Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem46.cn |
|- ( ph -> F e. ( dom F -cn-> CC ) ) |
2 |
|
fourierdlem46.rlim |
|- ( ( ph /\ x e. ( ( -u _pi [,) _pi ) \ dom F ) ) -> ( ( F |` ( x (,) +oo ) ) limCC x ) =/= (/) ) |
3 |
|
fourierdlem46.llim |
|- ( ( ph /\ x e. ( ( -u _pi (,] _pi ) \ dom F ) ) -> ( ( F |` ( -oo (,) x ) ) limCC x ) =/= (/) ) |
4 |
|
fourierdlem46.qiso |
|- ( ph -> Q Isom < , < ( ( 0 ... M ) , H ) ) |
5 |
|
fourierdlem46.qf |
|- ( ph -> Q : ( 0 ... M ) --> H ) |
6 |
|
fourierdlem46.i |
|- ( ph -> I e. ( 0 ..^ M ) ) |
7 |
|
fourierdlem46.10 |
|- ( ph -> ( Q ` I ) < ( Q ` ( I + 1 ) ) ) |
8 |
|
fourierdlem46.qiss |
|- ( ph -> ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) C_ ( -u _pi (,) _pi ) ) |
9 |
|
fourierdlem46.c |
|- ( ph -> C e. RR ) |
10 |
|
fourierdlem46.h |
|- H = ( { -u _pi , _pi , C } u. ( ( -u _pi [,] _pi ) \ dom F ) ) |
11 |
|
fourierdlem46.ranq |
|- ( ph -> ran Q = H ) |
12 |
|
pire |
|- _pi e. RR |
13 |
12
|
a1i |
|- ( ph -> _pi e. RR ) |
14 |
13
|
renegcld |
|- ( ph -> -u _pi e. RR ) |
15 |
|
tpssi |
|- ( ( -u _pi e. RR /\ _pi e. RR /\ C e. RR ) -> { -u _pi , _pi , C } C_ RR ) |
16 |
14 13 9 15
|
syl3anc |
|- ( ph -> { -u _pi , _pi , C } C_ RR ) |
17 |
14 13
|
iccssred |
|- ( ph -> ( -u _pi [,] _pi ) C_ RR ) |
18 |
17
|
ssdifssd |
|- ( ph -> ( ( -u _pi [,] _pi ) \ dom F ) C_ RR ) |
19 |
16 18
|
unssd |
|- ( ph -> ( { -u _pi , _pi , C } u. ( ( -u _pi [,] _pi ) \ dom F ) ) C_ RR ) |
20 |
10 19
|
eqsstrid |
|- ( ph -> H C_ RR ) |
21 |
|
elfzofz |
|- ( I e. ( 0 ..^ M ) -> I e. ( 0 ... M ) ) |
22 |
6 21
|
syl |
|- ( ph -> I e. ( 0 ... M ) ) |
23 |
5 22
|
ffvelrnd |
|- ( ph -> ( Q ` I ) e. H ) |
24 |
20 23
|
sseldd |
|- ( ph -> ( Q ` I ) e. RR ) |
25 |
24
|
adantr |
|- ( ( ph /\ ( Q ` I ) e. dom F ) -> ( Q ` I ) e. RR ) |
26 |
|
fzofzp1 |
|- ( I e. ( 0 ..^ M ) -> ( I + 1 ) e. ( 0 ... M ) ) |
27 |
6 26
|
syl |
|- ( ph -> ( I + 1 ) e. ( 0 ... M ) ) |
28 |
5 27
|
ffvelrnd |
|- ( ph -> ( Q ` ( I + 1 ) ) e. H ) |
29 |
20 28
|
sseldd |
|- ( ph -> ( Q ` ( I + 1 ) ) e. RR ) |
30 |
29
|
rexrd |
|- ( ph -> ( Q ` ( I + 1 ) ) e. RR* ) |
31 |
30
|
adantr |
|- ( ( ph /\ ( Q ` I ) e. dom F ) -> ( Q ` ( I + 1 ) ) e. RR* ) |
32 |
7
|
adantr |
|- ( ( ph /\ ( Q ` I ) e. dom F ) -> ( Q ` I ) < ( Q ` ( I + 1 ) ) ) |
33 |
|
simpr |
|- ( ( ( Q ` I ) e. dom F /\ x = ( Q ` I ) ) -> x = ( Q ` I ) ) |
34 |
|
simpl |
|- ( ( ( Q ` I ) e. dom F /\ x = ( Q ` I ) ) -> ( Q ` I ) e. dom F ) |
35 |
33 34
|
eqeltrd |
|- ( ( ( Q ` I ) e. dom F /\ x = ( Q ` I ) ) -> x e. dom F ) |
36 |
35
|
adantll |
|- ( ( ( ph /\ ( Q ` I ) e. dom F ) /\ x = ( Q ` I ) ) -> x e. dom F ) |
37 |
36
|
adantlr |
|- ( ( ( ( ph /\ ( Q ` I ) e. dom F ) /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) /\ x = ( Q ` I ) ) -> x e. dom F ) |
38 |
|
ssun2 |
|- ( ( -u _pi [,] _pi ) \ dom F ) C_ ( { -u _pi , _pi , C } u. ( ( -u _pi [,] _pi ) \ dom F ) ) |
39 |
38 10
|
sseqtrri |
|- ( ( -u _pi [,] _pi ) \ dom F ) C_ H |
40 |
|
ioossicc |
|- ( -u _pi (,) _pi ) C_ ( -u _pi [,] _pi ) |
41 |
8 40
|
sstrdi |
|- ( ph -> ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) C_ ( -u _pi [,] _pi ) ) |
42 |
41
|
sselda |
|- ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> x e. ( -u _pi [,] _pi ) ) |
43 |
42
|
adantr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ -. x e. dom F ) -> x e. ( -u _pi [,] _pi ) ) |
44 |
|
simpr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ -. x e. dom F ) -> -. x e. dom F ) |
45 |
43 44
|
eldifd |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ -. x e. dom F ) -> x e. ( ( -u _pi [,] _pi ) \ dom F ) ) |
46 |
39 45
|
sseldi |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ -. x e. dom F ) -> x e. H ) |
47 |
11
|
eqcomd |
|- ( ph -> H = ran Q ) |
48 |
47
|
ad2antrr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ -. x e. dom F ) -> H = ran Q ) |
49 |
46 48
|
eleqtrd |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ -. x e. dom F ) -> x e. ran Q ) |
50 |
|
simpr |
|- ( ( ph /\ x e. ran Q ) -> x e. ran Q ) |
51 |
|
ffn |
|- ( Q : ( 0 ... M ) --> H -> Q Fn ( 0 ... M ) ) |
52 |
5 51
|
syl |
|- ( ph -> Q Fn ( 0 ... M ) ) |
53 |
52
|
adantr |
|- ( ( ph /\ x e. ran Q ) -> Q Fn ( 0 ... M ) ) |
54 |
|
fvelrnb |
|- ( Q Fn ( 0 ... M ) -> ( x e. ran Q <-> E. j e. ( 0 ... M ) ( Q ` j ) = x ) ) |
55 |
53 54
|
syl |
|- ( ( ph /\ x e. ran Q ) -> ( x e. ran Q <-> E. j e. ( 0 ... M ) ( Q ` j ) = x ) ) |
56 |
50 55
|
mpbid |
|- ( ( ph /\ x e. ran Q ) -> E. j e. ( 0 ... M ) ( Q ` j ) = x ) |
57 |
56
|
adantlr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ x e. ran Q ) -> E. j e. ( 0 ... M ) ( Q ` j ) = x ) |
58 |
|
elfzelz |
|- ( j e. ( 0 ... M ) -> j e. ZZ ) |
59 |
58
|
ad2antlr |
|- ( ( ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ x e. ran Q ) /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) = x ) -> j e. ZZ ) |
60 |
|
simplll |
|- ( ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) = x ) -> ph ) |
61 |
|
simplr |
|- ( ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) = x ) -> j e. ( 0 ... M ) ) |
62 |
|
simpr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ ( Q ` j ) = x ) -> ( Q ` j ) = x ) |
63 |
|
simplr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ ( Q ` j ) = x ) -> x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) |
64 |
62 63
|
eqeltrd |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ ( Q ` j ) = x ) -> ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) |
65 |
64
|
adantlr |
|- ( ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) = x ) -> ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) |
66 |
|
elfzoelz |
|- ( I e. ( 0 ..^ M ) -> I e. ZZ ) |
67 |
6 66
|
syl |
|- ( ph -> I e. ZZ ) |
68 |
67
|
ad2antrr |
|- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> I e. ZZ ) |
69 |
24
|
rexrd |
|- ( ph -> ( Q ` I ) e. RR* ) |
70 |
69
|
ad2antrr |
|- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> ( Q ` I ) e. RR* ) |
71 |
30
|
ad2antrr |
|- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> ( Q ` ( I + 1 ) ) e. RR* ) |
72 |
|
simpr |
|- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) |
73 |
|
ioogtlb |
|- ( ( ( Q ` I ) e. RR* /\ ( Q ` ( I + 1 ) ) e. RR* /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> ( Q ` I ) < ( Q ` j ) ) |
74 |
70 71 72 73
|
syl3anc |
|- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> ( Q ` I ) < ( Q ` j ) ) |
75 |
4
|
ad2antrr |
|- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> Q Isom < , < ( ( 0 ... M ) , H ) ) |
76 |
22
|
ad2antrr |
|- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> I e. ( 0 ... M ) ) |
77 |
|
simplr |
|- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> j e. ( 0 ... M ) ) |
78 |
|
isorel |
|- ( ( Q Isom < , < ( ( 0 ... M ) , H ) /\ ( I e. ( 0 ... M ) /\ j e. ( 0 ... M ) ) ) -> ( I < j <-> ( Q ` I ) < ( Q ` j ) ) ) |
79 |
75 76 77 78
|
syl12anc |
|- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> ( I < j <-> ( Q ` I ) < ( Q ` j ) ) ) |
80 |
74 79
|
mpbird |
|- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> I < j ) |
81 |
|
iooltub |
|- ( ( ( Q ` I ) e. RR* /\ ( Q ` ( I + 1 ) ) e. RR* /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> ( Q ` j ) < ( Q ` ( I + 1 ) ) ) |
82 |
70 71 72 81
|
syl3anc |
|- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> ( Q ` j ) < ( Q ` ( I + 1 ) ) ) |
83 |
27
|
ad2antrr |
|- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> ( I + 1 ) e. ( 0 ... M ) ) |
84 |
|
isorel |
|- ( ( Q Isom < , < ( ( 0 ... M ) , H ) /\ ( j e. ( 0 ... M ) /\ ( I + 1 ) e. ( 0 ... M ) ) ) -> ( j < ( I + 1 ) <-> ( Q ` j ) < ( Q ` ( I + 1 ) ) ) ) |
85 |
75 77 83 84
|
syl12anc |
|- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> ( j < ( I + 1 ) <-> ( Q ` j ) < ( Q ` ( I + 1 ) ) ) ) |
86 |
82 85
|
mpbird |
|- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> j < ( I + 1 ) ) |
87 |
|
btwnnz |
|- ( ( I e. ZZ /\ I < j /\ j < ( I + 1 ) ) -> -. j e. ZZ ) |
88 |
68 80 86 87
|
syl3anc |
|- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> -. j e. ZZ ) |
89 |
60 61 65 88
|
syl21anc |
|- ( ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) = x ) -> -. j e. ZZ ) |
90 |
89
|
adantllr |
|- ( ( ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ x e. ran Q ) /\ j e. ( 0 ... M ) ) /\ ( Q ` j ) = x ) -> -. j e. ZZ ) |
91 |
59 90
|
pm2.65da |
|- ( ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ x e. ran Q ) /\ j e. ( 0 ... M ) ) -> -. ( Q ` j ) = x ) |
92 |
91
|
nrexdv |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ x e. ran Q ) -> -. E. j e. ( 0 ... M ) ( Q ` j ) = x ) |
93 |
57 92
|
pm2.65da |
|- ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> -. x e. ran Q ) |
94 |
93
|
adantr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) /\ -. x e. dom F ) -> -. x e. ran Q ) |
95 |
49 94
|
condan |
|- ( ( ph /\ x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) -> x e. dom F ) |
96 |
95
|
ralrimiva |
|- ( ph -> A. x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) x e. dom F ) |
97 |
|
dfss3 |
|- ( ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) C_ dom F <-> A. x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) x e. dom F ) |
98 |
96 97
|
sylibr |
|- ( ph -> ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) C_ dom F ) |
99 |
98
|
ad2antrr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` I ) ) -> ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) C_ dom F ) |
100 |
69
|
ad2antrr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` I ) ) -> ( Q ` I ) e. RR* ) |
101 |
30
|
ad2antrr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` I ) ) -> ( Q ` ( I + 1 ) ) e. RR* ) |
102 |
|
icossre |
|- ( ( ( Q ` I ) e. RR /\ ( Q ` ( I + 1 ) ) e. RR* ) -> ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) C_ RR ) |
103 |
24 30 102
|
syl2anc |
|- ( ph -> ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) C_ RR ) |
104 |
103
|
sselda |
|- ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) -> x e. RR ) |
105 |
104
|
adantr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` I ) ) -> x e. RR ) |
106 |
24
|
ad2antrr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` I ) ) -> ( Q ` I ) e. RR ) |
107 |
69
|
adantr |
|- ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) -> ( Q ` I ) e. RR* ) |
108 |
30
|
adantr |
|- ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) -> ( Q ` ( I + 1 ) ) e. RR* ) |
109 |
|
simpr |
|- ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) -> x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) |
110 |
|
icogelb |
|- ( ( ( Q ` I ) e. RR* /\ ( Q ` ( I + 1 ) ) e. RR* /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) -> ( Q ` I ) <_ x ) |
111 |
107 108 109 110
|
syl3anc |
|- ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) -> ( Q ` I ) <_ x ) |
112 |
111
|
adantr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` I ) ) -> ( Q ` I ) <_ x ) |
113 |
|
neqne |
|- ( -. x = ( Q ` I ) -> x =/= ( Q ` I ) ) |
114 |
113
|
adantl |
|- ( ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` I ) ) -> x =/= ( Q ` I ) ) |
115 |
106 105 112 114
|
leneltd |
|- ( ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` I ) ) -> ( Q ` I ) < x ) |
116 |
|
icoltub |
|- ( ( ( Q ` I ) e. RR* /\ ( Q ` ( I + 1 ) ) e. RR* /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) -> x < ( Q ` ( I + 1 ) ) ) |
117 |
107 108 109 116
|
syl3anc |
|- ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) -> x < ( Q ` ( I + 1 ) ) ) |
118 |
117
|
adantr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` I ) ) -> x < ( Q ` ( I + 1 ) ) ) |
119 |
100 101 105 115 118
|
eliood |
|- ( ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` I ) ) -> x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) |
120 |
99 119
|
sseldd |
|- ( ( ( ph /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` I ) ) -> x e. dom F ) |
121 |
120
|
adantllr |
|- ( ( ( ( ph /\ ( Q ` I ) e. dom F ) /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` I ) ) -> x e. dom F ) |
122 |
37 121
|
pm2.61dan |
|- ( ( ( ph /\ ( Q ` I ) e. dom F ) /\ x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) -> x e. dom F ) |
123 |
122
|
ralrimiva |
|- ( ( ph /\ ( Q ` I ) e. dom F ) -> A. x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) x e. dom F ) |
124 |
|
dfss3 |
|- ( ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) C_ dom F <-> A. x e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) x e. dom F ) |
125 |
123 124
|
sylibr |
|- ( ( ph /\ ( Q ` I ) e. dom F ) -> ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) C_ dom F ) |
126 |
1
|
adantr |
|- ( ( ph /\ ( Q ` I ) e. dom F ) -> F e. ( dom F -cn-> CC ) ) |
127 |
|
rescncf |
|- ( ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) C_ dom F -> ( F e. ( dom F -cn-> CC ) -> ( F |` ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) e. ( ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) -cn-> CC ) ) ) |
128 |
125 126 127
|
sylc |
|- ( ( ph /\ ( Q ` I ) e. dom F ) -> ( F |` ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) e. ( ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) -cn-> CC ) ) |
129 |
25 31 32 128
|
icocncflimc |
|- ( ( ph /\ ( Q ` I ) e. dom F ) -> ( ( F |` ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) ` ( Q ` I ) ) e. ( ( ( F |` ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) ) |
130 |
24
|
leidd |
|- ( ph -> ( Q ` I ) <_ ( Q ` I ) ) |
131 |
69 30 69 130 7
|
elicod |
|- ( ph -> ( Q ` I ) e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) |
132 |
|
fvres |
|- ( ( Q ` I ) e. ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) -> ( ( F |` ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) ` ( Q ` I ) ) = ( F ` ( Q ` I ) ) ) |
133 |
131 132
|
syl |
|- ( ph -> ( ( F |` ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) ` ( Q ` I ) ) = ( F ` ( Q ` I ) ) ) |
134 |
133
|
eqcomd |
|- ( ph -> ( F ` ( Q ` I ) ) = ( ( F |` ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) ` ( Q ` I ) ) ) |
135 |
134
|
adantr |
|- ( ( ph /\ ( Q ` I ) e. dom F ) -> ( F ` ( Q ` I ) ) = ( ( F |` ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) ` ( Q ` I ) ) ) |
136 |
|
ioossico |
|- ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) C_ ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) |
137 |
136
|
a1i |
|- ( ( ph /\ ( Q ` I ) e. dom F ) -> ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) C_ ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) |
138 |
137
|
resabs1d |
|- ( ( ph /\ ( Q ` I ) e. dom F ) -> ( ( F |` ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) = ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) ) |
139 |
138
|
eqcomd |
|- ( ( ph /\ ( Q ` I ) e. dom F ) -> ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) = ( ( F |` ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) ) |
140 |
139
|
oveq1d |
|- ( ( ph /\ ( Q ` I ) e. dom F ) -> ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) = ( ( ( F |` ( ( Q ` I ) [,) ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) ) |
141 |
129 135 140
|
3eltr4d |
|- ( ( ph /\ ( Q ` I ) e. dom F ) -> ( F ` ( Q ` I ) ) e. ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) ) |
142 |
141
|
ne0d |
|- ( ( ph /\ ( Q ` I ) e. dom F ) -> ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) =/= (/) ) |
143 |
|
pnfxr |
|- +oo e. RR* |
144 |
143
|
a1i |
|- ( ph -> +oo e. RR* ) |
145 |
29
|
ltpnfd |
|- ( ph -> ( Q ` ( I + 1 ) ) < +oo ) |
146 |
30 144 145
|
xrltled |
|- ( ph -> ( Q ` ( I + 1 ) ) <_ +oo ) |
147 |
|
iooss2 |
|- ( ( +oo e. RR* /\ ( Q ` ( I + 1 ) ) <_ +oo ) -> ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) C_ ( ( Q ` I ) (,) +oo ) ) |
148 |
143 146 147
|
sylancr |
|- ( ph -> ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) C_ ( ( Q ` I ) (,) +oo ) ) |
149 |
148
|
resabs1d |
|- ( ph -> ( ( F |` ( ( Q ` I ) (,) +oo ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) = ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) ) |
150 |
149
|
oveq1d |
|- ( ph -> ( ( ( F |` ( ( Q ` I ) (,) +oo ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) = ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) ) |
151 |
150
|
eqcomd |
|- ( ph -> ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) = ( ( ( F |` ( ( Q ` I ) (,) +oo ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) ) |
152 |
151
|
adantr |
|- ( ( ph /\ -. ( Q ` I ) e. dom F ) -> ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) = ( ( ( F |` ( ( Q ` I ) (,) +oo ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) ) |
153 |
|
limcresi |
|- ( ( F |` ( ( Q ` I ) (,) +oo ) ) limCC ( Q ` I ) ) C_ ( ( ( F |` ( ( Q ` I ) (,) +oo ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) |
154 |
24
|
adantr |
|- ( ( ph /\ -. ( Q ` I ) e. dom F ) -> ( Q ` I ) e. RR ) |
155 |
|
simpl |
|- ( ( ph /\ -. ( Q ` I ) e. dom F ) -> ph ) |
156 |
12
|
renegcli |
|- -u _pi e. RR |
157 |
156
|
rexri |
|- -u _pi e. RR* |
158 |
157
|
a1i |
|- ( ph -> -u _pi e. RR* ) |
159 |
12
|
rexri |
|- _pi e. RR* |
160 |
159
|
a1i |
|- ( ph -> _pi e. RR* ) |
161 |
14 13 24 29 7 8
|
fourierdlem10 |
|- ( ph -> ( -u _pi <_ ( Q ` I ) /\ ( Q ` ( I + 1 ) ) <_ _pi ) ) |
162 |
161
|
simpld |
|- ( ph -> -u _pi <_ ( Q ` I ) ) |
163 |
161
|
simprd |
|- ( ph -> ( Q ` ( I + 1 ) ) <_ _pi ) |
164 |
24 29 13 7 163
|
ltletrd |
|- ( ph -> ( Q ` I ) < _pi ) |
165 |
158 160 69 162 164
|
elicod |
|- ( ph -> ( Q ` I ) e. ( -u _pi [,) _pi ) ) |
166 |
165
|
adantr |
|- ( ( ph /\ -. ( Q ` I ) e. dom F ) -> ( Q ` I ) e. ( -u _pi [,) _pi ) ) |
167 |
|
simpr |
|- ( ( ph /\ -. ( Q ` I ) e. dom F ) -> -. ( Q ` I ) e. dom F ) |
168 |
166 167
|
eldifd |
|- ( ( ph /\ -. ( Q ` I ) e. dom F ) -> ( Q ` I ) e. ( ( -u _pi [,) _pi ) \ dom F ) ) |
169 |
155 168
|
jca |
|- ( ( ph /\ -. ( Q ` I ) e. dom F ) -> ( ph /\ ( Q ` I ) e. ( ( -u _pi [,) _pi ) \ dom F ) ) ) |
170 |
|
eleq1 |
|- ( x = ( Q ` I ) -> ( x e. ( ( -u _pi [,) _pi ) \ dom F ) <-> ( Q ` I ) e. ( ( -u _pi [,) _pi ) \ dom F ) ) ) |
171 |
170
|
anbi2d |
|- ( x = ( Q ` I ) -> ( ( ph /\ x e. ( ( -u _pi [,) _pi ) \ dom F ) ) <-> ( ph /\ ( Q ` I ) e. ( ( -u _pi [,) _pi ) \ dom F ) ) ) ) |
172 |
|
oveq1 |
|- ( x = ( Q ` I ) -> ( x (,) +oo ) = ( ( Q ` I ) (,) +oo ) ) |
173 |
172
|
reseq2d |
|- ( x = ( Q ` I ) -> ( F |` ( x (,) +oo ) ) = ( F |` ( ( Q ` I ) (,) +oo ) ) ) |
174 |
|
id |
|- ( x = ( Q ` I ) -> x = ( Q ` I ) ) |
175 |
173 174
|
oveq12d |
|- ( x = ( Q ` I ) -> ( ( F |` ( x (,) +oo ) ) limCC x ) = ( ( F |` ( ( Q ` I ) (,) +oo ) ) limCC ( Q ` I ) ) ) |
176 |
175
|
neeq1d |
|- ( x = ( Q ` I ) -> ( ( ( F |` ( x (,) +oo ) ) limCC x ) =/= (/) <-> ( ( F |` ( ( Q ` I ) (,) +oo ) ) limCC ( Q ` I ) ) =/= (/) ) ) |
177 |
171 176
|
imbi12d |
|- ( x = ( Q ` I ) -> ( ( ( ph /\ x e. ( ( -u _pi [,) _pi ) \ dom F ) ) -> ( ( F |` ( x (,) +oo ) ) limCC x ) =/= (/) ) <-> ( ( ph /\ ( Q ` I ) e. ( ( -u _pi [,) _pi ) \ dom F ) ) -> ( ( F |` ( ( Q ` I ) (,) +oo ) ) limCC ( Q ` I ) ) =/= (/) ) ) ) |
178 |
177 2
|
vtoclg |
|- ( ( Q ` I ) e. RR -> ( ( ph /\ ( Q ` I ) e. ( ( -u _pi [,) _pi ) \ dom F ) ) -> ( ( F |` ( ( Q ` I ) (,) +oo ) ) limCC ( Q ` I ) ) =/= (/) ) ) |
179 |
154 169 178
|
sylc |
|- ( ( ph /\ -. ( Q ` I ) e. dom F ) -> ( ( F |` ( ( Q ` I ) (,) +oo ) ) limCC ( Q ` I ) ) =/= (/) ) |
180 |
|
ssn0 |
|- ( ( ( ( F |` ( ( Q ` I ) (,) +oo ) ) limCC ( Q ` I ) ) C_ ( ( ( F |` ( ( Q ` I ) (,) +oo ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) /\ ( ( F |` ( ( Q ` I ) (,) +oo ) ) limCC ( Q ` I ) ) =/= (/) ) -> ( ( ( F |` ( ( Q ` I ) (,) +oo ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) =/= (/) ) |
181 |
153 179 180
|
sylancr |
|- ( ( ph /\ -. ( Q ` I ) e. dom F ) -> ( ( ( F |` ( ( Q ` I ) (,) +oo ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) =/= (/) ) |
182 |
152 181
|
eqnetrd |
|- ( ( ph /\ -. ( Q ` I ) e. dom F ) -> ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) =/= (/) ) |
183 |
142 182
|
pm2.61dan |
|- ( ph -> ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) =/= (/) ) |
184 |
69
|
adantr |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) e. dom F ) -> ( Q ` I ) e. RR* ) |
185 |
29
|
adantr |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) e. dom F ) -> ( Q ` ( I + 1 ) ) e. RR ) |
186 |
7
|
adantr |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) e. dom F ) -> ( Q ` I ) < ( Q ` ( I + 1 ) ) ) |
187 |
|
simpr |
|- ( ( ( Q ` ( I + 1 ) ) e. dom F /\ x = ( Q ` ( I + 1 ) ) ) -> x = ( Q ` ( I + 1 ) ) ) |
188 |
|
simpl |
|- ( ( ( Q ` ( I + 1 ) ) e. dom F /\ x = ( Q ` ( I + 1 ) ) ) -> ( Q ` ( I + 1 ) ) e. dom F ) |
189 |
187 188
|
eqeltrd |
|- ( ( ( Q ` ( I + 1 ) ) e. dom F /\ x = ( Q ` ( I + 1 ) ) ) -> x e. dom F ) |
190 |
189
|
adantll |
|- ( ( ( ph /\ ( Q ` ( I + 1 ) ) e. dom F ) /\ x = ( Q ` ( I + 1 ) ) ) -> x e. dom F ) |
191 |
190
|
adantlr |
|- ( ( ( ( ph /\ ( Q ` ( I + 1 ) ) e. dom F ) /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) /\ x = ( Q ` ( I + 1 ) ) ) -> x e. dom F ) |
192 |
98
|
ad2antrr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` ( I + 1 ) ) ) -> ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) C_ dom F ) |
193 |
69
|
ad2antrr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` ( I + 1 ) ) ) -> ( Q ` I ) e. RR* ) |
194 |
30
|
ad2antrr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` ( I + 1 ) ) ) -> ( Q ` ( I + 1 ) ) e. RR* ) |
195 |
69
|
adantr |
|- ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) -> ( Q ` I ) e. RR* ) |
196 |
29
|
adantr |
|- ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) -> ( Q ` ( I + 1 ) ) e. RR ) |
197 |
|
iocssre |
|- ( ( ( Q ` I ) e. RR* /\ ( Q ` ( I + 1 ) ) e. RR ) -> ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) C_ RR ) |
198 |
195 196 197
|
syl2anc |
|- ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) -> ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) C_ RR ) |
199 |
|
simpr |
|- ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) -> x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) |
200 |
198 199
|
sseldd |
|- ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) -> x e. RR ) |
201 |
200
|
adantr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` ( I + 1 ) ) ) -> x e. RR ) |
202 |
30
|
adantr |
|- ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) -> ( Q ` ( I + 1 ) ) e. RR* ) |
203 |
|
iocgtlb |
|- ( ( ( Q ` I ) e. RR* /\ ( Q ` ( I + 1 ) ) e. RR* /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) -> ( Q ` I ) < x ) |
204 |
195 202 199 203
|
syl3anc |
|- ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) -> ( Q ` I ) < x ) |
205 |
204
|
adantr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` ( I + 1 ) ) ) -> ( Q ` I ) < x ) |
206 |
29
|
ad2antrr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` ( I + 1 ) ) ) -> ( Q ` ( I + 1 ) ) e. RR ) |
207 |
|
iocleub |
|- ( ( ( Q ` I ) e. RR* /\ ( Q ` ( I + 1 ) ) e. RR* /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) -> x <_ ( Q ` ( I + 1 ) ) ) |
208 |
195 202 199 207
|
syl3anc |
|- ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) -> x <_ ( Q ` ( I + 1 ) ) ) |
209 |
208
|
adantr |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` ( I + 1 ) ) ) -> x <_ ( Q ` ( I + 1 ) ) ) |
210 |
|
neqne |
|- ( -. x = ( Q ` ( I + 1 ) ) -> x =/= ( Q ` ( I + 1 ) ) ) |
211 |
210
|
necomd |
|- ( -. x = ( Q ` ( I + 1 ) ) -> ( Q ` ( I + 1 ) ) =/= x ) |
212 |
211
|
adantl |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` ( I + 1 ) ) ) -> ( Q ` ( I + 1 ) ) =/= x ) |
213 |
201 206 209 212
|
leneltd |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` ( I + 1 ) ) ) -> x < ( Q ` ( I + 1 ) ) ) |
214 |
193 194 201 205 213
|
eliood |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` ( I + 1 ) ) ) -> x e. ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) |
215 |
192 214
|
sseldd |
|- ( ( ( ph /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` ( I + 1 ) ) ) -> x e. dom F ) |
216 |
215
|
adantllr |
|- ( ( ( ( ph /\ ( Q ` ( I + 1 ) ) e. dom F ) /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) /\ -. x = ( Q ` ( I + 1 ) ) ) -> x e. dom F ) |
217 |
191 216
|
pm2.61dan |
|- ( ( ( ph /\ ( Q ` ( I + 1 ) ) e. dom F ) /\ x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) -> x e. dom F ) |
218 |
217
|
ralrimiva |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) e. dom F ) -> A. x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) x e. dom F ) |
219 |
|
dfss3 |
|- ( ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) C_ dom F <-> A. x e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) x e. dom F ) |
220 |
218 219
|
sylibr |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) e. dom F ) -> ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) C_ dom F ) |
221 |
1
|
adantr |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) e. dom F ) -> F e. ( dom F -cn-> CC ) ) |
222 |
|
rescncf |
|- ( ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) C_ dom F -> ( F e. ( dom F -cn-> CC ) -> ( F |` ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) e. ( ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) -cn-> CC ) ) ) |
223 |
220 221 222
|
sylc |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) e. dom F ) -> ( F |` ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) e. ( ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) -cn-> CC ) ) |
224 |
184 185 186 223
|
ioccncflimc |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) e. dom F ) -> ( ( F |` ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) ` ( Q ` ( I + 1 ) ) ) e. ( ( ( F |` ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) ) |
225 |
29
|
leidd |
|- ( ph -> ( Q ` ( I + 1 ) ) <_ ( Q ` ( I + 1 ) ) ) |
226 |
69 30 30 7 225
|
eliocd |
|- ( ph -> ( Q ` ( I + 1 ) ) e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) |
227 |
|
fvres |
|- ( ( Q ` ( I + 1 ) ) e. ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) -> ( ( F |` ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) ` ( Q ` ( I + 1 ) ) ) = ( F ` ( Q ` ( I + 1 ) ) ) ) |
228 |
226 227
|
syl |
|- ( ph -> ( ( F |` ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) ` ( Q ` ( I + 1 ) ) ) = ( F ` ( Q ` ( I + 1 ) ) ) ) |
229 |
228
|
eqcomd |
|- ( ph -> ( F ` ( Q ` ( I + 1 ) ) ) = ( ( F |` ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) ` ( Q ` ( I + 1 ) ) ) ) |
230 |
|
ioossioc |
|- ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) C_ ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) |
231 |
|
resabs1 |
|- ( ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) C_ ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) -> ( ( F |` ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) = ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) ) |
232 |
230 231
|
ax-mp |
|- ( ( F |` ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) = ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) |
233 |
232
|
eqcomi |
|- ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) = ( ( F |` ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) |
234 |
233
|
oveq1i |
|- ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) = ( ( ( F |` ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) |
235 |
234
|
a1i |
|- ( ph -> ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) = ( ( ( F |` ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) ) |
236 |
229 235
|
eleq12d |
|- ( ph -> ( ( F ` ( Q ` ( I + 1 ) ) ) e. ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) <-> ( ( F |` ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) ` ( Q ` ( I + 1 ) ) ) e. ( ( ( F |` ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) ) ) |
237 |
236
|
adantr |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) e. dom F ) -> ( ( F ` ( Q ` ( I + 1 ) ) ) e. ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) <-> ( ( F |` ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) ` ( Q ` ( I + 1 ) ) ) e. ( ( ( F |` ( ( Q ` I ) (,] ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) ) ) |
238 |
224 237
|
mpbird |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) e. dom F ) -> ( F ` ( Q ` ( I + 1 ) ) ) e. ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) ) |
239 |
238
|
ne0d |
|- ( ( ph /\ ( Q ` ( I + 1 ) ) e. dom F ) -> ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) =/= (/) ) |
240 |
|
mnfxr |
|- -oo e. RR* |
241 |
240
|
a1i |
|- ( ph -> -oo e. RR* ) |
242 |
24
|
mnfltd |
|- ( ph -> -oo < ( Q ` I ) ) |
243 |
241 69 242
|
xrltled |
|- ( ph -> -oo <_ ( Q ` I ) ) |
244 |
|
iooss1 |
|- ( ( -oo e. RR* /\ -oo <_ ( Q ` I ) ) -> ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) C_ ( -oo (,) ( Q ` ( I + 1 ) ) ) ) |
245 |
240 243 244
|
sylancr |
|- ( ph -> ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) C_ ( -oo (,) ( Q ` ( I + 1 ) ) ) ) |
246 |
245
|
resabs1d |
|- ( ph -> ( ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) = ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) ) |
247 |
246
|
eqcomd |
|- ( ph -> ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) = ( ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) ) |
248 |
247
|
adantr |
|- ( ( ph /\ -. ( Q ` ( I + 1 ) ) e. dom F ) -> ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) = ( ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) ) |
249 |
248
|
oveq1d |
|- ( ( ph /\ -. ( Q ` ( I + 1 ) ) e. dom F ) -> ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) = ( ( ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) ) |
250 |
|
limcresi |
|- ( ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) C_ ( ( ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) |
251 |
29
|
adantr |
|- ( ( ph /\ -. ( Q ` ( I + 1 ) ) e. dom F ) -> ( Q ` ( I + 1 ) ) e. RR ) |
252 |
|
simpl |
|- ( ( ph /\ -. ( Q ` ( I + 1 ) ) e. dom F ) -> ph ) |
253 |
157
|
a1i |
|- ( ( ph /\ -. ( Q ` ( I + 1 ) ) e. dom F ) -> -u _pi e. RR* ) |
254 |
159
|
a1i |
|- ( ( ph /\ -. ( Q ` ( I + 1 ) ) e. dom F ) -> _pi e. RR* ) |
255 |
30
|
adantr |
|- ( ( ph /\ -. ( Q ` ( I + 1 ) ) e. dom F ) -> ( Q ` ( I + 1 ) ) e. RR* ) |
256 |
14 24 29 162 7
|
lelttrd |
|- ( ph -> -u _pi < ( Q ` ( I + 1 ) ) ) |
257 |
256
|
adantr |
|- ( ( ph /\ -. ( Q ` ( I + 1 ) ) e. dom F ) -> -u _pi < ( Q ` ( I + 1 ) ) ) |
258 |
163
|
adantr |
|- ( ( ph /\ -. ( Q ` ( I + 1 ) ) e. dom F ) -> ( Q ` ( I + 1 ) ) <_ _pi ) |
259 |
253 254 255 257 258
|
eliocd |
|- ( ( ph /\ -. ( Q ` ( I + 1 ) ) e. dom F ) -> ( Q ` ( I + 1 ) ) e. ( -u _pi (,] _pi ) ) |
260 |
|
simpr |
|- ( ( ph /\ -. ( Q ` ( I + 1 ) ) e. dom F ) -> -. ( Q ` ( I + 1 ) ) e. dom F ) |
261 |
259 260
|
eldifd |
|- ( ( ph /\ -. ( Q ` ( I + 1 ) ) e. dom F ) -> ( Q ` ( I + 1 ) ) e. ( ( -u _pi (,] _pi ) \ dom F ) ) |
262 |
252 261
|
jca |
|- ( ( ph /\ -. ( Q ` ( I + 1 ) ) e. dom F ) -> ( ph /\ ( Q ` ( I + 1 ) ) e. ( ( -u _pi (,] _pi ) \ dom F ) ) ) |
263 |
|
eleq1 |
|- ( x = ( Q ` ( I + 1 ) ) -> ( x e. ( ( -u _pi (,] _pi ) \ dom F ) <-> ( Q ` ( I + 1 ) ) e. ( ( -u _pi (,] _pi ) \ dom F ) ) ) |
264 |
263
|
anbi2d |
|- ( x = ( Q ` ( I + 1 ) ) -> ( ( ph /\ x e. ( ( -u _pi (,] _pi ) \ dom F ) ) <-> ( ph /\ ( Q ` ( I + 1 ) ) e. ( ( -u _pi (,] _pi ) \ dom F ) ) ) ) |
265 |
|
oveq2 |
|- ( x = ( Q ` ( I + 1 ) ) -> ( -oo (,) x ) = ( -oo (,) ( Q ` ( I + 1 ) ) ) ) |
266 |
265
|
reseq2d |
|- ( x = ( Q ` ( I + 1 ) ) -> ( F |` ( -oo (,) x ) ) = ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) ) |
267 |
|
id |
|- ( x = ( Q ` ( I + 1 ) ) -> x = ( Q ` ( I + 1 ) ) ) |
268 |
266 267
|
oveq12d |
|- ( x = ( Q ` ( I + 1 ) ) -> ( ( F |` ( -oo (,) x ) ) limCC x ) = ( ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) ) |
269 |
268
|
neeq1d |
|- ( x = ( Q ` ( I + 1 ) ) -> ( ( ( F |` ( -oo (,) x ) ) limCC x ) =/= (/) <-> ( ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) =/= (/) ) ) |
270 |
264 269
|
imbi12d |
|- ( x = ( Q ` ( I + 1 ) ) -> ( ( ( ph /\ x e. ( ( -u _pi (,] _pi ) \ dom F ) ) -> ( ( F |` ( -oo (,) x ) ) limCC x ) =/= (/) ) <-> ( ( ph /\ ( Q ` ( I + 1 ) ) e. ( ( -u _pi (,] _pi ) \ dom F ) ) -> ( ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) =/= (/) ) ) ) |
271 |
270 3
|
vtoclg |
|- ( ( Q ` ( I + 1 ) ) e. RR -> ( ( ph /\ ( Q ` ( I + 1 ) ) e. ( ( -u _pi (,] _pi ) \ dom F ) ) -> ( ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) =/= (/) ) ) |
272 |
251 262 271
|
sylc |
|- ( ( ph /\ -. ( Q ` ( I + 1 ) ) e. dom F ) -> ( ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) =/= (/) ) |
273 |
|
ssn0 |
|- ( ( ( ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) C_ ( ( ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) /\ ( ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) =/= (/) ) -> ( ( ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) =/= (/) ) |
274 |
250 272 273
|
sylancr |
|- ( ( ph /\ -. ( Q ` ( I + 1 ) ) e. dom F ) -> ( ( ( F |` ( -oo (,) ( Q ` ( I + 1 ) ) ) ) |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) =/= (/) ) |
275 |
249 274
|
eqnetrd |
|- ( ( ph /\ -. ( Q ` ( I + 1 ) ) e. dom F ) -> ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) =/= (/) ) |
276 |
239 275
|
pm2.61dan |
|- ( ph -> ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) =/= (/) ) |
277 |
183 276
|
jca |
|- ( ph -> ( ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` I ) ) =/= (/) /\ ( ( F |` ( ( Q ` I ) (,) ( Q ` ( I + 1 ) ) ) ) limCC ( Q ` ( I + 1 ) ) ) =/= (/) ) ) |