Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem59.f |
|- ( ph -> F : RR --> RR ) |
2 |
|
fourierdlem59.x |
|- ( ph -> X e. RR ) |
3 |
|
fourierdlem59.a |
|- ( ph -> A e. RR ) |
4 |
|
fourierdlem59.b |
|- ( ph -> B e. RR ) |
5 |
|
fourierdlem59.n0 |
|- ( ph -> -. 0 e. ( A (,) B ) ) |
6 |
|
fourierdlem59.fdv |
|- ( ph -> ( RR _D ( F |` ( ( X + A ) (,) ( X + B ) ) ) ) e. ( ( ( X + A ) (,) ( X + B ) ) -cn-> RR ) ) |
7 |
|
fourierdlem59.c |
|- ( ph -> C e. RR ) |
8 |
|
fourierdlem59.h |
|- H = ( s e. ( A (,) B ) |-> ( ( ( F ` ( X + s ) ) - C ) / s ) ) |
9 |
1
|
adantr |
|- ( ( ph /\ s e. ( A (,) B ) ) -> F : RR --> RR ) |
10 |
2
|
adantr |
|- ( ( ph /\ s e. ( A (,) B ) ) -> X e. RR ) |
11 |
|
elioore |
|- ( s e. ( A (,) B ) -> s e. RR ) |
12 |
11
|
adantl |
|- ( ( ph /\ s e. ( A (,) B ) ) -> s e. RR ) |
13 |
10 12
|
readdcld |
|- ( ( ph /\ s e. ( A (,) B ) ) -> ( X + s ) e. RR ) |
14 |
9 13
|
ffvelrnd |
|- ( ( ph /\ s e. ( A (,) B ) ) -> ( F ` ( X + s ) ) e. RR ) |
15 |
7
|
adantr |
|- ( ( ph /\ s e. ( A (,) B ) ) -> C e. RR ) |
16 |
14 15
|
resubcld |
|- ( ( ph /\ s e. ( A (,) B ) ) -> ( ( F ` ( X + s ) ) - C ) e. RR ) |
17 |
|
eqcom |
|- ( s = 0 <-> 0 = s ) |
18 |
17
|
biimpi |
|- ( s = 0 -> 0 = s ) |
19 |
18
|
adantl |
|- ( ( s e. ( A (,) B ) /\ s = 0 ) -> 0 = s ) |
20 |
|
simpl |
|- ( ( s e. ( A (,) B ) /\ s = 0 ) -> s e. ( A (,) B ) ) |
21 |
19 20
|
eqeltrd |
|- ( ( s e. ( A (,) B ) /\ s = 0 ) -> 0 e. ( A (,) B ) ) |
22 |
21
|
adantll |
|- ( ( ( ph /\ s e. ( A (,) B ) ) /\ s = 0 ) -> 0 e. ( A (,) B ) ) |
23 |
5
|
ad2antrr |
|- ( ( ( ph /\ s e. ( A (,) B ) ) /\ s = 0 ) -> -. 0 e. ( A (,) B ) ) |
24 |
22 23
|
pm2.65da |
|- ( ( ph /\ s e. ( A (,) B ) ) -> -. s = 0 ) |
25 |
24
|
neqned |
|- ( ( ph /\ s e. ( A (,) B ) ) -> s =/= 0 ) |
26 |
16 12 25
|
redivcld |
|- ( ( ph /\ s e. ( A (,) B ) ) -> ( ( ( F ` ( X + s ) ) - C ) / s ) e. RR ) |
27 |
26 8
|
fmptd |
|- ( ph -> H : ( A (,) B ) --> RR ) |
28 |
|
ioossre |
|- ( A (,) B ) C_ RR |
29 |
28
|
a1i |
|- ( ph -> ( A (,) B ) C_ RR ) |
30 |
|
dvfre |
|- ( ( H : ( A (,) B ) --> RR /\ ( A (,) B ) C_ RR ) -> ( RR _D H ) : dom ( RR _D H ) --> RR ) |
31 |
27 29 30
|
syl2anc |
|- ( ph -> ( RR _D H ) : dom ( RR _D H ) --> RR ) |
32 |
|
ovex |
|- ( A (,) B ) e. _V |
33 |
32
|
a1i |
|- ( ph -> ( A (,) B ) e. _V ) |
34 |
|
eqidd |
|- ( ph -> ( s e. ( A (,) B ) |-> ( ( F ` ( X + s ) ) - C ) ) = ( s e. ( A (,) B ) |-> ( ( F ` ( X + s ) ) - C ) ) ) |
35 |
|
eqidd |
|- ( ph -> ( s e. ( A (,) B ) |-> s ) = ( s e. ( A (,) B ) |-> s ) ) |
36 |
33 16 12 34 35
|
offval2 |
|- ( ph -> ( ( s e. ( A (,) B ) |-> ( ( F ` ( X + s ) ) - C ) ) oF / ( s e. ( A (,) B ) |-> s ) ) = ( s e. ( A (,) B ) |-> ( ( ( F ` ( X + s ) ) - C ) / s ) ) ) |
37 |
8 36
|
eqtr4id |
|- ( ph -> H = ( ( s e. ( A (,) B ) |-> ( ( F ` ( X + s ) ) - C ) ) oF / ( s e. ( A (,) B ) |-> s ) ) ) |
38 |
37
|
oveq2d |
|- ( ph -> ( RR _D H ) = ( RR _D ( ( s e. ( A (,) B ) |-> ( ( F ` ( X + s ) ) - C ) ) oF / ( s e. ( A (,) B ) |-> s ) ) ) ) |
39 |
|
reelprrecn |
|- RR e. { RR , CC } |
40 |
39
|
a1i |
|- ( ph -> RR e. { RR , CC } ) |
41 |
16
|
recnd |
|- ( ( ph /\ s e. ( A (,) B ) ) -> ( ( F ` ( X + s ) ) - C ) e. CC ) |
42 |
|
eqid |
|- ( s e. ( A (,) B ) |-> ( ( F ` ( X + s ) ) - C ) ) = ( s e. ( A (,) B ) |-> ( ( F ` ( X + s ) ) - C ) ) |
43 |
41 42
|
fmptd |
|- ( ph -> ( s e. ( A (,) B ) |-> ( ( F ` ( X + s ) ) - C ) ) : ( A (,) B ) --> CC ) |
44 |
12
|
recnd |
|- ( ( ph /\ s e. ( A (,) B ) ) -> s e. CC ) |
45 |
|
eldifsn |
|- ( s e. ( CC \ { 0 } ) <-> ( s e. CC /\ s =/= 0 ) ) |
46 |
44 25 45
|
sylanbrc |
|- ( ( ph /\ s e. ( A (,) B ) ) -> s e. ( CC \ { 0 } ) ) |
47 |
|
eqid |
|- ( s e. ( A (,) B ) |-> s ) = ( s e. ( A (,) B ) |-> s ) |
48 |
46 47
|
fmptd |
|- ( ph -> ( s e. ( A (,) B ) |-> s ) : ( A (,) B ) --> ( CC \ { 0 } ) ) |
49 |
|
eqidd |
|- ( ph -> ( s e. ( A (,) B ) |-> ( F ` ( X + s ) ) ) = ( s e. ( A (,) B ) |-> ( F ` ( X + s ) ) ) ) |
50 |
|
eqidd |
|- ( ph -> ( s e. ( A (,) B ) |-> C ) = ( s e. ( A (,) B ) |-> C ) ) |
51 |
33 14 15 49 50
|
offval2 |
|- ( ph -> ( ( s e. ( A (,) B ) |-> ( F ` ( X + s ) ) ) oF - ( s e. ( A (,) B ) |-> C ) ) = ( s e. ( A (,) B ) |-> ( ( F ` ( X + s ) ) - C ) ) ) |
52 |
51
|
eqcomd |
|- ( ph -> ( s e. ( A (,) B ) |-> ( ( F ` ( X + s ) ) - C ) ) = ( ( s e. ( A (,) B ) |-> ( F ` ( X + s ) ) ) oF - ( s e. ( A (,) B ) |-> C ) ) ) |
53 |
52
|
oveq2d |
|- ( ph -> ( RR _D ( s e. ( A (,) B ) |-> ( ( F ` ( X + s ) ) - C ) ) ) = ( RR _D ( ( s e. ( A (,) B ) |-> ( F ` ( X + s ) ) ) oF - ( s e. ( A (,) B ) |-> C ) ) ) ) |
54 |
14
|
recnd |
|- ( ( ph /\ s e. ( A (,) B ) ) -> ( F ` ( X + s ) ) e. CC ) |
55 |
|
eqid |
|- ( s e. ( A (,) B ) |-> ( F ` ( X + s ) ) ) = ( s e. ( A (,) B ) |-> ( F ` ( X + s ) ) ) |
56 |
54 55
|
fmptd |
|- ( ph -> ( s e. ( A (,) B ) |-> ( F ` ( X + s ) ) ) : ( A (,) B ) --> CC ) |
57 |
15
|
recnd |
|- ( ( ph /\ s e. ( A (,) B ) ) -> C e. CC ) |
58 |
|
eqid |
|- ( s e. ( A (,) B ) |-> C ) = ( s e. ( A (,) B ) |-> C ) |
59 |
57 58
|
fmptd |
|- ( ph -> ( s e. ( A (,) B ) |-> C ) : ( A (,) B ) --> CC ) |
60 |
|
eqid |
|- ( RR _D ( F |` ( ( X + A ) (,) ( X + B ) ) ) ) = ( RR _D ( F |` ( ( X + A ) (,) ( X + B ) ) ) ) |
61 |
|
cncff |
|- ( ( RR _D ( F |` ( ( X + A ) (,) ( X + B ) ) ) ) e. ( ( ( X + A ) (,) ( X + B ) ) -cn-> RR ) -> ( RR _D ( F |` ( ( X + A ) (,) ( X + B ) ) ) ) : ( ( X + A ) (,) ( X + B ) ) --> RR ) |
62 |
6 61
|
syl |
|- ( ph -> ( RR _D ( F |` ( ( X + A ) (,) ( X + B ) ) ) ) : ( ( X + A ) (,) ( X + B ) ) --> RR ) |
63 |
1 2 3 4 60 62
|
fourierdlem28 |
|- ( ph -> ( RR _D ( s e. ( A (,) B ) |-> ( F ` ( X + s ) ) ) ) = ( s e. ( A (,) B ) |-> ( ( RR _D ( F |` ( ( X + A ) (,) ( X + B ) ) ) ) ` ( X + s ) ) ) ) |
64 |
|
ioosscn |
|- ( ( X + A ) (,) ( X + B ) ) C_ CC |
65 |
64
|
a1i |
|- ( ph -> ( ( X + A ) (,) ( X + B ) ) C_ CC ) |
66 |
|
ax-resscn |
|- RR C_ CC |
67 |
66
|
a1i |
|- ( ph -> RR C_ CC ) |
68 |
62 67
|
fssd |
|- ( ph -> ( RR _D ( F |` ( ( X + A ) (,) ( X + B ) ) ) ) : ( ( X + A ) (,) ( X + B ) ) --> CC ) |
69 |
|
ssid |
|- CC C_ CC |
70 |
69
|
a1i |
|- ( ph -> CC C_ CC ) |
71 |
|
cncffvrn |
|- ( ( CC C_ CC /\ ( RR _D ( F |` ( ( X + A ) (,) ( X + B ) ) ) ) e. ( ( ( X + A ) (,) ( X + B ) ) -cn-> RR ) ) -> ( ( RR _D ( F |` ( ( X + A ) (,) ( X + B ) ) ) ) e. ( ( ( X + A ) (,) ( X + B ) ) -cn-> CC ) <-> ( RR _D ( F |` ( ( X + A ) (,) ( X + B ) ) ) ) : ( ( X + A ) (,) ( X + B ) ) --> CC ) ) |
72 |
70 6 71
|
syl2anc |
|- ( ph -> ( ( RR _D ( F |` ( ( X + A ) (,) ( X + B ) ) ) ) e. ( ( ( X + A ) (,) ( X + B ) ) -cn-> CC ) <-> ( RR _D ( F |` ( ( X + A ) (,) ( X + B ) ) ) ) : ( ( X + A ) (,) ( X + B ) ) --> CC ) ) |
73 |
68 72
|
mpbird |
|- ( ph -> ( RR _D ( F |` ( ( X + A ) (,) ( X + B ) ) ) ) e. ( ( ( X + A ) (,) ( X + B ) ) -cn-> CC ) ) |
74 |
|
ioosscn |
|- ( A (,) B ) C_ CC |
75 |
74
|
a1i |
|- ( ph -> ( A (,) B ) C_ CC ) |
76 |
2
|
recnd |
|- ( ph -> X e. CC ) |
77 |
2 3
|
readdcld |
|- ( ph -> ( X + A ) e. RR ) |
78 |
77
|
rexrd |
|- ( ph -> ( X + A ) e. RR* ) |
79 |
78
|
adantr |
|- ( ( ph /\ s e. ( A (,) B ) ) -> ( X + A ) e. RR* ) |
80 |
2 4
|
readdcld |
|- ( ph -> ( X + B ) e. RR ) |
81 |
80
|
rexrd |
|- ( ph -> ( X + B ) e. RR* ) |
82 |
81
|
adantr |
|- ( ( ph /\ s e. ( A (,) B ) ) -> ( X + B ) e. RR* ) |
83 |
3
|
adantr |
|- ( ( ph /\ s e. ( A (,) B ) ) -> A e. RR ) |
84 |
83
|
rexrd |
|- ( ( ph /\ s e. ( A (,) B ) ) -> A e. RR* ) |
85 |
4
|
rexrd |
|- ( ph -> B e. RR* ) |
86 |
85
|
adantr |
|- ( ( ph /\ s e. ( A (,) B ) ) -> B e. RR* ) |
87 |
|
simpr |
|- ( ( ph /\ s e. ( A (,) B ) ) -> s e. ( A (,) B ) ) |
88 |
|
ioogtlb |
|- ( ( A e. RR* /\ B e. RR* /\ s e. ( A (,) B ) ) -> A < s ) |
89 |
84 86 87 88
|
syl3anc |
|- ( ( ph /\ s e. ( A (,) B ) ) -> A < s ) |
90 |
83 12 10 89
|
ltadd2dd |
|- ( ( ph /\ s e. ( A (,) B ) ) -> ( X + A ) < ( X + s ) ) |
91 |
4
|
adantr |
|- ( ( ph /\ s e. ( A (,) B ) ) -> B e. RR ) |
92 |
|
iooltub |
|- ( ( A e. RR* /\ B e. RR* /\ s e. ( A (,) B ) ) -> s < B ) |
93 |
84 86 87 92
|
syl3anc |
|- ( ( ph /\ s e. ( A (,) B ) ) -> s < B ) |
94 |
12 91 10 93
|
ltadd2dd |
|- ( ( ph /\ s e. ( A (,) B ) ) -> ( X + s ) < ( X + B ) ) |
95 |
79 82 13 90 94
|
eliood |
|- ( ( ph /\ s e. ( A (,) B ) ) -> ( X + s ) e. ( ( X + A ) (,) ( X + B ) ) ) |
96 |
65 73 75 76 95
|
fourierdlem23 |
|- ( ph -> ( s e. ( A (,) B ) |-> ( ( RR _D ( F |` ( ( X + A ) (,) ( X + B ) ) ) ) ` ( X + s ) ) ) e. ( ( A (,) B ) -cn-> CC ) ) |
97 |
63 96
|
eqeltrd |
|- ( ph -> ( RR _D ( s e. ( A (,) B ) |-> ( F ` ( X + s ) ) ) ) e. ( ( A (,) B ) -cn-> CC ) ) |
98 |
|
iooretop |
|- ( A (,) B ) e. ( topGen ` ran (,) ) |
99 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
100 |
99
|
tgioo2 |
|- ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR ) |
101 |
98 100
|
eleqtri |
|- ( A (,) B ) e. ( ( TopOpen ` CCfld ) |`t RR ) |
102 |
101
|
a1i |
|- ( ph -> ( A (,) B ) e. ( ( TopOpen ` CCfld ) |`t RR ) ) |
103 |
7
|
recnd |
|- ( ph -> C e. CC ) |
104 |
40 102 103
|
dvmptconst |
|- ( ph -> ( RR _D ( s e. ( A (,) B ) |-> C ) ) = ( s e. ( A (,) B ) |-> 0 ) ) |
105 |
|
0cnd |
|- ( ph -> 0 e. CC ) |
106 |
75 105 70
|
constcncfg |
|- ( ph -> ( s e. ( A (,) B ) |-> 0 ) e. ( ( A (,) B ) -cn-> CC ) ) |
107 |
104 106
|
eqeltrd |
|- ( ph -> ( RR _D ( s e. ( A (,) B ) |-> C ) ) e. ( ( A (,) B ) -cn-> CC ) ) |
108 |
40 56 59 97 107
|
dvsubcncf |
|- ( ph -> ( RR _D ( ( s e. ( A (,) B ) |-> ( F ` ( X + s ) ) ) oF - ( s e. ( A (,) B ) |-> C ) ) ) e. ( ( A (,) B ) -cn-> CC ) ) |
109 |
53 108
|
eqeltrd |
|- ( ph -> ( RR _D ( s e. ( A (,) B ) |-> ( ( F ` ( X + s ) ) - C ) ) ) e. ( ( A (,) B ) -cn-> CC ) ) |
110 |
40 102
|
dvmptidg |
|- ( ph -> ( RR _D ( s e. ( A (,) B ) |-> s ) ) = ( s e. ( A (,) B ) |-> 1 ) ) |
111 |
|
1cnd |
|- ( ph -> 1 e. CC ) |
112 |
75 111 70
|
constcncfg |
|- ( ph -> ( s e. ( A (,) B ) |-> 1 ) e. ( ( A (,) B ) -cn-> CC ) ) |
113 |
110 112
|
eqeltrd |
|- ( ph -> ( RR _D ( s e. ( A (,) B ) |-> s ) ) e. ( ( A (,) B ) -cn-> CC ) ) |
114 |
40 43 48 109 113
|
dvdivcncf |
|- ( ph -> ( RR _D ( ( s e. ( A (,) B ) |-> ( ( F ` ( X + s ) ) - C ) ) oF / ( s e. ( A (,) B ) |-> s ) ) ) e. ( ( A (,) B ) -cn-> CC ) ) |
115 |
38 114
|
eqeltrd |
|- ( ph -> ( RR _D H ) e. ( ( A (,) B ) -cn-> CC ) ) |
116 |
|
cncff |
|- ( ( RR _D H ) e. ( ( A (,) B ) -cn-> CC ) -> ( RR _D H ) : ( A (,) B ) --> CC ) |
117 |
|
fdm |
|- ( ( RR _D H ) : ( A (,) B ) --> CC -> dom ( RR _D H ) = ( A (,) B ) ) |
118 |
115 116 117
|
3syl |
|- ( ph -> dom ( RR _D H ) = ( A (,) B ) ) |
119 |
118
|
feq2d |
|- ( ph -> ( ( RR _D H ) : dom ( RR _D H ) --> RR <-> ( RR _D H ) : ( A (,) B ) --> RR ) ) |
120 |
31 119
|
mpbid |
|- ( ph -> ( RR _D H ) : ( A (,) B ) --> RR ) |
121 |
|
cncffvrn |
|- ( ( RR C_ CC /\ ( RR _D H ) e. ( ( A (,) B ) -cn-> CC ) ) -> ( ( RR _D H ) e. ( ( A (,) B ) -cn-> RR ) <-> ( RR _D H ) : ( A (,) B ) --> RR ) ) |
122 |
67 115 121
|
syl2anc |
|- ( ph -> ( ( RR _D H ) e. ( ( A (,) B ) -cn-> RR ) <-> ( RR _D H ) : ( A (,) B ) --> RR ) ) |
123 |
120 122
|
mpbird |
|- ( ph -> ( RR _D H ) e. ( ( A (,) B ) -cn-> RR ) ) |