Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem11.p |
|- P = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = A /\ ( p ` m ) = B ) /\ A. i e. ( 0 ..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } ) |
2 |
|
fourierdlem11.m |
|- ( ph -> M e. NN ) |
3 |
|
fourierdlem11.q |
|- ( ph -> Q e. ( P ` M ) ) |
4 |
1
|
fourierdlem2 |
|- ( M e. NN -> ( Q e. ( P ` M ) <-> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) ) |
5 |
2 4
|
syl |
|- ( ph -> ( Q e. ( P ` M ) <-> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) ) |
6 |
3 5
|
mpbid |
|- ( ph -> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) |
7 |
6
|
simprd |
|- ( ph -> ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) |
8 |
7
|
simpld |
|- ( ph -> ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) ) |
9 |
8
|
simpld |
|- ( ph -> ( Q ` 0 ) = A ) |
10 |
6
|
simpld |
|- ( ph -> Q e. ( RR ^m ( 0 ... M ) ) ) |
11 |
|
elmapi |
|- ( Q e. ( RR ^m ( 0 ... M ) ) -> Q : ( 0 ... M ) --> RR ) |
12 |
10 11
|
syl |
|- ( ph -> Q : ( 0 ... M ) --> RR ) |
13 |
|
0red |
|- ( ph -> 0 e. RR ) |
14 |
13
|
leidd |
|- ( ph -> 0 <_ 0 ) |
15 |
2
|
nnred |
|- ( ph -> M e. RR ) |
16 |
2
|
nngt0d |
|- ( ph -> 0 < M ) |
17 |
13 15 16
|
ltled |
|- ( ph -> 0 <_ M ) |
18 |
|
0zd |
|- ( ph -> 0 e. ZZ ) |
19 |
2
|
nnzd |
|- ( ph -> M e. ZZ ) |
20 |
|
elfz |
|- ( ( 0 e. ZZ /\ 0 e. ZZ /\ M e. ZZ ) -> ( 0 e. ( 0 ... M ) <-> ( 0 <_ 0 /\ 0 <_ M ) ) ) |
21 |
18 18 19 20
|
syl3anc |
|- ( ph -> ( 0 e. ( 0 ... M ) <-> ( 0 <_ 0 /\ 0 <_ M ) ) ) |
22 |
14 17 21
|
mpbir2and |
|- ( ph -> 0 e. ( 0 ... M ) ) |
23 |
12 22
|
ffvelrnd |
|- ( ph -> ( Q ` 0 ) e. RR ) |
24 |
9 23
|
eqeltrrd |
|- ( ph -> A e. RR ) |
25 |
8
|
simprd |
|- ( ph -> ( Q ` M ) = B ) |
26 |
15
|
leidd |
|- ( ph -> M <_ M ) |
27 |
|
elfz |
|- ( ( M e. ZZ /\ 0 e. ZZ /\ M e. ZZ ) -> ( M e. ( 0 ... M ) <-> ( 0 <_ M /\ M <_ M ) ) ) |
28 |
19 18 19 27
|
syl3anc |
|- ( ph -> ( M e. ( 0 ... M ) <-> ( 0 <_ M /\ M <_ M ) ) ) |
29 |
17 26 28
|
mpbir2and |
|- ( ph -> M e. ( 0 ... M ) ) |
30 |
12 29
|
ffvelrnd |
|- ( ph -> ( Q ` M ) e. RR ) |
31 |
25 30
|
eqeltrrd |
|- ( ph -> B e. RR ) |
32 |
|
0le1 |
|- 0 <_ 1 |
33 |
32
|
a1i |
|- ( ph -> 0 <_ 1 ) |
34 |
2
|
nnge1d |
|- ( ph -> 1 <_ M ) |
35 |
|
1zzd |
|- ( ph -> 1 e. ZZ ) |
36 |
|
elfz |
|- ( ( 1 e. ZZ /\ 0 e. ZZ /\ M e. ZZ ) -> ( 1 e. ( 0 ... M ) <-> ( 0 <_ 1 /\ 1 <_ M ) ) ) |
37 |
35 18 19 36
|
syl3anc |
|- ( ph -> ( 1 e. ( 0 ... M ) <-> ( 0 <_ 1 /\ 1 <_ M ) ) ) |
38 |
33 34 37
|
mpbir2and |
|- ( ph -> 1 e. ( 0 ... M ) ) |
39 |
12 38
|
ffvelrnd |
|- ( ph -> ( Q ` 1 ) e. RR ) |
40 |
|
elfzo |
|- ( ( 0 e. ZZ /\ 0 e. ZZ /\ M e. ZZ ) -> ( 0 e. ( 0 ..^ M ) <-> ( 0 <_ 0 /\ 0 < M ) ) ) |
41 |
18 18 19 40
|
syl3anc |
|- ( ph -> ( 0 e. ( 0 ..^ M ) <-> ( 0 <_ 0 /\ 0 < M ) ) ) |
42 |
14 16 41
|
mpbir2and |
|- ( ph -> 0 e. ( 0 ..^ M ) ) |
43 |
|
0re |
|- 0 e. RR |
44 |
|
eleq1 |
|- ( i = 0 -> ( i e. ( 0 ..^ M ) <-> 0 e. ( 0 ..^ M ) ) ) |
45 |
44
|
anbi2d |
|- ( i = 0 -> ( ( ph /\ i e. ( 0 ..^ M ) ) <-> ( ph /\ 0 e. ( 0 ..^ M ) ) ) ) |
46 |
|
fveq2 |
|- ( i = 0 -> ( Q ` i ) = ( Q ` 0 ) ) |
47 |
|
oveq1 |
|- ( i = 0 -> ( i + 1 ) = ( 0 + 1 ) ) |
48 |
47
|
fveq2d |
|- ( i = 0 -> ( Q ` ( i + 1 ) ) = ( Q ` ( 0 + 1 ) ) ) |
49 |
46 48
|
breq12d |
|- ( i = 0 -> ( ( Q ` i ) < ( Q ` ( i + 1 ) ) <-> ( Q ` 0 ) < ( Q ` ( 0 + 1 ) ) ) ) |
50 |
45 49
|
imbi12d |
|- ( i = 0 -> ( ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` i ) < ( Q ` ( i + 1 ) ) ) <-> ( ( ph /\ 0 e. ( 0 ..^ M ) ) -> ( Q ` 0 ) < ( Q ` ( 0 + 1 ) ) ) ) ) |
51 |
7
|
simprd |
|- ( ph -> A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) |
52 |
51
|
r19.21bi |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` i ) < ( Q ` ( i + 1 ) ) ) |
53 |
50 52
|
vtoclg |
|- ( 0 e. RR -> ( ( ph /\ 0 e. ( 0 ..^ M ) ) -> ( Q ` 0 ) < ( Q ` ( 0 + 1 ) ) ) ) |
54 |
43 53
|
ax-mp |
|- ( ( ph /\ 0 e. ( 0 ..^ M ) ) -> ( Q ` 0 ) < ( Q ` ( 0 + 1 ) ) ) |
55 |
42 54
|
mpdan |
|- ( ph -> ( Q ` 0 ) < ( Q ` ( 0 + 1 ) ) ) |
56 |
|
0p1e1 |
|- ( 0 + 1 ) = 1 |
57 |
56
|
a1i |
|- ( ph -> ( 0 + 1 ) = 1 ) |
58 |
57
|
fveq2d |
|- ( ph -> ( Q ` ( 0 + 1 ) ) = ( Q ` 1 ) ) |
59 |
55 9 58
|
3brtr3d |
|- ( ph -> A < ( Q ` 1 ) ) |
60 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
61 |
2 60
|
eleqtrdi |
|- ( ph -> M e. ( ZZ>= ` 1 ) ) |
62 |
12
|
adantr |
|- ( ( ph /\ i e. ( 1 ... M ) ) -> Q : ( 0 ... M ) --> RR ) |
63 |
|
0red |
|- ( i e. ( 1 ... M ) -> 0 e. RR ) |
64 |
|
elfzelz |
|- ( i e. ( 1 ... M ) -> i e. ZZ ) |
65 |
64
|
zred |
|- ( i e. ( 1 ... M ) -> i e. RR ) |
66 |
|
1red |
|- ( i e. ( 1 ... M ) -> 1 e. RR ) |
67 |
|
0lt1 |
|- 0 < 1 |
68 |
67
|
a1i |
|- ( i e. ( 1 ... M ) -> 0 < 1 ) |
69 |
|
elfzle1 |
|- ( i e. ( 1 ... M ) -> 1 <_ i ) |
70 |
63 66 65 68 69
|
ltletrd |
|- ( i e. ( 1 ... M ) -> 0 < i ) |
71 |
63 65 70
|
ltled |
|- ( i e. ( 1 ... M ) -> 0 <_ i ) |
72 |
|
elfzle2 |
|- ( i e. ( 1 ... M ) -> i <_ M ) |
73 |
|
0zd |
|- ( i e. ( 1 ... M ) -> 0 e. ZZ ) |
74 |
|
elfzel2 |
|- ( i e. ( 1 ... M ) -> M e. ZZ ) |
75 |
|
elfz |
|- ( ( i e. ZZ /\ 0 e. ZZ /\ M e. ZZ ) -> ( i e. ( 0 ... M ) <-> ( 0 <_ i /\ i <_ M ) ) ) |
76 |
64 73 74 75
|
syl3anc |
|- ( i e. ( 1 ... M ) -> ( i e. ( 0 ... M ) <-> ( 0 <_ i /\ i <_ M ) ) ) |
77 |
71 72 76
|
mpbir2and |
|- ( i e. ( 1 ... M ) -> i e. ( 0 ... M ) ) |
78 |
77
|
adantl |
|- ( ( ph /\ i e. ( 1 ... M ) ) -> i e. ( 0 ... M ) ) |
79 |
62 78
|
ffvelrnd |
|- ( ( ph /\ i e. ( 1 ... M ) ) -> ( Q ` i ) e. RR ) |
80 |
12
|
adantr |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> Q : ( 0 ... M ) --> RR ) |
81 |
|
0red |
|- ( i e. ( 1 ... ( M - 1 ) ) -> 0 e. RR ) |
82 |
|
elfzelz |
|- ( i e. ( 1 ... ( M - 1 ) ) -> i e. ZZ ) |
83 |
82
|
zred |
|- ( i e. ( 1 ... ( M - 1 ) ) -> i e. RR ) |
84 |
|
1red |
|- ( i e. ( 1 ... ( M - 1 ) ) -> 1 e. RR ) |
85 |
67
|
a1i |
|- ( i e. ( 1 ... ( M - 1 ) ) -> 0 < 1 ) |
86 |
|
elfzle1 |
|- ( i e. ( 1 ... ( M - 1 ) ) -> 1 <_ i ) |
87 |
81 84 83 85 86
|
ltletrd |
|- ( i e. ( 1 ... ( M - 1 ) ) -> 0 < i ) |
88 |
81 83 87
|
ltled |
|- ( i e. ( 1 ... ( M - 1 ) ) -> 0 <_ i ) |
89 |
88
|
adantl |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> 0 <_ i ) |
90 |
83
|
adantl |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> i e. RR ) |
91 |
15
|
adantr |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> M e. RR ) |
92 |
|
peano2rem |
|- ( M e. RR -> ( M - 1 ) e. RR ) |
93 |
91 92
|
syl |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> ( M - 1 ) e. RR ) |
94 |
|
elfzle2 |
|- ( i e. ( 1 ... ( M - 1 ) ) -> i <_ ( M - 1 ) ) |
95 |
94
|
adantl |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> i <_ ( M - 1 ) ) |
96 |
91
|
ltm1d |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> ( M - 1 ) < M ) |
97 |
90 93 91 95 96
|
lelttrd |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> i < M ) |
98 |
90 91 97
|
ltled |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> i <_ M ) |
99 |
82
|
adantl |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> i e. ZZ ) |
100 |
|
0zd |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> 0 e. ZZ ) |
101 |
19
|
adantr |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> M e. ZZ ) |
102 |
99 100 101 75
|
syl3anc |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> ( i e. ( 0 ... M ) <-> ( 0 <_ i /\ i <_ M ) ) ) |
103 |
89 98 102
|
mpbir2and |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> i e. ( 0 ... M ) ) |
104 |
80 103
|
ffvelrnd |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> ( Q ` i ) e. RR ) |
105 |
|
0red |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> 0 e. RR ) |
106 |
|
peano2re |
|- ( i e. RR -> ( i + 1 ) e. RR ) |
107 |
90 106
|
syl |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> ( i + 1 ) e. RR ) |
108 |
|
1red |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> 1 e. RR ) |
109 |
67
|
a1i |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> 0 < 1 ) |
110 |
83 106
|
syl |
|- ( i e. ( 1 ... ( M - 1 ) ) -> ( i + 1 ) e. RR ) |
111 |
83
|
ltp1d |
|- ( i e. ( 1 ... ( M - 1 ) ) -> i < ( i + 1 ) ) |
112 |
84 83 110 86 111
|
lelttrd |
|- ( i e. ( 1 ... ( M - 1 ) ) -> 1 < ( i + 1 ) ) |
113 |
112
|
adantl |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> 1 < ( i + 1 ) ) |
114 |
105 108 107 109 113
|
lttrd |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> 0 < ( i + 1 ) ) |
115 |
105 107 114
|
ltled |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> 0 <_ ( i + 1 ) ) |
116 |
90 93 108 95
|
leadd1dd |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> ( i + 1 ) <_ ( ( M - 1 ) + 1 ) ) |
117 |
2
|
nncnd |
|- ( ph -> M e. CC ) |
118 |
|
1cnd |
|- ( ph -> 1 e. CC ) |
119 |
117 118
|
npcand |
|- ( ph -> ( ( M - 1 ) + 1 ) = M ) |
120 |
119
|
adantr |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> ( ( M - 1 ) + 1 ) = M ) |
121 |
116 120
|
breqtrd |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> ( i + 1 ) <_ M ) |
122 |
99
|
peano2zd |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> ( i + 1 ) e. ZZ ) |
123 |
|
elfz |
|- ( ( ( i + 1 ) e. ZZ /\ 0 e. ZZ /\ M e. ZZ ) -> ( ( i + 1 ) e. ( 0 ... M ) <-> ( 0 <_ ( i + 1 ) /\ ( i + 1 ) <_ M ) ) ) |
124 |
122 100 101 123
|
syl3anc |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> ( ( i + 1 ) e. ( 0 ... M ) <-> ( 0 <_ ( i + 1 ) /\ ( i + 1 ) <_ M ) ) ) |
125 |
115 121 124
|
mpbir2and |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> ( i + 1 ) e. ( 0 ... M ) ) |
126 |
80 125
|
ffvelrnd |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> ( Q ` ( i + 1 ) ) e. RR ) |
127 |
|
elfzo |
|- ( ( i e. ZZ /\ 0 e. ZZ /\ M e. ZZ ) -> ( i e. ( 0 ..^ M ) <-> ( 0 <_ i /\ i < M ) ) ) |
128 |
99 100 101 127
|
syl3anc |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> ( i e. ( 0 ..^ M ) <-> ( 0 <_ i /\ i < M ) ) ) |
129 |
89 97 128
|
mpbir2and |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> i e. ( 0 ..^ M ) ) |
130 |
129 52
|
syldan |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> ( Q ` i ) < ( Q ` ( i + 1 ) ) ) |
131 |
104 126 130
|
ltled |
|- ( ( ph /\ i e. ( 1 ... ( M - 1 ) ) ) -> ( Q ` i ) <_ ( Q ` ( i + 1 ) ) ) |
132 |
61 79 131
|
monoord |
|- ( ph -> ( Q ` 1 ) <_ ( Q ` M ) ) |
133 |
132 25
|
breqtrd |
|- ( ph -> ( Q ` 1 ) <_ B ) |
134 |
24 39 31 59 133
|
ltletrd |
|- ( ph -> A < B ) |
135 |
24 31 134
|
3jca |
|- ( ph -> ( A e. RR /\ B e. RR /\ A < B ) ) |