Step | Hyp | Ref
| Expression |
1 | | fmul01lt1lem2.1 |
. . 3
⊢
Ⅎ𝑖𝐵 |
2 | | fmul01lt1lem2.2 |
. . . 4
⊢
Ⅎ𝑖𝜑 |
3 | | nfv 1873 |
. . . 4
⊢
Ⅎ𝑖 𝐽 = 𝐿 |
4 | 2, 3 | nfan 1862 |
. . 3
⊢
Ⅎ𝑖(𝜑 ∧ 𝐽 = 𝐿) |
5 | | fmul01lt1lem2.3 |
. . 3
⊢ 𝐴 = seq𝐿( · , 𝐵) |
6 | | fmul01lt1lem2.4 |
. . . 4
⊢ (𝜑 → 𝐿 ∈ ℤ) |
7 | 6 | adantr 473 |
. . 3
⊢ ((𝜑 ∧ 𝐽 = 𝐿) → 𝐿 ∈ ℤ) |
8 | | fmul01lt1lem2.5 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐿)) |
9 | 8 | adantr 473 |
. . 3
⊢ ((𝜑 ∧ 𝐽 = 𝐿) → 𝑀 ∈ (ℤ≥‘𝐿)) |
10 | | fmul01lt1lem2.6 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) |
11 | 10 | adantlr 702 |
. . 3
⊢ (((𝜑 ∧ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) |
12 | | fmul01lt1lem2.7 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) |
13 | 12 | adantlr 702 |
. . 3
⊢ (((𝜑 ∧ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) |
14 | | fmul01lt1lem2.8 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1) |
15 | 14 | adantlr 702 |
. . 3
⊢ (((𝜑 ∧ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1) |
16 | | fmul01lt1lem2.9 |
. . . 4
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
17 | 16 | adantr 473 |
. . 3
⊢ ((𝜑 ∧ 𝐽 = 𝐿) → 𝐸 ∈
ℝ+) |
18 | | simpr 477 |
. . . . 5
⊢ ((𝜑 ∧ 𝐽 = 𝐿) → 𝐽 = 𝐿) |
19 | 18 | fveq2d 6500 |
. . . 4
⊢ ((𝜑 ∧ 𝐽 = 𝐿) → (𝐵‘𝐽) = (𝐵‘𝐿)) |
20 | | fmul01lt1lem2.11 |
. . . . 5
⊢ (𝜑 → (𝐵‘𝐽) < 𝐸) |
21 | 20 | adantr 473 |
. . . 4
⊢ ((𝜑 ∧ 𝐽 = 𝐿) → (𝐵‘𝐽) < 𝐸) |
22 | 19, 21 | eqbrtrrd 4949 |
. . 3
⊢ ((𝜑 ∧ 𝐽 = 𝐿) → (𝐵‘𝐿) < 𝐸) |
23 | 1, 4, 5, 7, 9, 11,
13, 15, 17, 22 | fmul01lt1lem1 41321 |
. 2
⊢ ((𝜑 ∧ 𝐽 = 𝐿) → (𝐴‘𝑀) < 𝐸) |
24 | 5 | fveq1i 6497 |
. . 3
⊢ (𝐴‘𝑀) = (seq𝐿( · , 𝐵)‘𝑀) |
25 | | nfv 1873 |
. . . . . . . . 9
⊢
Ⅎ𝑖 𝑎 ∈ (𝐿...𝑀) |
26 | 2, 25 | nfan 1862 |
. . . . . . . 8
⊢
Ⅎ𝑖(𝜑 ∧ 𝑎 ∈ (𝐿...𝑀)) |
27 | | nfcv 2926 |
. . . . . . . . . 10
⊢
Ⅎ𝑖𝑎 |
28 | 1, 27 | nffv 6506 |
. . . . . . . . 9
⊢
Ⅎ𝑖(𝐵‘𝑎) |
29 | 28 | nfel1 2940 |
. . . . . . . 8
⊢
Ⅎ𝑖(𝐵‘𝑎) ∈ ℝ |
30 | 26, 29 | nfim 1859 |
. . . . . . 7
⊢
Ⅎ𝑖((𝜑 ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ) |
31 | | eleq1w 2842 |
. . . . . . . . 9
⊢ (𝑖 = 𝑎 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝑎 ∈ (𝐿...𝑀))) |
32 | 31 | anbi2d 619 |
. . . . . . . 8
⊢ (𝑖 = 𝑎 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝑎 ∈ (𝐿...𝑀)))) |
33 | | fveq2 6496 |
. . . . . . . . 9
⊢ (𝑖 = 𝑎 → (𝐵‘𝑖) = (𝐵‘𝑎)) |
34 | 33 | eleq1d 2844 |
. . . . . . . 8
⊢ (𝑖 = 𝑎 → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘𝑎) ∈ ℝ)) |
35 | 32, 34 | imbi12d 337 |
. . . . . . 7
⊢ (𝑖 = 𝑎 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ ((𝜑 ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ))) |
36 | 30, 35, 10 | chvar 2326 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ) |
37 | | remulcl 10418 |
. . . . . . 7
⊢ ((𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑎 · 𝑗) ∈ ℝ) |
38 | 37 | adantl 474 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑎 · 𝑗) ∈ ℝ) |
39 | 8, 36, 38 | seqcl 13203 |
. . . . 5
⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝑀) ∈ ℝ) |
40 | 39 | adantr 473 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) ∈ ℝ) |
41 | | fmul01lt1lem2.10 |
. . . . . . 7
⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑀)) |
42 | | elfzuz3 12719 |
. . . . . . 7
⊢ (𝐽 ∈ (𝐿...𝑀) → 𝑀 ∈ (ℤ≥‘𝐽)) |
43 | 41, 42 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐽)) |
44 | | nfv 1873 |
. . . . . . . . 9
⊢
Ⅎ𝑖 𝑎 ∈ (𝐽...𝑀) |
45 | 2, 44 | nfan 1862 |
. . . . . . . 8
⊢
Ⅎ𝑖(𝜑 ∧ 𝑎 ∈ (𝐽...𝑀)) |
46 | 45, 29 | nfim 1859 |
. . . . . . 7
⊢
Ⅎ𝑖((𝜑 ∧ 𝑎 ∈ (𝐽...𝑀)) → (𝐵‘𝑎) ∈ ℝ) |
47 | | eleq1w 2842 |
. . . . . . . . 9
⊢ (𝑖 = 𝑎 → (𝑖 ∈ (𝐽...𝑀) ↔ 𝑎 ∈ (𝐽...𝑀))) |
48 | 47 | anbi2d 619 |
. . . . . . . 8
⊢ (𝑖 = 𝑎 → ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) ↔ (𝜑 ∧ 𝑎 ∈ (𝐽...𝑀)))) |
49 | 48, 34 | imbi12d 337 |
. . . . . . 7
⊢ (𝑖 = 𝑎 → (((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ ((𝜑 ∧ 𝑎 ∈ (𝐽...𝑀)) → (𝐵‘𝑎) ∈ ℝ))) |
50 | 6 | adantr 473 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐿 ∈ ℤ) |
51 | | eluzelz 12066 |
. . . . . . . . . . . 12
⊢ (𝑀 ∈
(ℤ≥‘𝐿) → 𝑀 ∈ ℤ) |
52 | 8, 51 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℤ) |
53 | 52 | adantr 473 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝑀 ∈ ℤ) |
54 | | elfzelz 12722 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ (𝐽...𝑀) → 𝑖 ∈ ℤ) |
55 | 54 | adantl 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝑖 ∈ ℤ) |
56 | 50, 53, 55 | 3jca 1108 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ)) |
57 | 6 | zred 11898 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐿 ∈ ℝ) |
58 | 57 | adantr 473 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐿 ∈ ℝ) |
59 | | elfzelz 12722 |
. . . . . . . . . . . . . 14
⊢ (𝐽 ∈ (𝐿...𝑀) → 𝐽 ∈ ℤ) |
60 | 41, 59 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐽 ∈ ℤ) |
61 | 60 | zred 11898 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐽 ∈ ℝ) |
62 | 61 | adantr 473 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐽 ∈ ℝ) |
63 | 54 | zred 11898 |
. . . . . . . . . . . 12
⊢ (𝑖 ∈ (𝐽...𝑀) → 𝑖 ∈ ℝ) |
64 | 63 | adantl 474 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝑖 ∈ ℝ) |
65 | | elfzle1 12724 |
. . . . . . . . . . . . 13
⊢ (𝐽 ∈ (𝐿...𝑀) → 𝐿 ≤ 𝐽) |
66 | 41, 65 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐿 ≤ 𝐽) |
67 | 66 | adantr 473 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐿 ≤ 𝐽) |
68 | | elfzle1 12724 |
. . . . . . . . . . . 12
⊢ (𝑖 ∈ (𝐽...𝑀) → 𝐽 ≤ 𝑖) |
69 | 68 | adantl 474 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐽 ≤ 𝑖) |
70 | 58, 62, 64, 67, 69 | letrd 10595 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝐿 ≤ 𝑖) |
71 | | elfzle2 12725 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ (𝐽...𝑀) → 𝑖 ≤ 𝑀) |
72 | 71 | adantl 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝑖 ≤ 𝑀) |
73 | 70, 72 | jca 504 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐿 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀)) |
74 | | elfz2 12713 |
. . . . . . . . 9
⊢ (𝑖 ∈ (𝐿...𝑀) ↔ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (𝐿 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀))) |
75 | 56, 73, 74 | sylanbrc 575 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 𝑖 ∈ (𝐿...𝑀)) |
76 | 75, 10 | syldan 582 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐵‘𝑖) ∈ ℝ) |
77 | 46, 49, 76 | chvar 2326 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ (𝐽...𝑀)) → (𝐵‘𝑎) ∈ ℝ) |
78 | 43, 77, 38 | seqcl 13203 |
. . . . 5
⊢ (𝜑 → (seq𝐽( · , 𝐵)‘𝑀) ∈ ℝ) |
79 | 78 | adantr 473 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐽( · , 𝐵)‘𝑀) ∈ ℝ) |
80 | 16 | rpred 12246 |
. . . . 5
⊢ (𝜑 → 𝐸 ∈ ℝ) |
81 | 80 | adantr 473 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐸 ∈ ℝ) |
82 | | remulcl 10418 |
. . . . . . . . 9
⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 · 𝑏) ∈ ℝ) |
83 | 82 | adantl 474 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ)) → (𝑎 · 𝑏) ∈ ℝ) |
84 | | simp1 1116 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑎 ∈
ℝ) |
85 | 84 | recnd 10466 |
. . . . . . . . . 10
⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑎 ∈
ℂ) |
86 | | simp2 1117 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑏 ∈
ℝ) |
87 | 86 | recnd 10466 |
. . . . . . . . . 10
⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑏 ∈
ℂ) |
88 | | simp3 1118 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑐 ∈
ℝ) |
89 | 88 | recnd 10466 |
. . . . . . . . . 10
⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → 𝑐 ∈
ℂ) |
90 | 85, 87, 89 | mulassd 10461 |
. . . . . . . . 9
⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → ((𝑎 · 𝑏) · 𝑐) = (𝑎 · (𝑏 · 𝑐))) |
91 | 90 | adantl 474 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ)) → ((𝑎 · 𝑏) · 𝑐) = (𝑎 · (𝑏 · 𝑐))) |
92 | 60 | zcnd 11899 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐽 ∈ ℂ) |
93 | | 1cnd 10432 |
. . . . . . . . . . . 12
⊢ (𝜑 → 1 ∈
ℂ) |
94 | 92, 93 | npcand 10800 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐽 − 1) + 1) = 𝐽) |
95 | 94 | fveq2d 6500 |
. . . . . . . . . 10
⊢ (𝜑 →
(ℤ≥‘((𝐽 − 1) + 1)) =
(ℤ≥‘𝐽)) |
96 | 43, 95 | eleqtrrd 2863 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈
(ℤ≥‘((𝐽 − 1) + 1))) |
97 | 96 | adantr 473 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝑀 ∈
(ℤ≥‘((𝐽 − 1) + 1))) |
98 | 6 | adantr 473 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐿 ∈ ℤ) |
99 | 60 | adantr 473 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐽 ∈ ℤ) |
100 | | 1zzd 11824 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 1 ∈ ℤ) |
101 | 99, 100 | zsubcld 11903 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐽 − 1) ∈ ℤ) |
102 | | simpr 477 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ¬ 𝐽 = 𝐿) |
103 | | eqcom 2779 |
. . . . . . . . . . . 12
⊢ (𝐽 = 𝐿 ↔ 𝐿 = 𝐽) |
104 | 102, 103 | sylnib 320 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ¬ 𝐿 = 𝐽) |
105 | 57, 61 | leloed 10581 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐿 ≤ 𝐽 ↔ (𝐿 < 𝐽 ∨ 𝐿 = 𝐽))) |
106 | 66, 105 | mpbid 224 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐿 < 𝐽 ∨ 𝐿 = 𝐽)) |
107 | 106 | adantr 473 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐿 < 𝐽 ∨ 𝐿 = 𝐽)) |
108 | | orel2 874 |
. . . . . . . . . . 11
⊢ (¬
𝐿 = 𝐽 → ((𝐿 < 𝐽 ∨ 𝐿 = 𝐽) → 𝐿 < 𝐽)) |
109 | 104, 107,
108 | sylc 65 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐿 < 𝐽) |
110 | | zltlem1 11846 |
. . . . . . . . . . . 12
⊢ ((𝐿 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐿 < 𝐽 ↔ 𝐿 ≤ (𝐽 − 1))) |
111 | 6, 60, 110 | syl2anc 576 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐿 < 𝐽 ↔ 𝐿 ≤ (𝐽 − 1))) |
112 | 111 | adantr 473 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐿 < 𝐽 ↔ 𝐿 ≤ (𝐽 − 1))) |
113 | 109, 112 | mpbid 224 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐿 ≤ (𝐽 − 1)) |
114 | | eluz2 12062 |
. . . . . . . . 9
⊢ ((𝐽 − 1) ∈
(ℤ≥‘𝐿) ↔ (𝐿 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ ∧ 𝐿 ≤ (𝐽 − 1))) |
115 | 98, 101, 113, 114 | syl3anbrc 1323 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐽 − 1) ∈
(ℤ≥‘𝐿)) |
116 | | nfv 1873 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑖 ¬ 𝐽 = 𝐿 |
117 | 2, 116 | nfan 1862 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖(𝜑 ∧ ¬ 𝐽 = 𝐿) |
118 | 117, 25 | nfan 1862 |
. . . . . . . . . 10
⊢
Ⅎ𝑖((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...𝑀)) |
119 | 118, 29 | nfim 1859 |
. . . . . . . . 9
⊢
Ⅎ𝑖(((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ) |
120 | 31 | anbi2d 619 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑎 → (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ ((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...𝑀)))) |
121 | 120, 34 | imbi12d 337 |
. . . . . . . . 9
⊢ (𝑖 = 𝑎 → ((((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ))) |
122 | 10 | adantlr 702 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) |
123 | 119, 121,
122 | chvar 2326 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...𝑀)) → (𝐵‘𝑎) ∈ ℝ) |
124 | 83, 91, 97, 115, 123 | seqsplit 13216 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) = ((seq𝐿( · , 𝐵)‘(𝐽 − 1)) · (seq((𝐽 − 1) + 1)( · ,
𝐵)‘𝑀))) |
125 | 94 | adantr 473 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ((𝐽 − 1) + 1) = 𝐽) |
126 | 125 | seqeq1d 13188 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → seq((𝐽 − 1) + 1)( · , 𝐵) = seq𝐽( · , 𝐵)) |
127 | 126 | fveq1d 6498 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq((𝐽 − 1) + 1)( · , 𝐵)‘𝑀) = (seq𝐽( · , 𝐵)‘𝑀)) |
128 | 127 | oveq2d 6990 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ((seq𝐿( · , 𝐵)‘(𝐽 − 1)) · (seq((𝐽 − 1) + 1)( · ,
𝐵)‘𝑀)) = ((seq𝐿( · , 𝐵)‘(𝐽 − 1)) · (seq𝐽( · , 𝐵)‘𝑀))) |
129 | 124, 128 | eqtrd 2808 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) = ((seq𝐿( · , 𝐵)‘(𝐽 − 1)) · (seq𝐽( · , 𝐵)‘𝑀))) |
130 | | nfv 1873 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖 𝑎 ∈ (𝐿...(𝐽 − 1)) |
131 | 117, 130 | nfan 1862 |
. . . . . . . . . 10
⊢
Ⅎ𝑖((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...(𝐽 − 1))) |
132 | 131, 29 | nfim 1859 |
. . . . . . . . 9
⊢
Ⅎ𝑖(((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑎) ∈ ℝ) |
133 | | eleq1w 2842 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑎 → (𝑖 ∈ (𝐿...(𝐽 − 1)) ↔ 𝑎 ∈ (𝐿...(𝐽 − 1)))) |
134 | 133 | anbi2d 619 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑎 → (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) ↔ ((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...(𝐽 − 1))))) |
135 | 134, 34 | imbi12d 337 |
. . . . . . . . 9
⊢ (𝑖 = 𝑎 → ((((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑖) ∈ ℝ) ↔ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑎) ∈ ℝ))) |
136 | 6 | adantr 473 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝐿 ∈ ℤ) |
137 | 52 | adantr 473 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑀 ∈ ℤ) |
138 | | elfzelz 12722 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (𝐿...(𝐽 − 1)) → 𝑖 ∈ ℤ) |
139 | 138 | adantl 474 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ∈ ℤ) |
140 | 136, 137,
139 | 3jca 1108 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ)) |
141 | | elfzle1 12724 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (𝐿...(𝐽 − 1)) → 𝐿 ≤ 𝑖) |
142 | 141 | adantl 474 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝐿 ≤ 𝑖) |
143 | 138 | zred 11898 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ (𝐿...(𝐽 − 1)) → 𝑖 ∈ ℝ) |
144 | 143 | adantl 474 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ∈ ℝ) |
145 | 61 | adantr 473 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝐽 ∈ ℝ) |
146 | 52 | zred 11898 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 ∈ ℝ) |
147 | 146 | adantr 473 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑀 ∈ ℝ) |
148 | | 1red 10438 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 1 ∈
ℝ) |
149 | 61, 148 | resubcld 10867 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐽 − 1) ∈ ℝ) |
150 | 149 | adantr 473 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐽 − 1) ∈ ℝ) |
151 | | elfzle2 12725 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (𝐿...(𝐽 − 1)) → 𝑖 ≤ (𝐽 − 1)) |
152 | 151 | adantl 474 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ≤ (𝐽 − 1)) |
153 | 61 | lem1d 11372 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐽 − 1) ≤ 𝐽) |
154 | 153 | adantr 473 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐽 − 1) ≤ 𝐽) |
155 | 144, 150,
145, 152, 154 | letrd 10595 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ≤ 𝐽) |
156 | | elfzle2 12725 |
. . . . . . . . . . . . . . . 16
⊢ (𝐽 ∈ (𝐿...𝑀) → 𝐽 ≤ 𝑀) |
157 | 41, 156 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐽 ≤ 𝑀) |
158 | 157 | adantr 473 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝐽 ≤ 𝑀) |
159 | 144, 145,
147, 155, 158 | letrd 10595 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ≤ 𝑀) |
160 | 142, 159 | jca 504 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐿 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀)) |
161 | 140, 160,
74 | sylanbrc 575 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → 𝑖 ∈ (𝐿...𝑀)) |
162 | 161, 10 | syldan 582 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑖) ∈ ℝ) |
163 | 162 | adantlr 702 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑖) ∈ ℝ) |
164 | 132, 135,
163 | chvar 2326 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑎 ∈ (𝐿...(𝐽 − 1))) → (𝐵‘𝑎) ∈ ℝ) |
165 | 37 | adantl 474 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ (𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑎 · 𝑗) ∈ ℝ) |
166 | 115, 164,
165 | seqcl 13203 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘(𝐽 − 1)) ∈
ℝ) |
167 | | 1red 10438 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 1 ∈ ℝ) |
168 | | eqid 2772 |
. . . . . . . . 9
⊢ seq𝐽( · , 𝐵) = seq𝐽( · , 𝐵) |
169 | 43 | adantr 473 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝑀 ∈ (ℤ≥‘𝐽)) |
170 | | eluzfz2 12729 |
. . . . . . . . . . 11
⊢ (𝑀 ∈
(ℤ≥‘𝐽) → 𝑀 ∈ (𝐽...𝑀)) |
171 | 43, 170 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ (𝐽...𝑀)) |
172 | 171 | adantr 473 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝑀 ∈ (𝐽...𝑀)) |
173 | 76 | adantlr 702 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐵‘𝑖) ∈ ℝ) |
174 | 75, 12 | syldan 582 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → 0 ≤ (𝐵‘𝑖)) |
175 | 174 | adantlr 702 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐽...𝑀)) → 0 ≤ (𝐵‘𝑖)) |
176 | 75, 14 | syldan 582 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐵‘𝑖) ≤ 1) |
177 | 176 | adantlr 702 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐽...𝑀)) → (𝐵‘𝑖) ≤ 1) |
178 | 1, 117, 168, 99, 169, 172, 173, 175, 177 | fmul01 41317 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (0 ≤ (seq𝐽( · , 𝐵)‘𝑀) ∧ (seq𝐽( · , 𝐵)‘𝑀) ≤ 1)) |
179 | 178 | simpld 487 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 0 ≤ (seq𝐽( · , 𝐵)‘𝑀)) |
180 | | eqid 2772 |
. . . . . . . . 9
⊢ seq𝐿( · , 𝐵) = seq𝐿( · , 𝐵) |
181 | 8 | adantr 473 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝑀 ∈ (ℤ≥‘𝐿)) |
182 | | 1zzd 11824 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 1 ∈
ℤ) |
183 | 60, 182 | zsubcld 11903 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐽 − 1) ∈ ℤ) |
184 | 6, 52, 183 | 3jca 1108 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝐽 − 1) ∈
ℤ)) |
185 | 184 | adantr 473 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝐽 − 1) ∈
ℤ)) |
186 | 149, 61, 146 | 3jca 1108 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐽 − 1) ∈ ℝ ∧ 𝐽 ∈ ℝ ∧ 𝑀 ∈
ℝ)) |
187 | 186 | adantr 473 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ((𝐽 − 1) ∈ ℝ ∧ 𝐽 ∈ ℝ ∧ 𝑀 ∈
ℝ)) |
188 | 61 | adantr 473 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐽 ∈ ℝ) |
189 | 188 | lem1d 11372 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐽 − 1) ≤ 𝐽) |
190 | 157 | adantr 473 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → 𝐽 ≤ 𝑀) |
191 | 189, 190 | jca 504 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ((𝐽 − 1) ≤ 𝐽 ∧ 𝐽 ≤ 𝑀)) |
192 | | letr 10532 |
. . . . . . . . . . . 12
⊢ (((𝐽 − 1) ∈ ℝ ∧
𝐽 ∈ ℝ ∧
𝑀 ∈ ℝ) →
(((𝐽 − 1) ≤ 𝐽 ∧ 𝐽 ≤ 𝑀) → (𝐽 − 1) ≤ 𝑀)) |
193 | 187, 191,
192 | sylc 65 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐽 − 1) ≤ 𝑀) |
194 | 113, 193 | jca 504 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐿 ≤ (𝐽 − 1) ∧ (𝐽 − 1) ≤ 𝑀)) |
195 | | elfz2 12713 |
. . . . . . . . . 10
⊢ ((𝐽 − 1) ∈ (𝐿...𝑀) ↔ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ) ∧ (𝐿 ≤ (𝐽 − 1) ∧ (𝐽 − 1) ≤ 𝑀))) |
196 | 185, 194,
195 | sylanbrc 575 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐽 − 1) ∈ (𝐿...𝑀)) |
197 | 12 | adantlr 702 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) |
198 | 14 | adantlr 702 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ¬ 𝐽 = 𝐿) ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1) |
199 | 1, 117, 180, 98, 181, 196, 122, 197, 198 | fmul01 41317 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (0 ≤ (seq𝐿( · , 𝐵)‘(𝐽 − 1)) ∧ (seq𝐿( · , 𝐵)‘(𝐽 − 1)) ≤ 1)) |
200 | 199 | simprd 488 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘(𝐽 − 1)) ≤ 1) |
201 | 166, 167,
79, 179, 200 | lemul1ad 11378 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → ((seq𝐿( · , 𝐵)‘(𝐽 − 1)) · (seq𝐽( · , 𝐵)‘𝑀)) ≤ (1 · (seq𝐽( · , 𝐵)‘𝑀))) |
202 | 129, 201 | eqbrtrd 4947 |
. . . . 5
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) ≤ (1 · (seq𝐽( · , 𝐵)‘𝑀))) |
203 | 79 | recnd 10466 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐽( · , 𝐵)‘𝑀) ∈ ℂ) |
204 | 203 | mulid2d 10456 |
. . . . 5
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (1 · (seq𝐽( · , 𝐵)‘𝑀)) = (seq𝐽( · , 𝐵)‘𝑀)) |
205 | 202, 204 | breqtrd 4951 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) ≤ (seq𝐽( · , 𝐵)‘𝑀)) |
206 | 1, 2, 168, 60, 43, 76, 174, 176, 16, 20 | fmul01lt1lem1 41321 |
. . . . 5
⊢ (𝜑 → (seq𝐽( · , 𝐵)‘𝑀) < 𝐸) |
207 | 206 | adantr 473 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐽( · , 𝐵)‘𝑀) < 𝐸) |
208 | 40, 79, 81, 205, 207 | lelttrd 10596 |
. . 3
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (seq𝐿( · , 𝐵)‘𝑀) < 𝐸) |
209 | 24, 208 | syl5eqbr 4960 |
. 2
⊢ ((𝜑 ∧ ¬ 𝐽 = 𝐿) → (𝐴‘𝑀) < 𝐸) |
210 | 23, 209 | pm2.61dan 800 |
1
⊢ (𝜑 → (𝐴‘𝑀) < 𝐸) |