Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem14.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
fourierdlem14.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
fourierdlem14.x |
⊢ ( 𝜑 → 𝑋 ∈ ℝ ) |
4 |
|
fourierdlem14.p |
⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = ( 𝐴 + 𝑋 ) ∧ ( 𝑝 ‘ 𝑚 ) = ( 𝐵 + 𝑋 ) ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) |
5 |
|
fourierdlem14.o |
⊢ 𝑂 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) |
6 |
|
fourierdlem14.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
7 |
|
fourierdlem14.v |
⊢ ( 𝜑 → 𝑉 ∈ ( 𝑃 ‘ 𝑀 ) ) |
8 |
|
fourierdlem14.q |
⊢ 𝑄 = ( 𝑖 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑉 ‘ 𝑖 ) − 𝑋 ) ) |
9 |
4
|
fourierdlem2 |
⊢ ( 𝑀 ∈ ℕ → ( 𝑉 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑉 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑉 ‘ 0 ) = ( 𝐴 + 𝑋 ) ∧ ( 𝑉 ‘ 𝑀 ) = ( 𝐵 + 𝑋 ) ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑉 ‘ 𝑖 ) < ( 𝑉 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
10 |
6 9
|
syl |
⊢ ( 𝜑 → ( 𝑉 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑉 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑉 ‘ 0 ) = ( 𝐴 + 𝑋 ) ∧ ( 𝑉 ‘ 𝑀 ) = ( 𝐵 + 𝑋 ) ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑉 ‘ 𝑖 ) < ( 𝑉 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
11 |
7 10
|
mpbid |
⊢ ( 𝜑 → ( 𝑉 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑉 ‘ 0 ) = ( 𝐴 + 𝑋 ) ∧ ( 𝑉 ‘ 𝑀 ) = ( 𝐵 + 𝑋 ) ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑉 ‘ 𝑖 ) < ( 𝑉 ‘ ( 𝑖 + 1 ) ) ) ) ) |
12 |
11
|
simpld |
⊢ ( 𝜑 → 𝑉 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ) |
13 |
|
elmapi |
⊢ ( 𝑉 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) → 𝑉 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
14 |
12 13
|
syl |
⊢ ( 𝜑 → 𝑉 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
15 |
14
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑀 ) ) → ( 𝑉 ‘ 𝑖 ) ∈ ℝ ) |
16 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑀 ) ) → 𝑋 ∈ ℝ ) |
17 |
15 16
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑀 ) ) → ( ( 𝑉 ‘ 𝑖 ) − 𝑋 ) ∈ ℝ ) |
18 |
17 8
|
fmptd |
⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
19 |
|
reex |
⊢ ℝ ∈ V |
20 |
19
|
a1i |
⊢ ( 𝜑 → ℝ ∈ V ) |
21 |
|
ovex |
⊢ ( 0 ... 𝑀 ) ∈ V |
22 |
21
|
a1i |
⊢ ( 𝜑 → ( 0 ... 𝑀 ) ∈ V ) |
23 |
20 22
|
elmapd |
⊢ ( 𝜑 → ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ↔ 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) ) |
24 |
18 23
|
mpbird |
⊢ ( 𝜑 → 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ) |
25 |
8
|
a1i |
⊢ ( 𝜑 → 𝑄 = ( 𝑖 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑉 ‘ 𝑖 ) − 𝑋 ) ) ) |
26 |
|
fveq2 |
⊢ ( 𝑖 = 0 → ( 𝑉 ‘ 𝑖 ) = ( 𝑉 ‘ 0 ) ) |
27 |
26
|
oveq1d |
⊢ ( 𝑖 = 0 → ( ( 𝑉 ‘ 𝑖 ) − 𝑋 ) = ( ( 𝑉 ‘ 0 ) − 𝑋 ) ) |
28 |
27
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 = 0 ) → ( ( 𝑉 ‘ 𝑖 ) − 𝑋 ) = ( ( 𝑉 ‘ 0 ) − 𝑋 ) ) |
29 |
|
0zd |
⊢ ( 𝜑 → 0 ∈ ℤ ) |
30 |
6
|
nnzd |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
31 |
|
0le0 |
⊢ 0 ≤ 0 |
32 |
31
|
a1i |
⊢ ( 𝜑 → 0 ≤ 0 ) |
33 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
34 |
6
|
nnred |
⊢ ( 𝜑 → 𝑀 ∈ ℝ ) |
35 |
6
|
nngt0d |
⊢ ( 𝜑 → 0 < 𝑀 ) |
36 |
33 34 35
|
ltled |
⊢ ( 𝜑 → 0 ≤ 𝑀 ) |
37 |
29 30 29 32 36
|
elfzd |
⊢ ( 𝜑 → 0 ∈ ( 0 ... 𝑀 ) ) |
38 |
14 37
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑉 ‘ 0 ) ∈ ℝ ) |
39 |
38 3
|
resubcld |
⊢ ( 𝜑 → ( ( 𝑉 ‘ 0 ) − 𝑋 ) ∈ ℝ ) |
40 |
25 28 37 39
|
fvmptd |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) = ( ( 𝑉 ‘ 0 ) − 𝑋 ) ) |
41 |
11
|
simprd |
⊢ ( 𝜑 → ( ( ( 𝑉 ‘ 0 ) = ( 𝐴 + 𝑋 ) ∧ ( 𝑉 ‘ 𝑀 ) = ( 𝐵 + 𝑋 ) ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑉 ‘ 𝑖 ) < ( 𝑉 ‘ ( 𝑖 + 1 ) ) ) ) |
42 |
41
|
simpld |
⊢ ( 𝜑 → ( ( 𝑉 ‘ 0 ) = ( 𝐴 + 𝑋 ) ∧ ( 𝑉 ‘ 𝑀 ) = ( 𝐵 + 𝑋 ) ) ) |
43 |
42
|
simpld |
⊢ ( 𝜑 → ( 𝑉 ‘ 0 ) = ( 𝐴 + 𝑋 ) ) |
44 |
43
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝑉 ‘ 0 ) − 𝑋 ) = ( ( 𝐴 + 𝑋 ) − 𝑋 ) ) |
45 |
1
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
46 |
3
|
recnd |
⊢ ( 𝜑 → 𝑋 ∈ ℂ ) |
47 |
45 46
|
pncand |
⊢ ( 𝜑 → ( ( 𝐴 + 𝑋 ) − 𝑋 ) = 𝐴 ) |
48 |
40 44 47
|
3eqtrd |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) = 𝐴 ) |
49 |
|
fveq2 |
⊢ ( 𝑖 = 𝑀 → ( 𝑉 ‘ 𝑖 ) = ( 𝑉 ‘ 𝑀 ) ) |
50 |
49
|
oveq1d |
⊢ ( 𝑖 = 𝑀 → ( ( 𝑉 ‘ 𝑖 ) − 𝑋 ) = ( ( 𝑉 ‘ 𝑀 ) − 𝑋 ) ) |
51 |
50
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 = 𝑀 ) → ( ( 𝑉 ‘ 𝑖 ) − 𝑋 ) = ( ( 𝑉 ‘ 𝑀 ) − 𝑋 ) ) |
52 |
34
|
leidd |
⊢ ( 𝜑 → 𝑀 ≤ 𝑀 ) |
53 |
29 30 30 36 52
|
elfzd |
⊢ ( 𝜑 → 𝑀 ∈ ( 0 ... 𝑀 ) ) |
54 |
14 53
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑉 ‘ 𝑀 ) ∈ ℝ ) |
55 |
54 3
|
resubcld |
⊢ ( 𝜑 → ( ( 𝑉 ‘ 𝑀 ) − 𝑋 ) ∈ ℝ ) |
56 |
25 51 53 55
|
fvmptd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) = ( ( 𝑉 ‘ 𝑀 ) − 𝑋 ) ) |
57 |
42
|
simprd |
⊢ ( 𝜑 → ( 𝑉 ‘ 𝑀 ) = ( 𝐵 + 𝑋 ) ) |
58 |
57
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝑉 ‘ 𝑀 ) − 𝑋 ) = ( ( 𝐵 + 𝑋 ) − 𝑋 ) ) |
59 |
2
|
recnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
60 |
59 46
|
pncand |
⊢ ( 𝜑 → ( ( 𝐵 + 𝑋 ) − 𝑋 ) = 𝐵 ) |
61 |
56 58 60
|
3eqtrd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) = 𝐵 ) |
62 |
48 61
|
jca |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ) |
63 |
|
elfzofz |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
64 |
63 15
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑉 ‘ 𝑖 ) ∈ ℝ ) |
65 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑉 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
66 |
|
fzofzp1 |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
67 |
66
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
68 |
65 67
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑉 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
69 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑋 ∈ ℝ ) |
70 |
41
|
simprd |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑉 ‘ 𝑖 ) < ( 𝑉 ‘ ( 𝑖 + 1 ) ) ) |
71 |
70
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑉 ‘ 𝑖 ) < ( 𝑉 ‘ ( 𝑖 + 1 ) ) ) |
72 |
64 68 69 71
|
ltsub1dd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑉 ‘ 𝑖 ) − 𝑋 ) < ( ( 𝑉 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) |
73 |
63
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
74 |
63 17
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑉 ‘ 𝑖 ) − 𝑋 ) ∈ ℝ ) |
75 |
8
|
fvmpt2 |
⊢ ( ( 𝑖 ∈ ( 0 ... 𝑀 ) ∧ ( ( 𝑉 ‘ 𝑖 ) − 𝑋 ) ∈ ℝ ) → ( 𝑄 ‘ 𝑖 ) = ( ( 𝑉 ‘ 𝑖 ) − 𝑋 ) ) |
76 |
73 74 75
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) = ( ( 𝑉 ‘ 𝑖 ) − 𝑋 ) ) |
77 |
|
fveq2 |
⊢ ( 𝑖 = 𝑗 → ( 𝑉 ‘ 𝑖 ) = ( 𝑉 ‘ 𝑗 ) ) |
78 |
77
|
oveq1d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝑉 ‘ 𝑖 ) − 𝑋 ) = ( ( 𝑉 ‘ 𝑗 ) − 𝑋 ) ) |
79 |
78
|
cbvmptv |
⊢ ( 𝑖 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑉 ‘ 𝑖 ) − 𝑋 ) ) = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑉 ‘ 𝑗 ) − 𝑋 ) ) |
80 |
8 79
|
eqtri |
⊢ 𝑄 = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑉 ‘ 𝑗 ) − 𝑋 ) ) |
81 |
80
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑉 ‘ 𝑗 ) − 𝑋 ) ) ) |
82 |
|
fveq2 |
⊢ ( 𝑗 = ( 𝑖 + 1 ) → ( 𝑉 ‘ 𝑗 ) = ( 𝑉 ‘ ( 𝑖 + 1 ) ) ) |
83 |
82
|
oveq1d |
⊢ ( 𝑗 = ( 𝑖 + 1 ) → ( ( 𝑉 ‘ 𝑗 ) − 𝑋 ) = ( ( 𝑉 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) |
84 |
83
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 = ( 𝑖 + 1 ) ) → ( ( 𝑉 ‘ 𝑗 ) − 𝑋 ) = ( ( 𝑉 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) |
85 |
68 69
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑉 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ∈ ℝ ) |
86 |
81 84 67 85
|
fvmptd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( ( 𝑉 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) |
87 |
72 76 86
|
3brtr4d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
88 |
87
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
89 |
24 62 88
|
jca32 |
⊢ ( 𝜑 → ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
90 |
5
|
fourierdlem2 |
⊢ ( 𝑀 ∈ ℕ → ( 𝑄 ∈ ( 𝑂 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
91 |
6 90
|
syl |
⊢ ( 𝜑 → ( 𝑄 ∈ ( 𝑂 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
92 |
89 91
|
mpbird |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝑂 ‘ 𝑀 ) ) |