Step | Hyp | Ref
| Expression |
1 | | simpr 476 |
. . . . 5
⊢ ((𝜑 ∧ 𝑀 = 𝐿) → 𝑀 = 𝐿) |
2 | 1 | fveq2d 6107 |
. . . 4
⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐴‘𝑀) = (𝐴‘𝐿)) |
3 | | fmul01lt1lem1.3 |
. . . . . 6
⊢ 𝐴 = seq𝐿( · , 𝐵) |
4 | 3 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑀 = 𝐿) → 𝐴 = seq𝐿( · , 𝐵)) |
5 | 4 | fveq1d 6105 |
. . . 4
⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐴‘𝐿) = (seq𝐿( · , 𝐵)‘𝐿)) |
6 | | fmul01lt1lem1.4 |
. . . . . 6
⊢ (𝜑 → 𝐿 ∈ ℤ) |
7 | | seq1 12676 |
. . . . . 6
⊢ (𝐿 ∈ ℤ → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿)) |
8 | 6, 7 | syl 17 |
. . . . 5
⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿)) |
9 | 8 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (seq𝐿( · , 𝐵)‘𝐿) = (𝐵‘𝐿)) |
10 | 2, 5, 9 | 3eqtrd 2648 |
. . 3
⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐴‘𝑀) = (𝐵‘𝐿)) |
11 | | fmul01lt1lem1.10 |
. . . 4
⊢ (𝜑 → (𝐵‘𝐿) < 𝐸) |
12 | 11 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐵‘𝐿) < 𝐸) |
13 | 10, 12 | eqbrtrd 4605 |
. 2
⊢ ((𝜑 ∧ 𝑀 = 𝐿) → (𝐴‘𝑀) < 𝐸) |
14 | | simpr 476 |
. . . . 5
⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → ¬ 𝑀 = 𝐿) |
15 | 14 | neqned 2789 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → 𝑀 ≠ 𝐿) |
16 | 6 | zred 11358 |
. . . . . . 7
⊢ (𝜑 → 𝐿 ∈ ℝ) |
17 | | fmul01lt1lem1.5 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐿)) |
18 | | eluzelz 11573 |
. . . . . . . . 9
⊢ (𝑀 ∈
(ℤ≥‘𝐿) → 𝑀 ∈ ℤ) |
19 | 17, 18 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℤ) |
20 | 19 | zred 11358 |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ ℝ) |
21 | | eluzle 11576 |
. . . . . . . 8
⊢ (𝑀 ∈
(ℤ≥‘𝐿) → 𝐿 ≤ 𝑀) |
22 | 17, 21 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐿 ≤ 𝑀) |
23 | 16, 20, 22 | 3jca 1235 |
. . . . . 6
⊢ (𝜑 → (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝐿 ≤ 𝑀)) |
24 | 23 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝐿 ≤ 𝑀)) |
25 | | leltne 10006 |
. . . . 5
⊢ ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝐿 ≤ 𝑀) → (𝐿 < 𝑀 ↔ 𝑀 ≠ 𝐿)) |
26 | 24, 25 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → (𝐿 < 𝑀 ↔ 𝑀 ≠ 𝐿)) |
27 | 15, 26 | mpbird 246 |
. . 3
⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → 𝐿 < 𝑀) |
28 | 3 | fveq1i 6104 |
. . . 4
⊢ (𝐴‘𝑀) = (seq𝐿( · , 𝐵)‘𝑀) |
29 | | remulcl 9900 |
. . . . . . 7
⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑗 · 𝑘) ∈ ℝ) |
30 | 29 | adantl 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ (𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (𝑗 · 𝑘) ∈ ℝ) |
31 | | recn 9905 |
. . . . . . . . 9
⊢ (𝑗 ∈ ℝ → 𝑗 ∈
ℂ) |
32 | 31 | 3ad2ant1 1075 |
. . . . . . . 8
⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑗 ∈
ℂ) |
33 | | recn 9905 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℝ → 𝑘 ∈
ℂ) |
34 | 33 | 3ad2ant2 1076 |
. . . . . . . 8
⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑘 ∈
ℂ) |
35 | | recn 9905 |
. . . . . . . . 9
⊢ (𝑙 ∈ ℝ → 𝑙 ∈
ℂ) |
36 | 35 | 3ad2ant3 1077 |
. . . . . . . 8
⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑙 ∈
ℂ) |
37 | 32, 34, 36 | mulassd 9942 |
. . . . . . 7
⊢ ((𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ) → ((𝑗 · 𝑘) · 𝑙) = (𝑗 · (𝑘 · 𝑙))) |
38 | 37 | adantl 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ (𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ)) → ((𝑗 · 𝑘) · 𝑙) = (𝑗 · (𝑘 · 𝑙))) |
39 | | simpr 476 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐿 < 𝑀) |
40 | 39 | olcd 407 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 < 𝐿 ∨ 𝐿 < 𝑀)) |
41 | 20, 16 | jca 553 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ)) |
42 | 41 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ)) |
43 | | lttri2 9999 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 ≠ 𝐿 ↔ (𝑀 < 𝐿 ∨ 𝐿 < 𝑀))) |
44 | 42, 43 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 ≠ 𝐿 ↔ (𝑀 < 𝐿 ∨ 𝐿 < 𝑀))) |
45 | 40, 44 | mpbird 246 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ≠ 𝐿) |
46 | 45 | neneqd 2787 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ¬ 𝑀 = 𝐿) |
47 | | uzp1 11597 |
. . . . . . . . . 10
⊢ (𝑀 ∈
(ℤ≥‘𝐿) → (𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ≥‘(𝐿 + 1)))) |
48 | 17, 47 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ≥‘(𝐿 + 1)))) |
49 | 48 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ≥‘(𝐿 + 1)))) |
50 | 49 | ord 391 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (¬ 𝑀 = 𝐿 → 𝑀 ∈ (ℤ≥‘(𝐿 + 1)))) |
51 | 46, 50 | mpd 15 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ (ℤ≥‘(𝐿 + 1))) |
52 | 6 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐿 ∈ ℤ) |
53 | | uzid 11578 |
. . . . . . 7
⊢ (𝐿 ∈ ℤ → 𝐿 ∈
(ℤ≥‘𝐿)) |
54 | 52, 53 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐿 ∈ (ℤ≥‘𝐿)) |
55 | | fmul01lt1lem1.2 |
. . . . . . . . . 10
⊢
Ⅎ𝑖𝜑 |
56 | | nfv 1830 |
. . . . . . . . . 10
⊢
Ⅎ𝑖 𝑗 ∈ (𝐿...𝑀) |
57 | 55, 56 | nfan 1816 |
. . . . . . . . 9
⊢
Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ (𝐿...𝑀)) |
58 | | fmul01lt1lem1.1 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖𝐵 |
59 | | nfcv 2751 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖𝑗 |
60 | 58, 59 | nffv 6110 |
. . . . . . . . . 10
⊢
Ⅎ𝑖(𝐵‘𝑗) |
61 | 60 | nfel1 2765 |
. . . . . . . . 9
⊢
Ⅎ𝑖(𝐵‘𝑗) ∈ ℝ |
62 | 57, 61 | nfim 1813 |
. . . . . . . 8
⊢
Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ (𝐿...𝑀)) → (𝐵‘𝑗) ∈ ℝ) |
63 | | eleq1 2676 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑗 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝑗 ∈ (𝐿...𝑀))) |
64 | 63 | anbi2d 736 |
. . . . . . . . 9
⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝑗 ∈ (𝐿...𝑀)))) |
65 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑗 → (𝐵‘𝑖) = (𝐵‘𝑗)) |
66 | 65 | eleq1d 2672 |
. . . . . . . . 9
⊢ (𝑖 = 𝑗 → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘𝑗) ∈ ℝ)) |
67 | 64, 66 | imbi12d 333 |
. . . . . . . 8
⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ ((𝜑 ∧ 𝑗 ∈ (𝐿...𝑀)) → (𝐵‘𝑗) ∈ ℝ))) |
68 | | fmul01lt1lem1.6 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) |
69 | 62, 67, 68 | chvar 2250 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (𝐿...𝑀)) → (𝐵‘𝑗) ∈ ℝ) |
70 | 69 | adantlr 747 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑗 ∈ (𝐿...𝑀)) → (𝐵‘𝑗) ∈ ℝ) |
71 | 30, 38, 51, 54, 70 | seqsplit 12696 |
. . . . 5
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq𝐿( · , 𝐵)‘𝑀) = ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀))) |
72 | | eluzfz1 12219 |
. . . . . . . . . . 11
⊢ (𝑀 ∈
(ℤ≥‘𝐿) → 𝐿 ∈ (𝐿...𝑀)) |
73 | 17, 72 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐿 ∈ (𝐿...𝑀)) |
74 | 73 | ancli 572 |
. . . . . . . . . 10
⊢ (𝜑 → (𝜑 ∧ 𝐿 ∈ (𝐿...𝑀))) |
75 | | nfv 1830 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑖 𝐿 ∈ (𝐿...𝑀) |
76 | 55, 75 | nfan 1816 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑖(𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) |
77 | | nfcv 2751 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑖𝐿 |
78 | 58, 77 | nffv 6110 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑖(𝐵‘𝐿) |
79 | 78 | nfel1 2765 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑖(𝐵‘𝐿) ∈ ℝ |
80 | 76, 79 | nfim 1813 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ∈ ℝ) |
81 | | eleq1 2676 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝐿 → (𝑖 ∈ (𝐿...𝑀) ↔ 𝐿 ∈ (𝐿...𝑀))) |
82 | 81 | anbi2d 736 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝐿 → ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) ↔ (𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)))) |
83 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝐿 → (𝐵‘𝑖) = (𝐵‘𝐿)) |
84 | 83 | eleq1d 2672 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝐿 → ((𝐵‘𝑖) ∈ ℝ ↔ (𝐵‘𝐿) ∈ ℝ)) |
85 | 82, 84 | imbi12d 333 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝐿 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) ↔ ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ∈ ℝ))) |
86 | 80, 85, 68 | vtoclg1f 3238 |
. . . . . . . . . 10
⊢ (𝐿 ∈ (𝐿...𝑀) → ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → (𝐵‘𝐿) ∈ ℝ)) |
87 | 73, 74, 86 | sylc 63 |
. . . . . . . . 9
⊢ (𝜑 → (𝐵‘𝐿) ∈ ℝ) |
88 | 8, 87 | eqeltrd 2688 |
. . . . . . . 8
⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) ∈ ℝ) |
89 | 88 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq𝐿( · , 𝐵)‘𝐿) ∈ ℝ) |
90 | 6 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℤ) |
91 | 19 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑀 ∈ ℤ) |
92 | | elfzelz 12213 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ((𝐿 + 1)...𝑀) → 𝑗 ∈ ℤ) |
93 | 92 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑗 ∈ ℤ) |
94 | 90, 91, 93 | 3jca 1235 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ)) |
95 | 16 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℝ) |
96 | | peano2re 10088 |
. . . . . . . . . . . . . . 15
⊢ (𝐿 ∈ ℝ → (𝐿 + 1) ∈
ℝ) |
97 | 16, 96 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐿 + 1) ∈ ℝ) |
98 | 97 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ∈ ℝ) |
99 | 92 | zred 11358 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈ ((𝐿 + 1)...𝑀) → 𝑗 ∈ ℝ) |
100 | 99 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑗 ∈ ℝ) |
101 | 16 | lep1d 10834 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐿 ≤ (𝐿 + 1)) |
102 | 101 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ (𝐿 + 1)) |
103 | | elfzle1 12215 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈ ((𝐿 + 1)...𝑀) → (𝐿 + 1) ≤ 𝑗) |
104 | 103 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ≤ 𝑗) |
105 | 95, 98, 100, 102, 104 | letrd 10073 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ 𝑗) |
106 | | elfzle2 12216 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ((𝐿 + 1)...𝑀) → 𝑗 ≤ 𝑀) |
107 | 106 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑗 ≤ 𝑀) |
108 | 105, 107 | jca 553 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)) |
109 | | elfz2 12204 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ (𝐿...𝑀) ↔ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (𝐿 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀))) |
110 | 94, 108, 109 | sylanbrc 695 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → 𝑗 ∈ (𝐿...𝑀)) |
111 | 110, 69 | syldan 486 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑗) ∈ ℝ) |
112 | 111 | adantlr 747 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑗 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑗) ∈ ℝ) |
113 | 51, 112, 30 | seqcl 12683 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq(𝐿 + 1)( · , 𝐵)‘𝑀) ∈ ℝ) |
114 | 89, 113 | remulcld 9949 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)) ∈ ℝ) |
115 | | fmul01lt1lem1.9 |
. . . . . . . 8
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
116 | 115 | rpred 11748 |
. . . . . . 7
⊢ (𝜑 → 𝐸 ∈ ℝ) |
117 | 116 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐸 ∈ ℝ) |
118 | | 1red 9934 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 1 ∈ ℝ) |
119 | | nfcv 2751 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑖0 |
120 | | nfcv 2751 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑖
≤ |
121 | 119, 120,
78 | nfbr 4629 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑖0 ≤
(𝐵‘𝐿) |
122 | 76, 121 | nfim 1813 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑖((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿)) |
123 | 83 | breq2d 4595 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝐿 → (0 ≤ (𝐵‘𝑖) ↔ 0 ≤ (𝐵‘𝐿))) |
124 | 82, 123 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝐿 → (((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) ↔ ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿)))) |
125 | | fmul01lt1lem1.7 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) |
126 | 122, 124,
125 | vtoclg1f 3238 |
. . . . . . . . . . 11
⊢ (𝐿 ∈ (𝐿...𝑀) → ((𝜑 ∧ 𝐿 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝐿))) |
127 | 73, 74, 126 | sylc 63 |
. . . . . . . . . 10
⊢ (𝜑 → 0 ≤ (𝐵‘𝐿)) |
128 | 127, 8 | breqtrrd 4611 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ (seq𝐿( · , 𝐵)‘𝐿)) |
129 | 128 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 0 ≤ (seq𝐿( · , 𝐵)‘𝐿)) |
130 | | nfv 1830 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖 𝐿 < 𝑀 |
131 | 55, 130 | nfan 1816 |
. . . . . . . . . 10
⊢
Ⅎ𝑖(𝜑 ∧ 𝐿 < 𝑀) |
132 | | eqid 2610 |
. . . . . . . . . 10
⊢ seq(𝐿 + 1)( · , 𝐵) = seq(𝐿 + 1)( · , 𝐵) |
133 | 6 | peano2zd 11361 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐿 + 1) ∈ ℤ) |
134 | 133 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝐿 + 1) ∈ ℤ) |
135 | 16 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝐿 ∈ ℝ) |
136 | 135, 39 | gtned 10051 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ≠ 𝐿) |
137 | 136 | neneqd 2787 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ¬ 𝑀 = 𝐿) |
138 | 17 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ (ℤ≥‘𝐿)) |
139 | 138, 47 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ≥‘(𝐿 + 1)))) |
140 | | orel1 396 |
. . . . . . . . . . 11
⊢ (¬
𝑀 = 𝐿 → ((𝑀 = 𝐿 ∨ 𝑀 ∈ (ℤ≥‘(𝐿 + 1))) → 𝑀 ∈ (ℤ≥‘(𝐿 + 1)))) |
141 | 137, 139,
140 | sylc 63 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ (ℤ≥‘(𝐿 + 1))) |
142 | 19 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ ℤ) |
143 | 134, 142,
142 | 3jca 1235 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((𝐿 + 1) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ)) |
144 | | zltp1le 11304 |
. . . . . . . . . . . . . 14
⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 < 𝑀 ↔ (𝐿 + 1) ≤ 𝑀)) |
145 | 52, 142, 144 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝐿 < 𝑀 ↔ (𝐿 + 1) ≤ 𝑀)) |
146 | 39, 145 | mpbid 221 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝐿 + 1) ≤ 𝑀) |
147 | 20 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ ℝ) |
148 | 147 | leidd 10473 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ≤ 𝑀) |
149 | 146, 148 | jca 553 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((𝐿 + 1) ≤ 𝑀 ∧ 𝑀 ≤ 𝑀)) |
150 | | elfz2 12204 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ ((𝐿 + 1)...𝑀) ↔ (((𝐿 + 1) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ ((𝐿 + 1) ≤ 𝑀 ∧ 𝑀 ≤ 𝑀))) |
151 | 143, 149,
150 | sylanbrc 695 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → 𝑀 ∈ ((𝐿 + 1)...𝑀)) |
152 | 6 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℤ) |
153 | 19 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑀 ∈ ℤ) |
154 | | elfzelz 12213 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ ((𝐿 + 1)...𝑀) → 𝑖 ∈ ℤ) |
155 | 154 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ ℤ) |
156 | 152, 153,
155 | 3jca 1235 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ)) |
157 | 16 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℝ) |
158 | 157, 96 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ∈ ℝ) |
159 | 154 | zred 11358 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ ((𝐿 + 1)...𝑀) → 𝑖 ∈ ℝ) |
160 | 159 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ ℝ) |
161 | 101 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ (𝐿 + 1)) |
162 | | elfzle1 12215 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ ((𝐿 + 1)...𝑀) → (𝐿 + 1) ≤ 𝑖) |
163 | 162 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ≤ 𝑖) |
164 | 157, 158,
160, 161, 163 | letrd 10073 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ 𝑖) |
165 | | elfzle2 12216 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ ((𝐿 + 1)...𝑀) → 𝑖 ≤ 𝑀) |
166 | 165 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ≤ 𝑀) |
167 | 164, 166 | jca 553 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀)) |
168 | | elfz2 12204 |
. . . . . . . . . . . . 13
⊢ (𝑖 ∈ (𝐿...𝑀) ↔ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (𝐿 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀))) |
169 | 156, 167,
168 | sylanbrc 695 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ (𝐿...𝑀)) |
170 | 169, 68 | syldan 486 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑖) ∈ ℝ) |
171 | 170 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑖) ∈ ℝ) |
172 | | simpll 786 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝜑) |
173 | 6 | ad2antrr 758 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℤ) |
174 | 19 | ad2antrr 758 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑀 ∈ ℤ) |
175 | 154 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ ℤ) |
176 | 173, 174,
175 | 3jca 1235 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ)) |
177 | 16 | ad2antrr 758 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ∈ ℝ) |
178 | 97 | ad2antrr 758 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ∈ ℝ) |
179 | 159 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ ℝ) |
180 | 101 | ad2antrr 758 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ (𝐿 + 1)) |
181 | 162 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 + 1) ≤ 𝑖) |
182 | 177, 178,
179, 180, 181 | letrd 10073 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝐿 ≤ 𝑖) |
183 | 165 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ≤ 𝑀) |
184 | 182, 183 | jca 553 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐿 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀)) |
185 | 176, 184,
168 | sylanbrc 695 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 𝑖 ∈ (𝐿...𝑀)) |
186 | 172, 185,
125 | syl2anc 691 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → 0 ≤ (𝐵‘𝑖)) |
187 | | fmul01lt1lem1.8 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1) |
188 | 172, 185,
187 | syl2anc 691 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐿 < 𝑀) ∧ 𝑖 ∈ ((𝐿 + 1)...𝑀)) → (𝐵‘𝑖) ≤ 1) |
189 | 58, 131, 132, 134, 141, 151, 171, 186, 188 | fmul01 38647 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (0 ≤ (seq(𝐿 + 1)( · , 𝐵)‘𝑀) ∧ (seq(𝐿 + 1)( · , 𝐵)‘𝑀) ≤ 1)) |
190 | 189 | simprd 478 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq(𝐿 + 1)( · , 𝐵)‘𝑀) ≤ 1) |
191 | 113, 118,
89, 129, 190 | lemul2ad 10843 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)) ≤ ((seq𝐿( · , 𝐵)‘𝐿) · 1)) |
192 | 88 | recnd 9947 |
. . . . . . . . 9
⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) ∈ ℂ) |
193 | 192 | mulid1d 9936 |
. . . . . . . 8
⊢ (𝜑 → ((seq𝐿( · , 𝐵)‘𝐿) · 1) = (seq𝐿( · , 𝐵)‘𝐿)) |
194 | 193 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · 1) = (seq𝐿( · , 𝐵)‘𝐿)) |
195 | 191, 194 | breqtrd 4609 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)) ≤ (seq𝐿( · , 𝐵)‘𝐿)) |
196 | 8, 11 | eqbrtrd 4605 |
. . . . . . 7
⊢ (𝜑 → (seq𝐿( · , 𝐵)‘𝐿) < 𝐸) |
197 | 196 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq𝐿( · , 𝐵)‘𝐿) < 𝐸) |
198 | 114, 89, 117, 195, 197 | lelttrd 10074 |
. . . . 5
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → ((seq𝐿( · , 𝐵)‘𝐿) · (seq(𝐿 + 1)( · , 𝐵)‘𝑀)) < 𝐸) |
199 | 71, 198 | eqbrtrd 4605 |
. . . 4
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (seq𝐿( · , 𝐵)‘𝑀) < 𝐸) |
200 | 28, 199 | syl5eqbr 4618 |
. . 3
⊢ ((𝜑 ∧ 𝐿 < 𝑀) → (𝐴‘𝑀) < 𝐸) |
201 | 27, 200 | syldan 486 |
. 2
⊢ ((𝜑 ∧ ¬ 𝑀 = 𝐿) → (𝐴‘𝑀) < 𝐸) |
202 | 13, 201 | pm2.61dan 828 |
1
⊢ (𝜑 → (𝐴‘𝑀) < 𝐸) |