Proof of Theorem fourierdlem11
Step | Hyp | Ref
| Expression |
1 | | fourierdlem11.q |
. . . . . . 7
⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) |
2 | | fourierdlem11.m |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℕ) |
3 | | fourierdlem11.p |
. . . . . . . . 9
⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
4 | 3 | fourierdlem2 42393 |
. . . . . . . 8
⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m
(0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))) |
5 | 2, 4 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m
(0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))) |
6 | 1, 5 | mpbid 234 |
. . . . . 6
⊢ (𝜑 → (𝑄 ∈ (ℝ ↑m
(0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))) |
7 | 6 | simprd 498 |
. . . . 5
⊢ (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))) |
8 | 7 | simpld 497 |
. . . 4
⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵)) |
9 | 8 | simpld 497 |
. . 3
⊢ (𝜑 → (𝑄‘0) = 𝐴) |
10 | 6 | simpld 497 |
. . . . 5
⊢ (𝜑 → 𝑄 ∈ (ℝ ↑m
(0...𝑀))) |
11 | | elmapi 8427 |
. . . . 5
⊢ (𝑄 ∈ (ℝ
↑m (0...𝑀))
→ 𝑄:(0...𝑀)⟶ℝ) |
12 | 10, 11 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ) |
13 | | 0red 10643 |
. . . . . 6
⊢ (𝜑 → 0 ∈
ℝ) |
14 | 13 | leidd 11205 |
. . . . 5
⊢ (𝜑 → 0 ≤ 0) |
15 | 2 | nnred 11652 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℝ) |
16 | 2 | nngt0d 11685 |
. . . . . 6
⊢ (𝜑 → 0 < 𝑀) |
17 | 13, 15, 16 | ltled 10787 |
. . . . 5
⊢ (𝜑 → 0 ≤ 𝑀) |
18 | | 0zd 11992 |
. . . . . 6
⊢ (𝜑 → 0 ∈
ℤ) |
19 | 2 | nnzd 12085 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℤ) |
20 | | elfz 12897 |
. . . . . 6
⊢ ((0
∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (0 ∈ (0...𝑀) ↔ (0 ≤ 0 ∧ 0 ≤
𝑀))) |
21 | 18, 18, 19, 20 | syl3anc 1367 |
. . . . 5
⊢ (𝜑 → (0 ∈ (0...𝑀) ↔ (0 ≤ 0 ∧ 0 ≤
𝑀))) |
22 | 14, 17, 21 | mpbir2and 711 |
. . . 4
⊢ (𝜑 → 0 ∈ (0...𝑀)) |
23 | 12, 22 | ffvelrnd 6851 |
. . 3
⊢ (𝜑 → (𝑄‘0) ∈ ℝ) |
24 | 9, 23 | eqeltrrd 2914 |
. 2
⊢ (𝜑 → 𝐴 ∈ ℝ) |
25 | 8 | simprd 498 |
. . 3
⊢ (𝜑 → (𝑄‘𝑀) = 𝐵) |
26 | 15 | leidd 11205 |
. . . . 5
⊢ (𝜑 → 𝑀 ≤ 𝑀) |
27 | | elfz 12897 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ 0 ∈
ℤ ∧ 𝑀 ∈
ℤ) → (𝑀 ∈
(0...𝑀) ↔ (0 ≤
𝑀 ∧ 𝑀 ≤ 𝑀))) |
28 | 19, 18, 19, 27 | syl3anc 1367 |
. . . . 5
⊢ (𝜑 → (𝑀 ∈ (0...𝑀) ↔ (0 ≤ 𝑀 ∧ 𝑀 ≤ 𝑀))) |
29 | 17, 26, 28 | mpbir2and 711 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ (0...𝑀)) |
30 | 12, 29 | ffvelrnd 6851 |
. . 3
⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ) |
31 | 25, 30 | eqeltrrd 2914 |
. 2
⊢ (𝜑 → 𝐵 ∈ ℝ) |
32 | | 0le1 11162 |
. . . . . 6
⊢ 0 ≤
1 |
33 | 32 | a1i 11 |
. . . . 5
⊢ (𝜑 → 0 ≤ 1) |
34 | 2 | nnge1d 11684 |
. . . . 5
⊢ (𝜑 → 1 ≤ 𝑀) |
35 | | 1zzd 12012 |
. . . . . 6
⊢ (𝜑 → 1 ∈
ℤ) |
36 | | elfz 12897 |
. . . . . 6
⊢ ((1
∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (1 ∈ (0...𝑀) ↔ (0 ≤ 1 ∧ 1 ≤
𝑀))) |
37 | 35, 18, 19, 36 | syl3anc 1367 |
. . . . 5
⊢ (𝜑 → (1 ∈ (0...𝑀) ↔ (0 ≤ 1 ∧ 1 ≤
𝑀))) |
38 | 33, 34, 37 | mpbir2and 711 |
. . . 4
⊢ (𝜑 → 1 ∈ (0...𝑀)) |
39 | 12, 38 | ffvelrnd 6851 |
. . 3
⊢ (𝜑 → (𝑄‘1) ∈ ℝ) |
40 | | elfzo 13039 |
. . . . . . 7
⊢ ((0
∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (0 ∈ (0..^𝑀) ↔ (0 ≤ 0 ∧ 0 <
𝑀))) |
41 | 18, 18, 19, 40 | syl3anc 1367 |
. . . . . 6
⊢ (𝜑 → (0 ∈ (0..^𝑀) ↔ (0 ≤ 0 ∧ 0 <
𝑀))) |
42 | 14, 16, 41 | mpbir2and 711 |
. . . . 5
⊢ (𝜑 → 0 ∈ (0..^𝑀)) |
43 | | 0re 10642 |
. . . . . 6
⊢ 0 ∈
ℝ |
44 | | eleq1 2900 |
. . . . . . . . 9
⊢ (𝑖 = 0 → (𝑖 ∈ (0..^𝑀) ↔ 0 ∈ (0..^𝑀))) |
45 | 44 | anbi2d 630 |
. . . . . . . 8
⊢ (𝑖 = 0 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 0 ∈ (0..^𝑀)))) |
46 | | fveq2 6669 |
. . . . . . . . 9
⊢ (𝑖 = 0 → (𝑄‘𝑖) = (𝑄‘0)) |
47 | | oveq1 7162 |
. . . . . . . . . 10
⊢ (𝑖 = 0 → (𝑖 + 1) = (0 + 1)) |
48 | 47 | fveq2d 6673 |
. . . . . . . . 9
⊢ (𝑖 = 0 → (𝑄‘(𝑖 + 1)) = (𝑄‘(0 + 1))) |
49 | 46, 48 | breq12d 5078 |
. . . . . . . 8
⊢ (𝑖 = 0 → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘0) < (𝑄‘(0 + 1)))) |
50 | 45, 49 | imbi12d 347 |
. . . . . . 7
⊢ (𝑖 = 0 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ 0 ∈ (0..^𝑀)) → (𝑄‘0) < (𝑄‘(0 + 1))))) |
51 | 7 | simprd 498 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) |
52 | 51 | r19.21bi 3208 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) |
53 | 50, 52 | vtoclg 3567 |
. . . . . 6
⊢ (0 ∈
ℝ → ((𝜑 ∧ 0
∈ (0..^𝑀)) →
(𝑄‘0) < (𝑄‘(0 +
1)))) |
54 | 43, 53 | ax-mp 5 |
. . . . 5
⊢ ((𝜑 ∧ 0 ∈ (0..^𝑀)) → (𝑄‘0) < (𝑄‘(0 + 1))) |
55 | 42, 54 | mpdan 685 |
. . . 4
⊢ (𝜑 → (𝑄‘0) < (𝑄‘(0 + 1))) |
56 | | 0p1e1 11758 |
. . . . . 6
⊢ (0 + 1) =
1 |
57 | 56 | a1i 11 |
. . . . 5
⊢ (𝜑 → (0 + 1) =
1) |
58 | 57 | fveq2d 6673 |
. . . 4
⊢ (𝜑 → (𝑄‘(0 + 1)) = (𝑄‘1)) |
59 | 55, 9, 58 | 3brtr3d 5096 |
. . 3
⊢ (𝜑 → 𝐴 < (𝑄‘1)) |
60 | | nnuz 12280 |
. . . . . 6
⊢ ℕ =
(ℤ≥‘1) |
61 | 2, 60 | eleqtrdi 2923 |
. . . . 5
⊢ (𝜑 → 𝑀 ∈
(ℤ≥‘1)) |
62 | 12 | adantr 483 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑄:(0...𝑀)⟶ℝ) |
63 | | 0red 10643 |
. . . . . . . . 9
⊢ (𝑖 ∈ (1...𝑀) → 0 ∈ ℝ) |
64 | | elfzelz 12907 |
. . . . . . . . . 10
⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℤ) |
65 | 64 | zred 12086 |
. . . . . . . . 9
⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℝ) |
66 | | 1red 10641 |
. . . . . . . . . 10
⊢ (𝑖 ∈ (1...𝑀) → 1 ∈ ℝ) |
67 | | 0lt1 11161 |
. . . . . . . . . . 11
⊢ 0 <
1 |
68 | 67 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑖 ∈ (1...𝑀) → 0 < 1) |
69 | | elfzle1 12909 |
. . . . . . . . . 10
⊢ (𝑖 ∈ (1...𝑀) → 1 ≤ 𝑖) |
70 | 63, 66, 65, 68, 69 | ltletrd 10799 |
. . . . . . . . 9
⊢ (𝑖 ∈ (1...𝑀) → 0 < 𝑖) |
71 | 63, 65, 70 | ltled 10787 |
. . . . . . . 8
⊢ (𝑖 ∈ (1...𝑀) → 0 ≤ 𝑖) |
72 | | elfzle2 12910 |
. . . . . . . 8
⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ≤ 𝑀) |
73 | | 0zd 11992 |
. . . . . . . . 9
⊢ (𝑖 ∈ (1...𝑀) → 0 ∈ ℤ) |
74 | | elfzel2 12905 |
. . . . . . . . 9
⊢ (𝑖 ∈ (1...𝑀) → 𝑀 ∈ ℤ) |
75 | | elfz 12897 |
. . . . . . . . 9
⊢ ((𝑖 ∈ ℤ ∧ 0 ∈
ℤ ∧ 𝑀 ∈
ℤ) → (𝑖 ∈
(0...𝑀) ↔ (0 ≤
𝑖 ∧ 𝑖 ≤ 𝑀))) |
76 | 64, 73, 74, 75 | syl3anc 1367 |
. . . . . . . 8
⊢ (𝑖 ∈ (1...𝑀) → (𝑖 ∈ (0...𝑀) ↔ (0 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀))) |
77 | 71, 72, 76 | mpbir2and 711 |
. . . . . . 7
⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ (0...𝑀)) |
78 | 77 | adantl 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (0...𝑀)) |
79 | 62, 78 | ffvelrnd 6851 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑄‘𝑖) ∈ ℝ) |
80 | 12 | adantr 483 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑄:(0...𝑀)⟶ℝ) |
81 | | 0red 10643 |
. . . . . . . . . 10
⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 0 ∈
ℝ) |
82 | | elfzelz 12907 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 ∈ ℤ) |
83 | 82 | zred 12086 |
. . . . . . . . . 10
⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 ∈ ℝ) |
84 | | 1red 10641 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 1 ∈
ℝ) |
85 | 67 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 0 <
1) |
86 | | elfzle1 12909 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 1 ≤ 𝑖) |
87 | 81, 84, 83, 85, 86 | ltletrd 10799 |
. . . . . . . . . 10
⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 0 < 𝑖) |
88 | 81, 83, 87 | ltled 10787 |
. . . . . . . . 9
⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 0 ≤ 𝑖) |
89 | 88 | adantl 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 ≤ 𝑖) |
90 | 83 | adantl 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ ℝ) |
91 | 15 | adantr 483 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑀 ∈ ℝ) |
92 | | peano2rem 10952 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈
ℝ) |
93 | 91, 92 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑀 − 1) ∈ ℝ) |
94 | | elfzle2 12910 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 ≤ (𝑀 − 1)) |
95 | 94 | adantl 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ≤ (𝑀 − 1)) |
96 | 91 | ltm1d 11571 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑀 − 1) < 𝑀) |
97 | 90, 93, 91, 95, 96 | lelttrd 10797 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 < 𝑀) |
98 | 90, 91, 97 | ltled 10787 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ≤ 𝑀) |
99 | 82 | adantl 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ ℤ) |
100 | | 0zd 11992 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 ∈
ℤ) |
101 | 19 | adantr 483 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑀 ∈ ℤ) |
102 | 99, 100, 101, 75 | syl3anc 1367 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 ∈ (0...𝑀) ↔ (0 ≤ 𝑖 ∧ 𝑖 ≤ 𝑀))) |
103 | 89, 98, 102 | mpbir2and 711 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ (0...𝑀)) |
104 | 80, 103 | ffvelrnd 6851 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑄‘𝑖) ∈ ℝ) |
105 | | 0red 10643 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 ∈
ℝ) |
106 | | peano2re 10812 |
. . . . . . . . . 10
⊢ (𝑖 ∈ ℝ → (𝑖 + 1) ∈
ℝ) |
107 | 90, 106 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ∈ ℝ) |
108 | | 1red 10641 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 1 ∈
ℝ) |
109 | 67 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 <
1) |
110 | 83, 106 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑖 ∈ (1...(𝑀 − 1)) → (𝑖 + 1) ∈ ℝ) |
111 | 83 | ltp1d 11569 |
. . . . . . . . . . . 12
⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 < (𝑖 + 1)) |
112 | 84, 83, 110, 86, 111 | lelttrd 10797 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 1 < (𝑖 + 1)) |
113 | 112 | adantl 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 1 < (𝑖 + 1)) |
114 | 105, 108,
107, 109, 113 | lttrd 10800 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 < (𝑖 + 1)) |
115 | 105, 107,
114 | ltled 10787 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 0 ≤ (𝑖 + 1)) |
116 | 90, 93, 108, 95 | leadd1dd 11253 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ≤ ((𝑀 − 1) + 1)) |
117 | 2 | nncnd 11653 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℂ) |
118 | | 1cnd 10635 |
. . . . . . . . . . 11
⊢ (𝜑 → 1 ∈
ℂ) |
119 | 117, 118 | npcand 11000 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀) |
120 | 119 | adantr 483 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀) |
121 | 116, 120 | breqtrd 5091 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ≤ 𝑀) |
122 | 99 | peano2zd 12089 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ∈ ℤ) |
123 | | elfz 12897 |
. . . . . . . . 9
⊢ (((𝑖 + 1) ∈ ℤ ∧ 0
∈ ℤ ∧ 𝑀
∈ ℤ) → ((𝑖
+ 1) ∈ (0...𝑀) ↔
(0 ≤ (𝑖 + 1) ∧
(𝑖 + 1) ≤ 𝑀))) |
124 | 122, 100,
101, 123 | syl3anc 1367 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → ((𝑖 + 1) ∈ (0...𝑀) ↔ (0 ≤ (𝑖 + 1) ∧ (𝑖 + 1) ≤ 𝑀))) |
125 | 115, 121,
124 | mpbir2and 711 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 + 1) ∈ (0...𝑀)) |
126 | 80, 125 | ffvelrnd 6851 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
127 | | elfzo 13039 |
. . . . . . . . 9
⊢ ((𝑖 ∈ ℤ ∧ 0 ∈
ℤ ∧ 𝑀 ∈
ℤ) → (𝑖 ∈
(0..^𝑀) ↔ (0 ≤
𝑖 ∧ 𝑖 < 𝑀))) |
128 | 99, 100, 101, 127 | syl3anc 1367 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 ∈ (0..^𝑀) ↔ (0 ≤ 𝑖 ∧ 𝑖 < 𝑀))) |
129 | 89, 97, 128 | mpbir2and 711 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ (0..^𝑀)) |
130 | 129, 52 | syldan 593 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) |
131 | 104, 126,
130 | ltled 10787 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑄‘𝑖) ≤ (𝑄‘(𝑖 + 1))) |
132 | 61, 79, 131 | monoord 13399 |
. . . 4
⊢ (𝜑 → (𝑄‘1) ≤ (𝑄‘𝑀)) |
133 | 132, 25 | breqtrd 5091 |
. . 3
⊢ (𝜑 → (𝑄‘1) ≤ 𝐵) |
134 | 24, 39, 31, 59, 133 | ltletrd 10799 |
. 2
⊢ (𝜑 → 𝐴 < 𝐵) |
135 | 24, 31, 134 | 3jca 1124 |
1
⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)) |