝑗...𝑁))) |
84 | 7 | rpcnd 11874 |
. . . . . . . 8
⊢ (𝜑 → 𝐸 ∈ ℂ) |
85 | 84 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → 𝐸 ∈ ℂ) |
86 | 12 | recnd 10068 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → ((𝑋‘𝑖)‘𝑡) ∈ ℂ) |
87 | 85, 86 | mulcld 10060 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℂ) |
88 | 62, 83, 6, 87 | fsumsplit 14471 |
. . . . 5
⊢ (𝜑 → Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = (Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) + Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))) |
89 | | fzfid 12772 |
. . . . . . 7
⊢ (𝜑 → (𝑗...𝑁) ∈ Fin) |
90 | 8 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → 𝐸 ∈ ℝ) |
91 | | 0zd 11389 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 ∈
ℤ) |
92 | | 0red 10041 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 ∈
ℝ) |
93 | | 0le1 10551 |
. . . . . . . . . . . . . . 15
⊢ 0 ≤
1 |
94 | 93 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 ≤ 1) |
95 | 19 | simp3d 1075 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 1 ≤ 𝑗) |
96 | 92, 26, 21, 94, 95 | letrd 10194 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 ≤ 𝑗) |
97 | | eluz2 11693 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈
(ℤ≥‘0) ↔ (0 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 0 ≤
𝑗)) |
98 | 91, 20, 96, 97 | syl3anbrc 1246 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑗 ∈
(ℤ≥‘0)) |
99 | | fzss1 12380 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈
(ℤ≥‘0) → (𝑗...𝑁) ⊆ (0...𝑁)) |
100 | 98, 99 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑗...𝑁) ⊆ (0...𝑁)) |
101 | 100 | sselda 3603 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → 𝑖 ∈ (0...𝑁)) |
102 | 101, 10 | syldan 487 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → (𝑋‘𝑖):𝑇⟶ℝ) |
103 | 1 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → 𝑡 ∈ 𝑇) |
104 | 102, 103 | ffvelrnd 6360 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ) |
105 | 90, 104 | remulcld 10070 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ) |
106 | 89, 105 | fsumrecl 14465 |
. . . . . 6
⊢ (𝜑 → Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ) |
107 | | eluzfz2 12349 |
. . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘𝑗) → 𝑁 ∈ (𝑗...𝑁)) |
108 | | ne0i 3921 |
. . . . . . . . 9
⊢ (𝑁 ∈ (𝑗...𝑁) → (𝑗...𝑁) ≠ ∅) |
109 | 15, 47, 107, 108 | 4syl 19 |
. . . . . . . 8
⊢ (𝜑 → (𝑗...𝑁) ≠ ∅) |
110 | 23 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → 𝑁 ∈ ℕ) |
111 | 90, 110 | nndivred 11069 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → (𝐸 / 𝑁) ∈ ℝ) |
112 | 90, 111 | remulcld 10070 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → (𝐸 · (𝐸 / 𝑁)) ∈ ℝ) |
113 | | stoweidlem11.6 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → ((𝑋‘𝑖)‘𝑡) < (𝐸 / 𝑁)) |
114 | 7 | rpgt0d 11875 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 < 𝐸) |
115 | 114 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → 0 < 𝐸) |
116 | | ltmul2 10874 |
. . . . . . . . . 10
⊢ ((((𝑋‘𝑖)‘𝑡) ∈ ℝ ∧ (𝐸 / 𝑁) ∈ ℝ ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → (((𝑋‘𝑖)‘𝑡) < (𝐸 / 𝑁) ↔ (𝐸 · ((𝑋‘𝑖)‘𝑡)) < (𝐸 · (𝐸 / 𝑁)))) |
117 | 104, 111,
90, 115, 116 | syl112anc 1330 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → (((𝑋‘𝑖)‘𝑡) < (𝐸 / 𝑁) ↔ (𝐸 · ((𝑋‘𝑖)‘𝑡)) < (𝐸 · (𝐸 / 𝑁)))) |
118 | 113, 117 | mpbid 222 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) < (𝐸 · (𝐸 / 𝑁))) |
119 | 89, 109, 105, 112, 118 | fsumlt 14532 |
. . . . . . 7
⊢ (𝜑 → Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) < Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · (𝐸 / 𝑁))) |
120 | 23 | nnne0d 11065 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ≠ 0) |
121 | 84, 64, 120 | divcld 10801 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐸 / 𝑁) ∈ ℂ) |
122 | 84, 121 | mulcld 10060 |
. . . . . . . . 9
⊢ (𝜑 → (𝐸 · (𝐸 / 𝑁)) ∈ ℂ) |
123 | | fsumconst 14522 |
. . . . . . . . 9
⊢ (((𝑗...𝑁) ∈ Fin ∧ (𝐸 · (𝐸 / 𝑁)) ∈ ℂ) → Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · (𝐸 / 𝑁)) = ((#‘(𝑗...𝑁)) · (𝐸 · (𝐸 / 𝑁)))) |
124 | 89, 122, 123 | syl2anc 693 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · (𝐸 / 𝑁)) = ((#‘(𝑗...𝑁)) · (𝐸 · (𝐸 / 𝑁)))) |
125 | | hashfz 13214 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘𝑗) → (#‘(𝑗...𝑁)) = ((𝑁 − 𝑗) + 1)) |
126 | 15, 47, 125 | 3syl 18 |
. . . . . . . . 9
⊢ (𝜑 → (#‘(𝑗...𝑁)) = ((𝑁 − 𝑗) + 1)) |
127 | 126 | oveq1d 6665 |
. . . . . . . 8
⊢ (𝜑 → ((#‘(𝑗...𝑁)) · (𝐸 · (𝐸 / 𝑁))) = (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))) |
128 | 124, 127 | eqtrd 2656 |
. . . . . . 7
⊢ (𝜑 → Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · (𝐸 / 𝑁)) = (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))) |
129 | 119, 128 | breqtrd 4679 |
. . . . . 6
⊢ (𝜑 → Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) < (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))) |
130 | 106, 30, 58, 129 | ltadd2dd 10196 |
. . . . 5
⊢ (𝜑 → (Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) + Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) < (Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁))))) |
131 | 88, 130 | eqbrtrd 4675 |
. . . 4
⊢ (𝜑 → Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) < (Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁))))) |
132 | | stoweidlem11.5 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → ((𝑋‘𝑖)‘𝑡) ≤ 1) |
133 | 55, 132 | syldan 487 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → ((𝑋‘𝑖)‘𝑡) ≤ 1) |
134 | | 1red 10055 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → 1 ∈
ℝ) |
135 | 114 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → 0 < 𝐸) |
136 | | lemul2 10876 |
. . . . . . . . . 10
⊢ ((((𝑋‘𝑖)‘𝑡) ∈ ℝ ∧ 1 ∈ ℝ ∧
(𝐸 ∈ ℝ ∧ 0
< 𝐸)) → (((𝑋‘𝑖)‘𝑡) ≤ 1 ↔ (𝐸 · ((𝑋‘𝑖)‘𝑡)) ≤ (𝐸 · 1))) |
137 | 56, 134, 40, 135, 136 | syl112anc 1330 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → (((𝑋‘𝑖)‘𝑡) ≤ 1 ↔ (𝐸 · ((𝑋‘𝑖)‘𝑡)) ≤ (𝐸 · 1))) |
138 | 133, 137 | mpbid 222 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ≤ (𝐸 · 1)) |
139 | 84 | mulid1d 10057 |
. . . . . . . . 9
⊢ (𝜑 → (𝐸 · 1) = 𝐸) |
140 | 139 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → (𝐸 · 1) = 𝐸) |
141 | 138, 140 | breqtrd 4679 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ≤ 𝐸) |
142 | 39, 57, 40, 141 | fsumle 14531 |
. . . . . 6
⊢ (𝜑 → Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) ≤ Σ𝑖 ∈ (0...(𝑗 − 1))𝐸) |
143 | | fsumconst 14522 |
. . . . . . . 8
⊢
(((0...(𝑗 −
1)) ∈ Fin ∧ 𝐸
∈ ℂ) → Σ𝑖 ∈ (0...(𝑗 − 1))𝐸 = ((#‘(0...(𝑗 − 1))) · 𝐸)) |
144 | 39, 84, 143 | syl2anc 693 |
. . . . . . 7
⊢ (𝜑 → Σ𝑖 ∈ (0...(𝑗 − 1))𝐸 = ((#‘(0...(𝑗 − 1))) · 𝐸)) |
145 | | 0z 11388 |
. . . . . . . . . . 11
⊢ 0 ∈
ℤ |
146 | | 1e0p1 11552 |
. . . . . . . . . . . . 13
⊢ 1 = (0 +
1) |
147 | 146 | fveq2i 6194 |
. . . . . . . . . . . 12
⊢
(ℤ≥‘1) = (ℤ≥‘(0 +
1)) |
148 | 17, 147 | syl6eleq 2711 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑗 ∈ (ℤ≥‘(0 +
1))) |
149 | | eluzp1m1 11711 |
. . . . . . . . . . 11
⊢ ((0
∈ ℤ ∧ 𝑗
∈ (ℤ≥‘(0 + 1))) → (𝑗 − 1) ∈
(ℤ≥‘0)) |
150 | 145, 148,
149 | sylancr 695 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑗 − 1) ∈
(ℤ≥‘0)) |
151 | | hashfz 13214 |
. . . . . . . . . 10
⊢ ((𝑗 − 1) ∈
(ℤ≥‘0) → (#‘(0...(𝑗 − 1))) = (((𝑗 − 1) − 0) + 1)) |
152 | 150, 151 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (#‘(0...(𝑗 − 1))) = (((𝑗 − 1) − 0) +
1)) |
153 | 79, 65 | subcld 10392 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑗 − 1) ∈ ℂ) |
154 | 153 | subid1d 10381 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑗 − 1) − 0) = (𝑗 − 1)) |
155 | 154 | oveq1d 6665 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑗 − 1) − 0) + 1) = ((𝑗 − 1) +
1)) |
156 | 152, 155,
80 | 3eqtrd 2660 |
. . . . . . . 8
⊢ (𝜑 → (#‘(0...(𝑗 − 1))) = 𝑗) |
157 | 156 | oveq1d 6665 |
. . . . . . 7
⊢ (𝜑 → ((#‘(0...(𝑗 − 1))) · 𝐸) = (𝑗 · 𝐸)) |
158 | 79, 84 | mulcomd 10061 |
. . . . . . 7
⊢ (𝜑 → (𝑗 · 𝐸) = (𝐸 · 𝑗)) |
159 | 144, 157,
158 | 3eqtrd 2660 |
. . . . . 6
⊢ (𝜑 → Σ𝑖 ∈ (0...(𝑗 − 1))𝐸 = (𝐸 · 𝑗)) |
160 | 142, 159 | breqtrd 4679 |
. . . . 5
⊢ (𝜑 → Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) ≤ (𝐸 · 𝑗)) |
161 | 58, 22, 30, 160 | leadd1dd 10641 |
. . . 4
⊢ (𝜑 → (Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))) ≤ ((𝐸 · 𝑗) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁))))) |
162 | 14, 59, 31, 131, 161 | ltletrd 10197 |
. . 3
⊢ (𝜑 → Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) < ((𝐸 · 𝑗) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁))))) |
163 | 8, 8 | remulcld 10070 |
. . . . 5
⊢ (𝜑 → (𝐸 · 𝐸) ∈ ℝ) |
164 | 22, 163 | readdcld 10069 |
. . . 4
⊢ (𝜑 → ((𝐸 · 𝑗) + (𝐸 · 𝐸)) ∈ ℝ) |
165 | 64, 79 | subcld 10392 |
. . . . . . . 8
⊢ (𝜑 → (𝑁 − 𝑗) ∈ ℂ) |
166 | 165, 65 | addcld 10059 |
. . . . . . 7
⊢ (𝜑 → ((𝑁 − 𝑗) + 1) ∈ ℂ) |
167 | 84, 166, 121 | mul12d 10245 |
. . . . . 6
⊢ (𝜑 → (𝐸 · (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁))) = (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))) |
168 | 167 | oveq2d 6666 |
. . . . 5
⊢ (𝜑 → ((𝐸 · 𝑗) + (𝐸 · (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁)))) = ((𝐸 · 𝑗) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁))))) |
169 | 27, 28 | remulcld 10070 |
. . . . . . 7
⊢ (𝜑 → (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁)) ∈ ℝ) |
170 | 8, 169 | remulcld 10070 |
. . . . . 6
⊢ (𝜑 → (𝐸 · (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁))) ∈ ℝ) |
171 | 166, 84, 64, 120 | div12d 10837 |
. . . . . . . 8
⊢ (𝜑 → (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁)) = (𝐸 · (((𝑁 − 𝑗) + 1) / 𝑁))) |
172 | 26, 21 | resubcld 10458 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1 − 𝑗) ∈
ℝ) |
173 | | elfzle1 12344 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ (1...𝑁) → 1 ≤ 𝑗) |
174 | 15, 173 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 1 ≤ 𝑗) |
175 | 26, 21 | suble0d 10618 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((1 − 𝑗) ≤ 0 ↔ 1 ≤ 𝑗)) |
176 | 174, 175 | mpbird 247 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1 − 𝑗) ≤ 0) |
177 | 172, 92, 24, 176 | leadd2dd 10642 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 + (1 − 𝑗)) ≤ (𝑁 + 0)) |
178 | 64, 65, 79 | addsub12d 10415 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑁 + (1 − 𝑗)) = (1 + (𝑁 − 𝑗))) |
179 | 65, 165 | addcomd 10238 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1 + (𝑁 − 𝑗)) = ((𝑁 − 𝑗) + 1)) |
180 | 178, 179 | eqtrd 2656 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 + (1 − 𝑗)) = ((𝑁 − 𝑗) + 1)) |
181 | 64 | addid1d 10236 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 + 0) = 𝑁) |
182 | 177, 180,
181 | 3brtr3d 4684 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑁 − 𝑗) + 1) ≤ 𝑁) |
183 | 23 | nngt0d 11064 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 < 𝑁) |
184 | | lediv1 10888 |
. . . . . . . . . . . . 13
⊢ ((((𝑁 − 𝑗) + 1) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (((𝑁 − 𝑗) + 1) ≤ 𝑁 ↔ (((𝑁 − 𝑗) + 1) / 𝑁) ≤ (𝑁 / 𝑁))) |
185 | 27, 24, 24, 183, 184 | syl112anc 1330 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((𝑁 − 𝑗) + 1) ≤ 𝑁 ↔ (((𝑁 − 𝑗) + 1) / 𝑁) ≤ (𝑁 / 𝑁))) |
186 | 182, 185 | mpbid 222 |
. . . . . . . . . . 11
⊢ (𝜑 → (((𝑁 − 𝑗) + 1) / 𝑁) ≤ (𝑁 / 𝑁)) |
187 | 64, 120 | dividd 10799 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁 / 𝑁) = 1) |
188 | 186, 187 | breqtrd 4679 |
. . . . . . . . . 10
⊢ (𝜑 → (((𝑁 − 𝑗) + 1) / 𝑁) ≤ 1) |
189 | 27, 23 | nndivred 11069 |
. . . . . . . . . . 11
⊢ (𝜑 → (((𝑁 − 𝑗) + 1) / 𝑁) ∈ ℝ) |
190 | 189, 26, 7 | lemul2d 11916 |
. . . . . . . . . 10
⊢ (𝜑 → ((((𝑁 − 𝑗) + 1) / 𝑁) ≤ 1 ↔ (𝐸 · (((𝑁 − 𝑗) + 1) / 𝑁)) ≤ (𝐸 · 1))) |
191 | 188, 190 | mpbid 222 |
. . . . . . . . 9
⊢ (𝜑 → (𝐸 · (((𝑁 − 𝑗) + 1) / 𝑁)) ≤ (𝐸 · 1)) |
192 | 191, 139 | breqtrd 4679 |
. . . . . . . 8
⊢ (𝜑 → (𝐸 · (((𝑁 − 𝑗) + 1) / 𝑁)) ≤ 𝐸) |
193 | 171, 192 | eqbrtrd 4675 |
. . . . . . 7
⊢ (𝜑 → (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁)) ≤ 𝐸) |
194 | 169, 8, 7 | lemul2d 11916 |
. . . . . . 7
⊢ (𝜑 → ((((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁)) ≤ 𝐸 ↔ (𝐸 · (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁))) ≤ (𝐸 · 𝐸))) |
195 | 193, 194 | mpbid 222 |
. . . . . 6
⊢ (𝜑 → (𝐸 · (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁))) ≤ (𝐸 · 𝐸)) |
196 | 170, 163,
22, 195 | leadd2dd 10642 |
. . . . 5
⊢ (𝜑 → ((𝐸 · 𝑗) + (𝐸 · (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁)))) ≤ ((𝐸 · 𝑗) + (𝐸 · 𝐸))) |
197 | 168, 196 | eqbrtrrd 4677 |
. . . 4
⊢ (𝜑 → ((𝐸 · 𝑗) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))) ≤ ((𝐸 · 𝑗) + (𝐸 · 𝐸))) |
198 | 84, 79 | mulcomd 10061 |
. . . . . . 7
⊢ (𝜑 → (𝐸 · 𝑗) = (𝑗 · 𝐸)) |
199 | 198 | oveq1d 6665 |
. . . . . 6
⊢ (𝜑 → ((𝐸 · 𝑗) + (𝐸 · 𝐸)) = ((𝑗 · 𝐸) + (𝐸 · 𝐸))) |
200 | 79, 84, 84 | adddird 10065 |
. . . . . 6
⊢ (𝜑 → ((𝑗 + 𝐸) · 𝐸) = ((𝑗 · 𝐸) + (𝐸 · 𝐸))) |
201 | 199, 200 | eqtr4d 2659 |
. . . . 5
⊢ (𝜑 → ((𝐸 · 𝑗) + (𝐸 · 𝐸)) = ((𝑗 + 𝐸) · 𝐸)) |
202 | 21, 8 | readdcld 10069 |
. . . . . 6
⊢ (𝜑 → (𝑗 + 𝐸) ∈ ℝ) |
203 | | stoweidlem11.8 |
. . . . . . 7
⊢ (𝜑 → 𝐸 < (1 / 3)) |
204 | 8, 36, 21, 203 | ltadd2dd 10196 |
. . . . . 6
⊢ (𝜑 → (𝑗 + 𝐸) < (𝑗 + (1 / 3))) |
205 | 202, 37, 7, 204 | ltmul1dd 11927 |
. . . . 5
⊢ (𝜑 → ((𝑗 + 𝐸) · 𝐸) < ((𝑗 + (1 / 3)) · 𝐸)) |
206 | 201, 205 | eqbrtrd 4675 |
. . . 4
⊢ (𝜑 → ((𝐸 · 𝑗) + (𝐸 · 𝐸)) < ((𝑗 + (1 / 3)) · 𝐸)) |
207 | 31, 164, 38, 197, 206 | lelttrd 10195 |
. . 3
⊢ (𝜑 → ((𝐸 · 𝑗) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))) < ((𝑗 + (1 / 3)) · 𝐸)) |
208 | 14, 31, 38, 162, 207 | lttrd 10198 |
. 2
⊢ (𝜑 → Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) < ((𝑗 + (1 / 3)) · 𝐸)) |
209 | 5, 208 | eqbrtrd 4675 |
1
⊢ (𝜑 → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸)) |