Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem25.m |
|- ( ph -> M e. NN ) |
2 |
|
fourierdlem25.qf |
|- ( ph -> Q : ( 0 ... M ) --> RR ) |
3 |
|
fourierdlem25.cel |
|- ( ph -> C e. ( ( Q ` 0 ) [,] ( Q ` M ) ) ) |
4 |
|
fourierdlem25.cnel |
|- ( ph -> -. C e. ran Q ) |
5 |
|
fourierdlem25.i |
|- I = sup ( { k e. ( 0 ..^ M ) | ( Q ` k ) < C } , RR , < ) |
6 |
|
ssrab2 |
|- { k e. ( 0 ..^ M ) | ( Q ` k ) < C } C_ ( 0 ..^ M ) |
7 |
|
ltso |
|- < Or RR |
8 |
7
|
a1i |
|- ( ph -> < Or RR ) |
9 |
|
fzofi |
|- ( 0 ..^ M ) e. Fin |
10 |
|
ssfi |
|- ( ( ( 0 ..^ M ) e. Fin /\ { k e. ( 0 ..^ M ) | ( Q ` k ) < C } C_ ( 0 ..^ M ) ) -> { k e. ( 0 ..^ M ) | ( Q ` k ) < C } e. Fin ) |
11 |
9 6 10
|
mp2an |
|- { k e. ( 0 ..^ M ) | ( Q ` k ) < C } e. Fin |
12 |
11
|
a1i |
|- ( ph -> { k e. ( 0 ..^ M ) | ( Q ` k ) < C } e. Fin ) |
13 |
|
0zd |
|- ( ph -> 0 e. ZZ ) |
14 |
1
|
nnzd |
|- ( ph -> M e. ZZ ) |
15 |
1
|
nngt0d |
|- ( ph -> 0 < M ) |
16 |
|
fzolb |
|- ( 0 e. ( 0 ..^ M ) <-> ( 0 e. ZZ /\ M e. ZZ /\ 0 < M ) ) |
17 |
13 14 15 16
|
syl3anbrc |
|- ( ph -> 0 e. ( 0 ..^ M ) ) |
18 |
|
elfzofz |
|- ( 0 e. ( 0 ..^ M ) -> 0 e. ( 0 ... M ) ) |
19 |
17 18
|
syl |
|- ( ph -> 0 e. ( 0 ... M ) ) |
20 |
2 19
|
ffvelrnd |
|- ( ph -> ( Q ` 0 ) e. RR ) |
21 |
1
|
nnnn0d |
|- ( ph -> M e. NN0 ) |
22 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
23 |
21 22
|
eleqtrdi |
|- ( ph -> M e. ( ZZ>= ` 0 ) ) |
24 |
|
eluzfz2 |
|- ( M e. ( ZZ>= ` 0 ) -> M e. ( 0 ... M ) ) |
25 |
23 24
|
syl |
|- ( ph -> M e. ( 0 ... M ) ) |
26 |
2 25
|
ffvelrnd |
|- ( ph -> ( Q ` M ) e. RR ) |
27 |
20 26
|
iccssred |
|- ( ph -> ( ( Q ` 0 ) [,] ( Q ` M ) ) C_ RR ) |
28 |
27 3
|
sseldd |
|- ( ph -> C e. RR ) |
29 |
20
|
rexrd |
|- ( ph -> ( Q ` 0 ) e. RR* ) |
30 |
26
|
rexrd |
|- ( ph -> ( Q ` M ) e. RR* ) |
31 |
|
iccgelb |
|- ( ( ( Q ` 0 ) e. RR* /\ ( Q ` M ) e. RR* /\ C e. ( ( Q ` 0 ) [,] ( Q ` M ) ) ) -> ( Q ` 0 ) <_ C ) |
32 |
29 30 3 31
|
syl3anc |
|- ( ph -> ( Q ` 0 ) <_ C ) |
33 |
|
simpr |
|- ( ( ph /\ C = ( Q ` 0 ) ) -> C = ( Q ` 0 ) ) |
34 |
2
|
ffnd |
|- ( ph -> Q Fn ( 0 ... M ) ) |
35 |
34
|
adantr |
|- ( ( ph /\ C = ( Q ` 0 ) ) -> Q Fn ( 0 ... M ) ) |
36 |
19
|
adantr |
|- ( ( ph /\ C = ( Q ` 0 ) ) -> 0 e. ( 0 ... M ) ) |
37 |
|
fnfvelrn |
|- ( ( Q Fn ( 0 ... M ) /\ 0 e. ( 0 ... M ) ) -> ( Q ` 0 ) e. ran Q ) |
38 |
35 36 37
|
syl2anc |
|- ( ( ph /\ C = ( Q ` 0 ) ) -> ( Q ` 0 ) e. ran Q ) |
39 |
33 38
|
eqeltrd |
|- ( ( ph /\ C = ( Q ` 0 ) ) -> C e. ran Q ) |
40 |
4 39
|
mtand |
|- ( ph -> -. C = ( Q ` 0 ) ) |
41 |
40
|
neqned |
|- ( ph -> C =/= ( Q ` 0 ) ) |
42 |
20 28 32 41
|
leneltd |
|- ( ph -> ( Q ` 0 ) < C ) |
43 |
|
fveq2 |
|- ( k = 0 -> ( Q ` k ) = ( Q ` 0 ) ) |
44 |
43
|
breq1d |
|- ( k = 0 -> ( ( Q ` k ) < C <-> ( Q ` 0 ) < C ) ) |
45 |
44
|
elrab |
|- ( 0 e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } <-> ( 0 e. ( 0 ..^ M ) /\ ( Q ` 0 ) < C ) ) |
46 |
17 42 45
|
sylanbrc |
|- ( ph -> 0 e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } ) |
47 |
46
|
ne0d |
|- ( ph -> { k e. ( 0 ..^ M ) | ( Q ` k ) < C } =/= (/) ) |
48 |
|
fzossfz |
|- ( 0 ..^ M ) C_ ( 0 ... M ) |
49 |
|
fzssz |
|- ( 0 ... M ) C_ ZZ |
50 |
|
zssre |
|- ZZ C_ RR |
51 |
49 50
|
sstri |
|- ( 0 ... M ) C_ RR |
52 |
48 51
|
sstri |
|- ( 0 ..^ M ) C_ RR |
53 |
6 52
|
sstri |
|- { k e. ( 0 ..^ M ) | ( Q ` k ) < C } C_ RR |
54 |
53
|
a1i |
|- ( ph -> { k e. ( 0 ..^ M ) | ( Q ` k ) < C } C_ RR ) |
55 |
|
fisupcl |
|- ( ( < Or RR /\ ( { k e. ( 0 ..^ M ) | ( Q ` k ) < C } e. Fin /\ { k e. ( 0 ..^ M ) | ( Q ` k ) < C } =/= (/) /\ { k e. ( 0 ..^ M ) | ( Q ` k ) < C } C_ RR ) ) -> sup ( { k e. ( 0 ..^ M ) | ( Q ` k ) < C } , RR , < ) e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } ) |
56 |
8 12 47 54 55
|
syl13anc |
|- ( ph -> sup ( { k e. ( 0 ..^ M ) | ( Q ` k ) < C } , RR , < ) e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } ) |
57 |
6 56
|
sselid |
|- ( ph -> sup ( { k e. ( 0 ..^ M ) | ( Q ` k ) < C } , RR , < ) e. ( 0 ..^ M ) ) |
58 |
5 57
|
eqeltrid |
|- ( ph -> I e. ( 0 ..^ M ) ) |
59 |
48 58
|
sselid |
|- ( ph -> I e. ( 0 ... M ) ) |
60 |
2 59
|
ffvelrnd |
|- ( ph -> ( Q ` I ) e. RR ) |
61 |
60
|
rexrd |
|- ( ph -> ( Q ` I ) e. RR* ) |
62 |
|
fzofzp1 |
|- ( I e. ( 0 ..^ M ) -> ( I + 1 ) e. ( 0 ... M ) ) |
63 |
58 62
|
syl |
|- ( ph -> ( I + 1 ) e. ( 0 ... M ) ) |
64 |
2 63
|
ffvelrnd |
|- ( ph -> ( Q ` ( I + 1 ) ) e. RR ) |
65 |
64
|
rexrd |
|- ( ph -> ( Q ` ( I + 1 ) ) e. RR* ) |
66 |
5 56
|
eqeltrid |
|- ( ph -> I e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } ) |
67 |
|
fveq2 |
|- ( k = I -> ( Q ` k ) = ( Q ` I ) ) |
68 |
67
|
breq1d |
|- ( k = I -> ( ( Q ` k ) < C <-> ( Q ` I ) < C ) ) |
69 |
68
|
elrab |
|- ( I e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } <-> ( I e. ( 0 ..^ M ) /\ ( Q ` I ) < C ) ) |
70 |
66 69
|
sylib |
|- ( ph -> ( I e. ( 0 ..^ M ) /\ ( Q ` I ) < C ) ) |
71 |
70
|
simprd |
|- ( ph -> ( Q ` I ) < C ) |
72 |
52 58
|
sselid |
|- ( ph -> I e. RR ) |
73 |
|
ltp1 |
|- ( I e. RR -> I < ( I + 1 ) ) |
74 |
|
id |
|- ( I e. RR -> I e. RR ) |
75 |
|
peano2re |
|- ( I e. RR -> ( I + 1 ) e. RR ) |
76 |
74 75
|
ltnled |
|- ( I e. RR -> ( I < ( I + 1 ) <-> -. ( I + 1 ) <_ I ) ) |
77 |
73 76
|
mpbid |
|- ( I e. RR -> -. ( I + 1 ) <_ I ) |
78 |
72 77
|
syl |
|- ( ph -> -. ( I + 1 ) <_ I ) |
79 |
48 49
|
sstri |
|- ( 0 ..^ M ) C_ ZZ |
80 |
6 79
|
sstri |
|- { k e. ( 0 ..^ M ) | ( Q ` k ) < C } C_ ZZ |
81 |
80
|
a1i |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> { k e. ( 0 ..^ M ) | ( Q ` k ) < C } C_ ZZ ) |
82 |
|
elrabi |
|- ( h e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } -> h e. ( 0 ..^ M ) ) |
83 |
|
elfzo0le |
|- ( h e. ( 0 ..^ M ) -> h <_ M ) |
84 |
82 83
|
syl |
|- ( h e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } -> h <_ M ) |
85 |
84
|
adantl |
|- ( ( ph /\ h e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } ) -> h <_ M ) |
86 |
85
|
ralrimiva |
|- ( ph -> A. h e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } h <_ M ) |
87 |
|
breq2 |
|- ( m = M -> ( h <_ m <-> h <_ M ) ) |
88 |
87
|
ralbidv |
|- ( m = M -> ( A. h e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } h <_ m <-> A. h e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } h <_ M ) ) |
89 |
88
|
rspcev |
|- ( ( M e. ZZ /\ A. h e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } h <_ M ) -> E. m e. ZZ A. h e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } h <_ m ) |
90 |
14 86 89
|
syl2anc |
|- ( ph -> E. m e. ZZ A. h e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } h <_ m ) |
91 |
90
|
adantr |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> E. m e. ZZ A. h e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } h <_ m ) |
92 |
|
elfzuz |
|- ( ( I + 1 ) e. ( 0 ... M ) -> ( I + 1 ) e. ( ZZ>= ` 0 ) ) |
93 |
63 92
|
syl |
|- ( ph -> ( I + 1 ) e. ( ZZ>= ` 0 ) ) |
94 |
93
|
adantr |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> ( I + 1 ) e. ( ZZ>= ` 0 ) ) |
95 |
14
|
adantr |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> M e. ZZ ) |
96 |
51 63
|
sselid |
|- ( ph -> ( I + 1 ) e. RR ) |
97 |
96
|
adantr |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> ( I + 1 ) e. RR ) |
98 |
95
|
zred |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> M e. RR ) |
99 |
|
elfzle2 |
|- ( ( I + 1 ) e. ( 0 ... M ) -> ( I + 1 ) <_ M ) |
100 |
63 99
|
syl |
|- ( ph -> ( I + 1 ) <_ M ) |
101 |
100
|
adantr |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> ( I + 1 ) <_ M ) |
102 |
|
simpr |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> ( Q ` ( I + 1 ) ) < C ) |
103 |
64
|
adantr |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> ( Q ` ( I + 1 ) ) e. RR ) |
104 |
28
|
adantr |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> C e. RR ) |
105 |
103 104
|
ltnled |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> ( ( Q ` ( I + 1 ) ) < C <-> -. C <_ ( Q ` ( I + 1 ) ) ) ) |
106 |
102 105
|
mpbid |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> -. C <_ ( Q ` ( I + 1 ) ) ) |
107 |
|
iccleub |
|- ( ( ( Q ` 0 ) e. RR* /\ ( Q ` M ) e. RR* /\ C e. ( ( Q ` 0 ) [,] ( Q ` M ) ) ) -> C <_ ( Q ` M ) ) |
108 |
29 30 3 107
|
syl3anc |
|- ( ph -> C <_ ( Q ` M ) ) |
109 |
108
|
adantr |
|- ( ( ph /\ M = ( I + 1 ) ) -> C <_ ( Q ` M ) ) |
110 |
|
fveq2 |
|- ( M = ( I + 1 ) -> ( Q ` M ) = ( Q ` ( I + 1 ) ) ) |
111 |
110
|
adantl |
|- ( ( ph /\ M = ( I + 1 ) ) -> ( Q ` M ) = ( Q ` ( I + 1 ) ) ) |
112 |
109 111
|
breqtrd |
|- ( ( ph /\ M = ( I + 1 ) ) -> C <_ ( Q ` ( I + 1 ) ) ) |
113 |
112
|
adantlr |
|- ( ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) /\ M = ( I + 1 ) ) -> C <_ ( Q ` ( I + 1 ) ) ) |
114 |
106 113
|
mtand |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> -. M = ( I + 1 ) ) |
115 |
114
|
neqned |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> M =/= ( I + 1 ) ) |
116 |
97 98 101 115
|
leneltd |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> ( I + 1 ) < M ) |
117 |
|
elfzo2 |
|- ( ( I + 1 ) e. ( 0 ..^ M ) <-> ( ( I + 1 ) e. ( ZZ>= ` 0 ) /\ M e. ZZ /\ ( I + 1 ) < M ) ) |
118 |
94 95 116 117
|
syl3anbrc |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> ( I + 1 ) e. ( 0 ..^ M ) ) |
119 |
|
fveq2 |
|- ( k = ( I + 1 ) -> ( Q ` k ) = ( Q ` ( I + 1 ) ) ) |
120 |
119
|
breq1d |
|- ( k = ( I + 1 ) -> ( ( Q ` k ) < C <-> ( Q ` ( I + 1 ) ) < C ) ) |
121 |
120
|
elrab |
|- ( ( I + 1 ) e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } <-> ( ( I + 1 ) e. ( 0 ..^ M ) /\ ( Q ` ( I + 1 ) ) < C ) ) |
122 |
118 102 121
|
sylanbrc |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> ( I + 1 ) e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } ) |
123 |
|
suprzub |
|- ( ( { k e. ( 0 ..^ M ) | ( Q ` k ) < C } C_ ZZ /\ E. m e. ZZ A. h e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } h <_ m /\ ( I + 1 ) e. { k e. ( 0 ..^ M ) | ( Q ` k ) < C } ) -> ( I + 1 ) <_ sup ( { k e. ( 0 ..^ M ) | ( Q ` k ) < C } , RR , < ) ) |
124 |
81 91 122 123
|
syl3anc |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> ( I + 1 ) <_ sup ( { k e. ( 0 ..^ M ) | ( Q ` k ) < C } , RR , < ) ) |
125 |
124 5
|
breqtrrdi |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) < C ) -> ( I + 1 ) <_ I ) |
126 |
78 125
|
mtand |
|- ( ph -> -. ( Q ` ( I + 1 ) ) < C ) |
127 |
|
eqcom |
|- ( ( Q ` ( I + 1 ) ) = C <-> C = ( Q ` ( I + 1 ) ) ) |
128 |
127
|
biimpi |
|- ( ( Q ` ( I + 1 ) ) = C -> C = ( Q ` ( I + 1 ) ) ) |
129 |
128
|
adantl |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) = C ) -> C = ( Q ` ( I + 1 ) ) ) |
130 |
34
|
adantr |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) = C ) -> Q Fn ( 0 ... M ) ) |
131 |
63
|
adantr |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) = C ) -> ( I + 1 ) e. ( 0 ... M ) ) |
132 |
|
fnfvelrn |
|- ( ( Q Fn ( 0 ... M ) /\ ( I + 1 ) e. ( 0 ... M ) ) -> ( Q ` ( I + 1 ) ) e. ran Q ) |
133 |
130 131 132
|
syl2anc |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) = C ) -> ( Q ` ( I + 1 ) ) e. ran Q ) |
134 |
129 133
|
eqeltrd |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) = C ) -> C e. ran Q ) |
135 |
4 134
|
mtand |
|- ( ph -> -. ( Q ` ( I + 1 ) ) = C ) |
136 |
126 135
|
jca |
|- ( ph -> ( -. ( Q ` ( I + 1 ) ) < C /\ -. ( Q ` ( I + 1 ) ) = C ) ) |
137 |
|
pm4.56 |
|- ( ( -. ( Q ` ( I + 1 ) ) < C /\ -. ( Q ` ( I + 1 ) ) = C ) <-> -. ( ( Q ` ( I + 1 ) ) < C \/ ( Q ` ( I + 1 ) ) = C ) ) |
138 |
136 137
|
sylib |
|- ( ph -> -. ( ( Q ` ( I + 1 ) ) < C \/ ( Q ` ( I + 1 ) ) = C ) ) |
139 |
64 28
|
leloed |
|- ( ph -> ( ( Q ` ( I + 1 ) ) <_ C <-> ( ( Q ` ( I + 1 ) ) < C \/ ( Q ` ( I + 1 ) ) = C ) ) ) |
140 |
138 139
|
mtbird |
|- ( ph -> -. ( Q ` ( I + 1 ) ) <_ C ) |
141 |
28 64
|
ltnled |
|- ( ph -> ( C < ( Q ` ( I + 1 ) ) <-> -. ( Q ` ( I + 1 ) ) <_ C ) ) |
142 |
140 141
|
mpbird |
|- ( ph -> C < ( Q ` ( I + 1 ) ) ) |
143 |
61 65 28 71 142
|
eliood |
|- ( ph -> C e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) |
144 |
|
fveq2 |
|- ( j = I -> ( Q ` j ) = ( Q ` I ) ) |
145 |
|
oveq1 |
|- ( j = I -> ( j + 1 ) = ( I + 1 ) ) |
146 |
145
|
fveq2d |
|- ( j = I -> ( Q ` ( j + 1 ) ) = ( Q ` ( I + 1 ) ) ) |
147 |
144 146
|
oveq12d |
|- ( j = I -> ( ( Q ` j ) (,) ( Q ` ( j + 1 ) ) ) = ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) |
148 |
147
|
eleq2d |
|- ( j = I -> ( C e. ( ( Q ` j ) (,) ( Q ` ( j + 1 ) ) ) <-> C e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) ) |
149 |
148
|
rspcev |
|- ( ( I e. ( 0 ..^ M ) /\ C e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> E. j e. ( 0 ..^ M ) C e. ( ( Q ` j ) (,) ( Q ` ( j + 1 ) ) ) ) |
150 |
58 143 149
|
syl2anc |
|- ( ph -> E. j e. ( 0 ..^ M ) C e. ( ( Q ` j ) (,) ( Q ` ( j + 1 ) ) ) ) |