Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem93.1 |
|- P = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = -u _pi /\ ( p ` m ) = _pi ) /\ A. i e. ( 0 ..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } ) |
2 |
|
fourierdlem93.2 |
|- H = ( i e. ( 0 ... M ) |-> ( ( Q ` i ) - X ) ) |
3 |
|
fourierdlem93.3 |
|- ( ph -> M e. NN ) |
4 |
|
fourierdlem93.4 |
|- ( ph -> Q e. ( P ` M ) ) |
5 |
|
fourierdlem93.5 |
|- ( ph -> X e. RR ) |
6 |
|
fourierdlem93.6 |
|- ( ph -> F : ( -u _pi [,] _pi ) --> CC ) |
7 |
|
fourierdlem93.7 |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) ) |
8 |
|
fourierdlem93.8 |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> R e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) ) |
9 |
|
fourierdlem93.9 |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> L e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` ( i + 1 ) ) ) ) |
10 |
1
|
fourierdlem2 |
|- ( M e. NN -> ( Q e. ( P ` M ) <-> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = -u _pi /\ ( Q ` M ) = _pi ) /\ A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) ) |
11 |
3 10
|
syl |
|- ( ph -> ( Q e. ( P ` M ) <-> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = -u _pi /\ ( Q ` M ) = _pi ) /\ A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) ) |
12 |
4 11
|
mpbid |
|- ( ph -> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = -u _pi /\ ( Q ` M ) = _pi ) /\ A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) |
13 |
12
|
simprd |
|- ( ph -> ( ( ( Q ` 0 ) = -u _pi /\ ( Q ` M ) = _pi ) /\ A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) |
14 |
13
|
simplld |
|- ( ph -> ( Q ` 0 ) = -u _pi ) |
15 |
14
|
eqcomd |
|- ( ph -> -u _pi = ( Q ` 0 ) ) |
16 |
13
|
simplrd |
|- ( ph -> ( Q ` M ) = _pi ) |
17 |
16
|
eqcomd |
|- ( ph -> _pi = ( Q ` M ) ) |
18 |
15 17
|
oveq12d |
|- ( ph -> ( -u _pi [,] _pi ) = ( ( Q ` 0 ) [,] ( Q ` M ) ) ) |
19 |
18
|
itgeq1d |
|- ( ph -> S. ( -u _pi [,] _pi ) ( F ` t ) _d t = S. ( ( Q ` 0 ) [,] ( Q ` M ) ) ( F ` t ) _d t ) |
20 |
|
0zd |
|- ( ph -> 0 e. ZZ ) |
21 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
22 |
3 21
|
eleqtrdi |
|- ( ph -> M e. ( ZZ>= ` 1 ) ) |
23 |
|
1e0p1 |
|- 1 = ( 0 + 1 ) |
24 |
23
|
a1i |
|- ( ph -> 1 = ( 0 + 1 ) ) |
25 |
24
|
fveq2d |
|- ( ph -> ( ZZ>= ` 1 ) = ( ZZ>= ` ( 0 + 1 ) ) ) |
26 |
22 25
|
eleqtrd |
|- ( ph -> M e. ( ZZ>= ` ( 0 + 1 ) ) ) |
27 |
1 3 4
|
fourierdlem15 |
|- ( ph -> Q : ( 0 ... M ) --> ( -u _pi [,] _pi ) ) |
28 |
|
pire |
|- _pi e. RR |
29 |
28
|
renegcli |
|- -u _pi e. RR |
30 |
|
iccssre |
|- ( ( -u _pi e. RR /\ _pi e. RR ) -> ( -u _pi [,] _pi ) C_ RR ) |
31 |
29 28 30
|
mp2an |
|- ( -u _pi [,] _pi ) C_ RR |
32 |
31
|
a1i |
|- ( ph -> ( -u _pi [,] _pi ) C_ RR ) |
33 |
27 32
|
fssd |
|- ( ph -> Q : ( 0 ... M ) --> RR ) |
34 |
13
|
simprd |
|- ( ph -> A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) |
35 |
34
|
r19.21bi |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` i ) < ( Q ` ( i + 1 ) ) ) |
36 |
6
|
adantr |
|- ( ( ph /\ t e. ( ( Q ` 0 ) [,] ( Q ` M ) ) ) -> F : ( -u _pi [,] _pi ) --> CC ) |
37 |
|
simpr |
|- ( ( ph /\ t e. ( ( Q ` 0 ) [,] ( Q ` M ) ) ) -> t e. ( ( Q ` 0 ) [,] ( Q ` M ) ) ) |
38 |
18
|
eqcomd |
|- ( ph -> ( ( Q ` 0 ) [,] ( Q ` M ) ) = ( -u _pi [,] _pi ) ) |
39 |
38
|
adantr |
|- ( ( ph /\ t e. ( ( Q ` 0 ) [,] ( Q ` M ) ) ) -> ( ( Q ` 0 ) [,] ( Q ` M ) ) = ( -u _pi [,] _pi ) ) |
40 |
37 39
|
eleqtrd |
|- ( ( ph /\ t e. ( ( Q ` 0 ) [,] ( Q ` M ) ) ) -> t e. ( -u _pi [,] _pi ) ) |
41 |
36 40
|
ffvelrnd |
|- ( ( ph /\ t e. ( ( Q ` 0 ) [,] ( Q ` M ) ) ) -> ( F ` t ) e. CC ) |
42 |
33
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> Q : ( 0 ... M ) --> RR ) |
43 |
|
elfzofz |
|- ( i e. ( 0 ..^ M ) -> i e. ( 0 ... M ) ) |
44 |
43
|
adantl |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> i e. ( 0 ... M ) ) |
45 |
42 44
|
ffvelrnd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` i ) e. RR ) |
46 |
|
fzofzp1 |
|- ( i e. ( 0 ..^ M ) -> ( i + 1 ) e. ( 0 ... M ) ) |
47 |
46
|
adantl |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( i + 1 ) e. ( 0 ... M ) ) |
48 |
42 47
|
ffvelrnd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` ( i + 1 ) ) e. RR ) |
49 |
6
|
feqmptd |
|- ( ph -> F = ( t e. ( -u _pi [,] _pi ) |-> ( F ` t ) ) ) |
50 |
49
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> F = ( t e. ( -u _pi [,] _pi ) |-> ( F ` t ) ) ) |
51 |
50
|
reseq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( ( t e. ( -u _pi [,] _pi ) |-> ( F ` t ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) |
52 |
|
ioossicc |
|- ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) |
53 |
52
|
a1i |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) |
54 |
29
|
rexri |
|- -u _pi e. RR* |
55 |
54
|
a1i |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) -> -u _pi e. RR* ) |
56 |
28
|
rexri |
|- _pi e. RR* |
57 |
56
|
a1i |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) -> _pi e. RR* ) |
58 |
27
|
ad2antrr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) -> Q : ( 0 ... M ) --> ( -u _pi [,] _pi ) ) |
59 |
|
simplr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) -> i e. ( 0 ..^ M ) ) |
60 |
|
simpr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) -> t e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) |
61 |
55 57 58 59 60
|
fourierdlem1 |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) -> t e. ( -u _pi [,] _pi ) ) |
62 |
61
|
ralrimiva |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> A. t e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) t e. ( -u _pi [,] _pi ) ) |
63 |
|
dfss3 |
|- ( ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) C_ ( -u _pi [,] _pi ) <-> A. t e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) t e. ( -u _pi [,] _pi ) ) |
64 |
62 63
|
sylibr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) C_ ( -u _pi [,] _pi ) ) |
65 |
53 64
|
sstrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ ( -u _pi [,] _pi ) ) |
66 |
65
|
resmptd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( t e. ( -u _pi [,] _pi ) |-> ( F ` t ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( F ` t ) ) ) |
67 |
51 66
|
eqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( F ` t ) ) ) |
68 |
67
|
eqcomd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( F ` t ) ) = ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) |
69 |
68 7
|
eqeltrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( F ` t ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) ) |
70 |
67
|
oveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` ( i + 1 ) ) ) = ( ( t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( F ` t ) ) limCC ( Q ` ( i + 1 ) ) ) ) |
71 |
9 70
|
eleqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> L e. ( ( t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( F ` t ) ) limCC ( Q ` ( i + 1 ) ) ) ) |
72 |
67
|
oveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) = ( ( t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( F ` t ) ) limCC ( Q ` i ) ) ) |
73 |
8 72
|
eleqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> R e. ( ( t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( F ` t ) ) limCC ( Q ` i ) ) ) |
74 |
45 48 69 71 73
|
iblcncfioo |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( F ` t ) ) e. L^1 ) |
75 |
6
|
ad2antrr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) -> F : ( -u _pi [,] _pi ) --> CC ) |
76 |
75 61
|
ffvelrnd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) -> ( F ` t ) e. CC ) |
77 |
45 48 74 76
|
ibliooicc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( t e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) |-> ( F ` t ) ) e. L^1 ) |
78 |
20 26 33 35 41 77
|
itgspltprt |
|- ( ph -> S. ( ( Q ` 0 ) [,] ( Q ` M ) ) ( F ` t ) _d t = sum_ i e. ( 0 ..^ M ) S. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ( F ` t ) _d t ) |
79 |
|
fvres |
|- ( t e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) -> ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) ` t ) = ( F ` t ) ) |
80 |
79
|
eqcomd |
|- ( t e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) -> ( F ` t ) = ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) ` t ) ) |
81 |
80
|
adantl |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) -> ( F ` t ) = ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) ` t ) ) |
82 |
81
|
itgeq2dv |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> S. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ( F ` t ) _d t = S. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) ` t ) _d t ) |
83 |
|
eqid |
|- ( x e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) |-> if ( x = ( Q ` i ) , R , if ( x = ( Q ` ( i + 1 ) ) , L , ( ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` x ) ) ) ) = ( x e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) |-> if ( x = ( Q ` i ) , R , if ( x = ( Q ` ( i + 1 ) ) , L , ( ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` x ) ) ) ) |
84 |
6
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> F : ( -u _pi [,] _pi ) --> CC ) |
85 |
84 64
|
fssresd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) : ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) --> CC ) |
86 |
53
|
resabs1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) |
87 |
86 7
|
eqeltrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) ) |
88 |
86
|
oveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` ( i + 1 ) ) ) = ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` ( i + 1 ) ) ) ) |
89 |
45 48 35 85
|
limcicciooub |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` ( i + 1 ) ) ) = ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` ( i + 1 ) ) ) ) |
90 |
88 89
|
eqtr3d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` ( i + 1 ) ) ) = ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` ( i + 1 ) ) ) ) |
91 |
9 90
|
eleqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> L e. ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` ( i + 1 ) ) ) ) |
92 |
86
|
eqcomd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) |
93 |
92
|
oveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) = ( ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) ) |
94 |
45 48 35 85
|
limciccioolb |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) = ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) ) |
95 |
93 94
|
eqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) = ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) ) |
96 |
8 95
|
eleqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> R e. ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) ) |
97 |
5
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> X e. RR ) |
98 |
83 45 48 35 85 87 91 96 97
|
fourierdlem82 |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> S. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) ` t ) _d t = S. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) ` ( X + t ) ) _d t ) |
99 |
45
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> ( Q ` i ) e. RR ) |
100 |
48
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> ( Q ` ( i + 1 ) ) e. RR ) |
101 |
5
|
ad2antrr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> X e. RR ) |
102 |
99 101
|
resubcld |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> ( ( Q ` i ) - X ) e. RR ) |
103 |
100 101
|
resubcld |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> ( ( Q ` ( i + 1 ) ) - X ) e. RR ) |
104 |
|
simpr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) |
105 |
|
eliccre |
|- ( ( ( ( Q ` i ) - X ) e. RR /\ ( ( Q ` ( i + 1 ) ) - X ) e. RR /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> t e. RR ) |
106 |
102 103 104 105
|
syl3anc |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> t e. RR ) |
107 |
101 106
|
readdcld |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> ( X + t ) e. RR ) |
108 |
|
elicc2 |
|- ( ( ( ( Q ` i ) - X ) e. RR /\ ( ( Q ` ( i + 1 ) ) - X ) e. RR ) -> ( t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) <-> ( t e. RR /\ ( ( Q ` i ) - X ) <_ t /\ t <_ ( ( Q ` ( i + 1 ) ) - X ) ) ) ) |
109 |
102 103 108
|
syl2anc |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> ( t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) <-> ( t e. RR /\ ( ( Q ` i ) - X ) <_ t /\ t <_ ( ( Q ` ( i + 1 ) ) - X ) ) ) ) |
110 |
104 109
|
mpbid |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> ( t e. RR /\ ( ( Q ` i ) - X ) <_ t /\ t <_ ( ( Q ` ( i + 1 ) ) - X ) ) ) |
111 |
110
|
simp2d |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> ( ( Q ` i ) - X ) <_ t ) |
112 |
99 101 106
|
lesubadd2d |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> ( ( ( Q ` i ) - X ) <_ t <-> ( Q ` i ) <_ ( X + t ) ) ) |
113 |
111 112
|
mpbid |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> ( Q ` i ) <_ ( X + t ) ) |
114 |
110
|
simp3d |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> t <_ ( ( Q ` ( i + 1 ) ) - X ) ) |
115 |
101 106 100
|
leaddsub2d |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> ( ( X + t ) <_ ( Q ` ( i + 1 ) ) <-> t <_ ( ( Q ` ( i + 1 ) ) - X ) ) ) |
116 |
114 115
|
mpbird |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> ( X + t ) <_ ( Q ` ( i + 1 ) ) ) |
117 |
99 100 107 113 116
|
eliccd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> ( X + t ) e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) |
118 |
|
fvres |
|- ( ( X + t ) e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) -> ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) ` ( X + t ) ) = ( F ` ( X + t ) ) ) |
119 |
117 118
|
syl |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) -> ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) ` ( X + t ) ) = ( F ` ( X + t ) ) ) |
120 |
119
|
itgeq2dv |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> S. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ( ( F |` ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) ` ( X + t ) ) _d t = S. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ( F ` ( X + t ) ) _d t ) |
121 |
82 98 120
|
3eqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> S. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ( F ` t ) _d t = S. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ( F ` ( X + t ) ) _d t ) |
122 |
121
|
sumeq2dv |
|- ( ph -> sum_ i e. ( 0 ..^ M ) S. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ( F ` t ) _d t = sum_ i e. ( 0 ..^ M ) S. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ( F ` ( X + t ) ) _d t ) |
123 |
|
oveq2 |
|- ( s = t -> ( X + s ) = ( X + t ) ) |
124 |
123
|
fveq2d |
|- ( s = t -> ( F ` ( X + s ) ) = ( F ` ( X + t ) ) ) |
125 |
124
|
cbvitgv |
|- S. ( ( -u _pi - X ) [,] ( _pi - X ) ) ( F ` ( X + s ) ) _d s = S. ( ( -u _pi - X ) [,] ( _pi - X ) ) ( F ` ( X + t ) ) _d t |
126 |
125
|
a1i |
|- ( ph -> S. ( ( -u _pi - X ) [,] ( _pi - X ) ) ( F ` ( X + s ) ) _d s = S. ( ( -u _pi - X ) [,] ( _pi - X ) ) ( F ` ( X + t ) ) _d t ) |
127 |
2
|
a1i |
|- ( ph -> H = ( i e. ( 0 ... M ) |-> ( ( Q ` i ) - X ) ) ) |
128 |
|
fveq2 |
|- ( i = 0 -> ( Q ` i ) = ( Q ` 0 ) ) |
129 |
128
|
oveq1d |
|- ( i = 0 -> ( ( Q ` i ) - X ) = ( ( Q ` 0 ) - X ) ) |
130 |
129
|
adantl |
|- ( ( ph /\ i = 0 ) -> ( ( Q ` i ) - X ) = ( ( Q ` 0 ) - X ) ) |
131 |
3
|
nnzd |
|- ( ph -> M e. ZZ ) |
132 |
20 131 20
|
3jca |
|- ( ph -> ( 0 e. ZZ /\ M e. ZZ /\ 0 e. ZZ ) ) |
133 |
|
0le0 |
|- 0 <_ 0 |
134 |
133
|
a1i |
|- ( ph -> 0 <_ 0 ) |
135 |
|
0red |
|- ( ph -> 0 e. RR ) |
136 |
3
|
nnred |
|- ( ph -> M e. RR ) |
137 |
3
|
nngt0d |
|- ( ph -> 0 < M ) |
138 |
135 136 137
|
ltled |
|- ( ph -> 0 <_ M ) |
139 |
134 138
|
jca |
|- ( ph -> ( 0 <_ 0 /\ 0 <_ M ) ) |
140 |
|
elfz2 |
|- ( 0 e. ( 0 ... M ) <-> ( ( 0 e. ZZ /\ M e. ZZ /\ 0 e. ZZ ) /\ ( 0 <_ 0 /\ 0 <_ M ) ) ) |
141 |
132 139 140
|
sylanbrc |
|- ( ph -> 0 e. ( 0 ... M ) ) |
142 |
14 29
|
eqeltrdi |
|- ( ph -> ( Q ` 0 ) e. RR ) |
143 |
142 5
|
resubcld |
|- ( ph -> ( ( Q ` 0 ) - X ) e. RR ) |
144 |
127 130 141 143
|
fvmptd |
|- ( ph -> ( H ` 0 ) = ( ( Q ` 0 ) - X ) ) |
145 |
14
|
oveq1d |
|- ( ph -> ( ( Q ` 0 ) - X ) = ( -u _pi - X ) ) |
146 |
144 145
|
eqtr2d |
|- ( ph -> ( -u _pi - X ) = ( H ` 0 ) ) |
147 |
|
fveq2 |
|- ( i = M -> ( Q ` i ) = ( Q ` M ) ) |
148 |
147
|
oveq1d |
|- ( i = M -> ( ( Q ` i ) - X ) = ( ( Q ` M ) - X ) ) |
149 |
148
|
adantl |
|- ( ( ph /\ i = M ) -> ( ( Q ` i ) - X ) = ( ( Q ` M ) - X ) ) |
150 |
20 131 131
|
3jca |
|- ( ph -> ( 0 e. ZZ /\ M e. ZZ /\ M e. ZZ ) ) |
151 |
136
|
leidd |
|- ( ph -> M <_ M ) |
152 |
138 151
|
jca |
|- ( ph -> ( 0 <_ M /\ M <_ M ) ) |
153 |
|
elfz2 |
|- ( M e. ( 0 ... M ) <-> ( ( 0 e. ZZ /\ M e. ZZ /\ M e. ZZ ) /\ ( 0 <_ M /\ M <_ M ) ) ) |
154 |
150 152 153
|
sylanbrc |
|- ( ph -> M e. ( 0 ... M ) ) |
155 |
16 28
|
eqeltrdi |
|- ( ph -> ( Q ` M ) e. RR ) |
156 |
155 5
|
resubcld |
|- ( ph -> ( ( Q ` M ) - X ) e. RR ) |
157 |
127 149 154 156
|
fvmptd |
|- ( ph -> ( H ` M ) = ( ( Q ` M ) - X ) ) |
158 |
16
|
oveq1d |
|- ( ph -> ( ( Q ` M ) - X ) = ( _pi - X ) ) |
159 |
157 158
|
eqtr2d |
|- ( ph -> ( _pi - X ) = ( H ` M ) ) |
160 |
146 159
|
oveq12d |
|- ( ph -> ( ( -u _pi - X ) [,] ( _pi - X ) ) = ( ( H ` 0 ) [,] ( H ` M ) ) ) |
161 |
160
|
itgeq1d |
|- ( ph -> S. ( ( -u _pi - X ) [,] ( _pi - X ) ) ( F ` ( X + t ) ) _d t = S. ( ( H ` 0 ) [,] ( H ` M ) ) ( F ` ( X + t ) ) _d t ) |
162 |
33
|
ffvelrnda |
|- ( ( ph /\ i e. ( 0 ... M ) ) -> ( Q ` i ) e. RR ) |
163 |
5
|
adantr |
|- ( ( ph /\ i e. ( 0 ... M ) ) -> X e. RR ) |
164 |
162 163
|
resubcld |
|- ( ( ph /\ i e. ( 0 ... M ) ) -> ( ( Q ` i ) - X ) e. RR ) |
165 |
164 2
|
fmptd |
|- ( ph -> H : ( 0 ... M ) --> RR ) |
166 |
45 48 97 35
|
ltsub1dd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( Q ` i ) - X ) < ( ( Q ` ( i + 1 ) ) - X ) ) |
167 |
44 164
|
syldan |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( Q ` i ) - X ) e. RR ) |
168 |
2
|
fvmpt2 |
|- ( ( i e. ( 0 ... M ) /\ ( ( Q ` i ) - X ) e. RR ) -> ( H ` i ) = ( ( Q ` i ) - X ) ) |
169 |
44 167 168
|
syl2anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( H ` i ) = ( ( Q ` i ) - X ) ) |
170 |
|
fveq2 |
|- ( i = j -> ( Q ` i ) = ( Q ` j ) ) |
171 |
170
|
oveq1d |
|- ( i = j -> ( ( Q ` i ) - X ) = ( ( Q ` j ) - X ) ) |
172 |
171
|
cbvmptv |
|- ( i e. ( 0 ... M ) |-> ( ( Q ` i ) - X ) ) = ( j e. ( 0 ... M ) |-> ( ( Q ` j ) - X ) ) |
173 |
2 172
|
eqtri |
|- H = ( j e. ( 0 ... M ) |-> ( ( Q ` j ) - X ) ) |
174 |
173
|
a1i |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> H = ( j e. ( 0 ... M ) |-> ( ( Q ` j ) - X ) ) ) |
175 |
|
fveq2 |
|- ( j = ( i + 1 ) -> ( Q ` j ) = ( Q ` ( i + 1 ) ) ) |
176 |
175
|
oveq1d |
|- ( j = ( i + 1 ) -> ( ( Q ` j ) - X ) = ( ( Q ` ( i + 1 ) ) - X ) ) |
177 |
176
|
adantl |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ j = ( i + 1 ) ) -> ( ( Q ` j ) - X ) = ( ( Q ` ( i + 1 ) ) - X ) ) |
178 |
48 97
|
resubcld |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( Q ` ( i + 1 ) ) - X ) e. RR ) |
179 |
174 177 47 178
|
fvmptd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( H ` ( i + 1 ) ) = ( ( Q ` ( i + 1 ) ) - X ) ) |
180 |
166 169 179
|
3brtr4d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( H ` i ) < ( H ` ( i + 1 ) ) ) |
181 |
|
frn |
|- ( F : ( -u _pi [,] _pi ) --> CC -> ran F C_ CC ) |
182 |
6 181
|
syl |
|- ( ph -> ran F C_ CC ) |
183 |
182
|
adantr |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> ran F C_ CC ) |
184 |
|
ffun |
|- ( F : ( -u _pi [,] _pi ) --> CC -> Fun F ) |
185 |
6 184
|
syl |
|- ( ph -> Fun F ) |
186 |
185
|
adantr |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> Fun F ) |
187 |
29
|
a1i |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> -u _pi e. RR ) |
188 |
28
|
a1i |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> _pi e. RR ) |
189 |
5
|
adantr |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> X e. RR ) |
190 |
144 143
|
eqeltrd |
|- ( ph -> ( H ` 0 ) e. RR ) |
191 |
190
|
adantr |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> ( H ` 0 ) e. RR ) |
192 |
157 156
|
eqeltrd |
|- ( ph -> ( H ` M ) e. RR ) |
193 |
192
|
adantr |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> ( H ` M ) e. RR ) |
194 |
|
simpr |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) |
195 |
|
eliccre |
|- ( ( ( H ` 0 ) e. RR /\ ( H ` M ) e. RR /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> t e. RR ) |
196 |
191 193 194 195
|
syl3anc |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> t e. RR ) |
197 |
189 196
|
readdcld |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> ( X + t ) e. RR ) |
198 |
128
|
adantl |
|- ( ( ph /\ i = 0 ) -> ( Q ` i ) = ( Q ` 0 ) ) |
199 |
198
|
oveq1d |
|- ( ( ph /\ i = 0 ) -> ( ( Q ` i ) - X ) = ( ( Q ` 0 ) - X ) ) |
200 |
127 199 141 143
|
fvmptd |
|- ( ph -> ( H ` 0 ) = ( ( Q ` 0 ) - X ) ) |
201 |
200
|
oveq2d |
|- ( ph -> ( X + ( H ` 0 ) ) = ( X + ( ( Q ` 0 ) - X ) ) ) |
202 |
5
|
recnd |
|- ( ph -> X e. CC ) |
203 |
142
|
recnd |
|- ( ph -> ( Q ` 0 ) e. CC ) |
204 |
202 203
|
pncan3d |
|- ( ph -> ( X + ( ( Q ` 0 ) - X ) ) = ( Q ` 0 ) ) |
205 |
201 204 14
|
3eqtrrd |
|- ( ph -> -u _pi = ( X + ( H ` 0 ) ) ) |
206 |
205
|
adantr |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> -u _pi = ( X + ( H ` 0 ) ) ) |
207 |
|
elicc2 |
|- ( ( ( H ` 0 ) e. RR /\ ( H ` M ) e. RR ) -> ( t e. ( ( H ` 0 ) [,] ( H ` M ) ) <-> ( t e. RR /\ ( H ` 0 ) <_ t /\ t <_ ( H ` M ) ) ) ) |
208 |
191 193 207
|
syl2anc |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> ( t e. ( ( H ` 0 ) [,] ( H ` M ) ) <-> ( t e. RR /\ ( H ` 0 ) <_ t /\ t <_ ( H ` M ) ) ) ) |
209 |
194 208
|
mpbid |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> ( t e. RR /\ ( H ` 0 ) <_ t /\ t <_ ( H ` M ) ) ) |
210 |
209
|
simp2d |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> ( H ` 0 ) <_ t ) |
211 |
191 196 189 210
|
leadd2dd |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> ( X + ( H ` 0 ) ) <_ ( X + t ) ) |
212 |
206 211
|
eqbrtrd |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> -u _pi <_ ( X + t ) ) |
213 |
209
|
simp3d |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> t <_ ( H ` M ) ) |
214 |
196 193 189 213
|
leadd2dd |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> ( X + t ) <_ ( X + ( H ` M ) ) ) |
215 |
157
|
oveq2d |
|- ( ph -> ( X + ( H ` M ) ) = ( X + ( ( Q ` M ) - X ) ) ) |
216 |
155
|
recnd |
|- ( ph -> ( Q ` M ) e. CC ) |
217 |
202 216
|
pncan3d |
|- ( ph -> ( X + ( ( Q ` M ) - X ) ) = ( Q ` M ) ) |
218 |
215 217 16
|
3eqtrrd |
|- ( ph -> _pi = ( X + ( H ` M ) ) ) |
219 |
218
|
adantr |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> _pi = ( X + ( H ` M ) ) ) |
220 |
214 219
|
breqtrrd |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> ( X + t ) <_ _pi ) |
221 |
187 188 197 212 220
|
eliccd |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> ( X + t ) e. ( -u _pi [,] _pi ) ) |
222 |
|
fdm |
|- ( F : ( -u _pi [,] _pi ) --> CC -> dom F = ( -u _pi [,] _pi ) ) |
223 |
6 222
|
syl |
|- ( ph -> dom F = ( -u _pi [,] _pi ) ) |
224 |
223
|
eqcomd |
|- ( ph -> ( -u _pi [,] _pi ) = dom F ) |
225 |
224
|
adantr |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> ( -u _pi [,] _pi ) = dom F ) |
226 |
221 225
|
eleqtrd |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> ( X + t ) e. dom F ) |
227 |
|
fvelrn |
|- ( ( Fun F /\ ( X + t ) e. dom F ) -> ( F ` ( X + t ) ) e. ran F ) |
228 |
186 226 227
|
syl2anc |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> ( F ` ( X + t ) ) e. ran F ) |
229 |
183 228
|
sseldd |
|- ( ( ph /\ t e. ( ( H ` 0 ) [,] ( H ` M ) ) ) -> ( F ` ( X + t ) ) e. CC ) |
230 |
169 167
|
eqeltrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( H ` i ) e. RR ) |
231 |
179 178
|
eqeltrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( H ` ( i + 1 ) ) e. RR ) |
232 |
84 65
|
fssresd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) : ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) --> CC ) |
233 |
45
|
rexrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` i ) e. RR* ) |
234 |
233
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( Q ` i ) e. RR* ) |
235 |
48
|
rexrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` ( i + 1 ) ) e. RR* ) |
236 |
235
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( Q ` ( i + 1 ) ) e. RR* ) |
237 |
5
|
ad2antrr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> X e. RR ) |
238 |
|
elioore |
|- ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) -> t e. RR ) |
239 |
238
|
adantl |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> t e. RR ) |
240 |
237 239
|
readdcld |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( X + t ) e. RR ) |
241 |
169
|
oveq2d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( X + ( H ` i ) ) = ( X + ( ( Q ` i ) - X ) ) ) |
242 |
202
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> X e. CC ) |
243 |
45
|
recnd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` i ) e. CC ) |
244 |
242 243
|
pncan3d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( X + ( ( Q ` i ) - X ) ) = ( Q ` i ) ) |
245 |
|
eqidd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` i ) = ( Q ` i ) ) |
246 |
241 244 245
|
3eqtrrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` i ) = ( X + ( H ` i ) ) ) |
247 |
246
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( Q ` i ) = ( X + ( H ` i ) ) ) |
248 |
230
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( H ` i ) e. RR ) |
249 |
|
simpr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) |
250 |
248
|
rexrd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( H ` i ) e. RR* ) |
251 |
231
|
rexrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( H ` ( i + 1 ) ) e. RR* ) |
252 |
251
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( H ` ( i + 1 ) ) e. RR* ) |
253 |
|
elioo2 |
|- ( ( ( H ` i ) e. RR* /\ ( H ` ( i + 1 ) ) e. RR* ) -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) <-> ( t e. RR /\ ( H ` i ) < t /\ t < ( H ` ( i + 1 ) ) ) ) ) |
254 |
250 252 253
|
syl2anc |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) <-> ( t e. RR /\ ( H ` i ) < t /\ t < ( H ` ( i + 1 ) ) ) ) ) |
255 |
249 254
|
mpbid |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( t e. RR /\ ( H ` i ) < t /\ t < ( H ` ( i + 1 ) ) ) ) |
256 |
255
|
simp2d |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( H ` i ) < t ) |
257 |
248 239 237 256
|
ltadd2dd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( X + ( H ` i ) ) < ( X + t ) ) |
258 |
247 257
|
eqbrtrd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( Q ` i ) < ( X + t ) ) |
259 |
231
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( H ` ( i + 1 ) ) e. RR ) |
260 |
255
|
simp3d |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> t < ( H ` ( i + 1 ) ) ) |
261 |
239 259 237 260
|
ltadd2dd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( X + t ) < ( X + ( H ` ( i + 1 ) ) ) ) |
262 |
179
|
oveq2d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( X + ( H ` ( i + 1 ) ) ) = ( X + ( ( Q ` ( i + 1 ) ) - X ) ) ) |
263 |
48
|
recnd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` ( i + 1 ) ) e. CC ) |
264 |
242 263
|
pncan3d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( X + ( ( Q ` ( i + 1 ) ) - X ) ) = ( Q ` ( i + 1 ) ) ) |
265 |
262 264
|
eqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( X + ( H ` ( i + 1 ) ) ) = ( Q ` ( i + 1 ) ) ) |
266 |
265
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( X + ( H ` ( i + 1 ) ) ) = ( Q ` ( i + 1 ) ) ) |
267 |
261 266
|
breqtrd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( X + t ) < ( Q ` ( i + 1 ) ) ) |
268 |
234 236 240 258 267
|
eliood |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( X + t ) e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) |
269 |
|
eqid |
|- ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) = ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) |
270 |
268 269
|
fmptd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) : ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) --> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) |
271 |
|
fcompt |
|- ( ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) : ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) --> CC /\ ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) : ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) --> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) o. ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) = ( s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) ) ) |
272 |
232 270 271
|
syl2anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) o. ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) = ( s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) ) ) |
273 |
|
oveq2 |
|- ( t = r -> ( X + t ) = ( X + r ) ) |
274 |
273
|
cbvmptv |
|- ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) = ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) |
275 |
274
|
fveq1i |
|- ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) = ( ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ` s ) |
276 |
275
|
fveq2i |
|- ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) = ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ` s ) ) |
277 |
276
|
mpteq2i |
|- ( s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) ) = ( s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ` s ) ) ) |
278 |
277
|
a1i |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) ) = ( s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ` s ) ) ) ) |
279 |
|
fveq2 |
|- ( s = t -> ( ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ` s ) = ( ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ` t ) ) |
280 |
279
|
fveq2d |
|- ( s = t -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ` s ) ) = ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ` t ) ) ) |
281 |
280
|
cbvmptv |
|- ( s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ` s ) ) ) = ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ` t ) ) ) |
282 |
281
|
a1i |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ` s ) ) ) = ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ` t ) ) ) ) |
283 |
|
eqidd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) = ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ) |
284 |
|
oveq2 |
|- ( r = t -> ( X + r ) = ( X + t ) ) |
285 |
284
|
adantl |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) /\ r = t ) -> ( X + r ) = ( X + t ) ) |
286 |
283 285 249 240
|
fvmptd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ` t ) = ( X + t ) ) |
287 |
286
|
fveq2d |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ` t ) ) = ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( X + t ) ) ) |
288 |
|
fvres |
|- ( ( X + t ) e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( X + t ) ) = ( F ` ( X + t ) ) ) |
289 |
268 288
|
syl |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( X + t ) ) = ( F ` ( X + t ) ) ) |
290 |
287 289
|
eqtrd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ` t ) ) = ( F ` ( X + t ) ) ) |
291 |
290
|
mpteq2dva |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( r e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + r ) ) ` t ) ) ) = ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( F ` ( X + t ) ) ) ) |
292 |
278 282 291
|
3eqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) ) = ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( F ` ( X + t ) ) ) ) |
293 |
272 292
|
eqtr2d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( F ` ( X + t ) ) ) = ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) o. ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) ) |
294 |
|
eqid |
|- ( t e. CC |-> ( X + t ) ) = ( t e. CC |-> ( X + t ) ) |
295 |
|
ssid |
|- CC C_ CC |
296 |
295
|
a1i |
|- ( X e. CC -> CC C_ CC ) |
297 |
|
id |
|- ( X e. CC -> X e. CC ) |
298 |
296 297 296
|
constcncfg |
|- ( X e. CC -> ( t e. CC |-> X ) e. ( CC -cn-> CC ) ) |
299 |
|
cncfmptid |
|- ( ( CC C_ CC /\ CC C_ CC ) -> ( t e. CC |-> t ) e. ( CC -cn-> CC ) ) |
300 |
295 295 299
|
mp2an |
|- ( t e. CC |-> t ) e. ( CC -cn-> CC ) |
301 |
300
|
a1i |
|- ( X e. CC -> ( t e. CC |-> t ) e. ( CC -cn-> CC ) ) |
302 |
298 301
|
addcncf |
|- ( X e. CC -> ( t e. CC |-> ( X + t ) ) e. ( CC -cn-> CC ) ) |
303 |
242 302
|
syl |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( t e. CC |-> ( X + t ) ) e. ( CC -cn-> CC ) ) |
304 |
|
ioosscn |
|- ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) C_ CC |
305 |
304
|
a1i |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) C_ CC ) |
306 |
|
ioosscn |
|- ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ CC |
307 |
306
|
a1i |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ CC ) |
308 |
294 303 305 307 268
|
cncfmptssg |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) -cn-> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) |
309 |
308 7
|
cncfco |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) o. ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) -cn-> CC ) ) |
310 |
293 309
|
eqeltrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( F ` ( X + t ) ) ) e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) -cn-> CC ) ) |
311 |
233
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> ( Q ` i ) e. RR* ) |
312 |
235
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> ( Q ` ( i + 1 ) ) e. RR* ) |
313 |
|
simpr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) |
314 |
|
vex |
|- r e. _V |
315 |
269
|
elrnmpt |
|- ( r e. _V -> ( r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) <-> E. t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) r = ( X + t ) ) ) |
316 |
314 315
|
ax-mp |
|- ( r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) <-> E. t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) r = ( X + t ) ) |
317 |
313 316
|
sylib |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> E. t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) r = ( X + t ) ) |
318 |
|
nfv |
|- F/ t ( ph /\ i e. ( 0 ..^ M ) ) |
319 |
|
nfmpt1 |
|- F/_ t ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) |
320 |
319
|
nfrn |
|- F/_ t ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) |
321 |
320
|
nfcri |
|- F/ t r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) |
322 |
318 321
|
nfan |
|- F/ t ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) |
323 |
|
nfv |
|- F/ t r e. RR |
324 |
|
simp3 |
|- ( ( ph /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) /\ r = ( X + t ) ) -> r = ( X + t ) ) |
325 |
5
|
3ad2ant1 |
|- ( ( ph /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) /\ r = ( X + t ) ) -> X e. RR ) |
326 |
238
|
3ad2ant2 |
|- ( ( ph /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) /\ r = ( X + t ) ) -> t e. RR ) |
327 |
325 326
|
readdcld |
|- ( ( ph /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) /\ r = ( X + t ) ) -> ( X + t ) e. RR ) |
328 |
324 327
|
eqeltrd |
|- ( ( ph /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) /\ r = ( X + t ) ) -> r e. RR ) |
329 |
328
|
3exp |
|- ( ph -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) -> ( r = ( X + t ) -> r e. RR ) ) ) |
330 |
329
|
ad2antrr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) -> ( r = ( X + t ) -> r e. RR ) ) ) |
331 |
322 323 330
|
rexlimd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> ( E. t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) r = ( X + t ) -> r e. RR ) ) |
332 |
317 331
|
mpd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> r e. RR ) |
333 |
|
nfv |
|- F/ t ( Q ` i ) < r |
334 |
258
|
3adant3 |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) /\ r = ( X + t ) ) -> ( Q ` i ) < ( X + t ) ) |
335 |
|
simp3 |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) /\ r = ( X + t ) ) -> r = ( X + t ) ) |
336 |
334 335
|
breqtrrd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) /\ r = ( X + t ) ) -> ( Q ` i ) < r ) |
337 |
336
|
3exp |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) -> ( r = ( X + t ) -> ( Q ` i ) < r ) ) ) |
338 |
337
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) -> ( r = ( X + t ) -> ( Q ` i ) < r ) ) ) |
339 |
322 333 338
|
rexlimd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> ( E. t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) r = ( X + t ) -> ( Q ` i ) < r ) ) |
340 |
317 339
|
mpd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> ( Q ` i ) < r ) |
341 |
|
nfv |
|- F/ t r < ( Q ` ( i + 1 ) ) |
342 |
267
|
3adant3 |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) /\ r = ( X + t ) ) -> ( X + t ) < ( Q ` ( i + 1 ) ) ) |
343 |
335 342
|
eqbrtrd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) /\ r = ( X + t ) ) -> r < ( Q ` ( i + 1 ) ) ) |
344 |
343
|
3exp |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) -> ( r = ( X + t ) -> r < ( Q ` ( i + 1 ) ) ) ) ) |
345 |
344
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) -> ( r = ( X + t ) -> r < ( Q ` ( i + 1 ) ) ) ) ) |
346 |
322 341 345
|
rexlimd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> ( E. t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) r = ( X + t ) -> r < ( Q ` ( i + 1 ) ) ) ) |
347 |
317 346
|
mpd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> r < ( Q ` ( i + 1 ) ) ) |
348 |
311 312 332 340 347
|
eliood |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> r e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) |
349 |
223
|
ineq2d |
|- ( ph -> ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) i^i dom F ) = ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) i^i ( -u _pi [,] _pi ) ) ) |
350 |
349
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) i^i dom F ) = ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) i^i ( -u _pi [,] _pi ) ) ) |
351 |
|
dmres |
|- dom ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) i^i dom F ) |
352 |
351
|
a1i |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> dom ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) i^i dom F ) ) |
353 |
|
dfss |
|- ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ ( -u _pi [,] _pi ) <-> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) = ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) i^i ( -u _pi [,] _pi ) ) ) |
354 |
65 353
|
sylib |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) = ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) i^i ( -u _pi [,] _pi ) ) ) |
355 |
350 352 354
|
3eqtr4d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> dom ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) |
356 |
355
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> dom ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) |
357 |
348 356
|
eleqtrrd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> r e. dom ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) |
358 |
332 347
|
ltned |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> r =/= ( Q ` ( i + 1 ) ) ) |
359 |
358
|
neneqd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> -. r = ( Q ` ( i + 1 ) ) ) |
360 |
|
velsn |
|- ( r e. { ( Q ` ( i + 1 ) ) } <-> r = ( Q ` ( i + 1 ) ) ) |
361 |
359 360
|
sylnibr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> -. r e. { ( Q ` ( i + 1 ) ) } ) |
362 |
357 361
|
eldifd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> r e. ( dom ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) \ { ( Q ` ( i + 1 ) ) } ) ) |
363 |
362
|
ralrimiva |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> A. r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) r e. ( dom ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) \ { ( Q ` ( i + 1 ) ) } ) ) |
364 |
|
dfss3 |
|- ( ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) C_ ( dom ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) \ { ( Q ` ( i + 1 ) ) } ) <-> A. r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) r e. ( dom ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) \ { ( Q ` ( i + 1 ) ) } ) ) |
365 |
363 364
|
sylibr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) C_ ( dom ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) \ { ( Q ` ( i + 1 ) ) } ) ) |
366 |
|
eqid |
|- ( s e. CC |-> ( X + s ) ) = ( s e. CC |-> ( X + s ) ) |
367 |
202
|
adantr |
|- ( ( ph /\ s e. CC ) -> X e. CC ) |
368 |
|
simpr |
|- ( ( ph /\ s e. CC ) -> s e. CC ) |
369 |
367 368
|
addcomd |
|- ( ( ph /\ s e. CC ) -> ( X + s ) = ( s + X ) ) |
370 |
369
|
mpteq2dva |
|- ( ph -> ( s e. CC |-> ( X + s ) ) = ( s e. CC |-> ( s + X ) ) ) |
371 |
|
eqid |
|- ( s e. CC |-> ( s + X ) ) = ( s e. CC |-> ( s + X ) ) |
372 |
371
|
addccncf |
|- ( X e. CC -> ( s e. CC |-> ( s + X ) ) e. ( CC -cn-> CC ) ) |
373 |
202 372
|
syl |
|- ( ph -> ( s e. CC |-> ( s + X ) ) e. ( CC -cn-> CC ) ) |
374 |
370 373
|
eqeltrd |
|- ( ph -> ( s e. CC |-> ( X + s ) ) e. ( CC -cn-> CC ) ) |
375 |
374
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( s e. CC |-> ( X + s ) ) e. ( CC -cn-> CC ) ) |
376 |
230
|
rexrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( H ` i ) e. RR* ) |
377 |
|
iocssre |
|- ( ( ( H ` i ) e. RR* /\ ( H ` ( i + 1 ) ) e. RR ) -> ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) C_ RR ) |
378 |
376 231 377
|
syl2anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) C_ RR ) |
379 |
|
ax-resscn |
|- RR C_ CC |
380 |
378 379
|
sstrdi |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) C_ CC ) |
381 |
295
|
a1i |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> CC C_ CC ) |
382 |
202
|
ad2antrr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) -> X e. CC ) |
383 |
380
|
sselda |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) -> s e. CC ) |
384 |
382 383
|
addcld |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) -> ( X + s ) e. CC ) |
385 |
366 375 380 381 384
|
cncfmptssg |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) -cn-> CC ) ) |
386 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
387 |
|
eqid |
|- ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) = ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) |
388 |
386
|
cnfldtop |
|- ( TopOpen ` CCfld ) e. Top |
389 |
|
unicntop |
|- CC = U. ( TopOpen ` CCfld ) |
390 |
389
|
restid |
|- ( ( TopOpen ` CCfld ) e. Top -> ( ( TopOpen ` CCfld ) |`t CC ) = ( TopOpen ` CCfld ) ) |
391 |
388 390
|
ax-mp |
|- ( ( TopOpen ` CCfld ) |`t CC ) = ( TopOpen ` CCfld ) |
392 |
391
|
eqcomi |
|- ( TopOpen ` CCfld ) = ( ( TopOpen ` CCfld ) |`t CC ) |
393 |
386 387 392
|
cncfcn |
|- ( ( ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) C_ CC /\ CC C_ CC ) -> ( ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) -cn-> CC ) = ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) Cn ( TopOpen ` CCfld ) ) ) |
394 |
380 381 393
|
syl2anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) -cn-> CC ) = ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) Cn ( TopOpen ` CCfld ) ) ) |
395 |
385 394
|
eleqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) Cn ( TopOpen ` CCfld ) ) ) |
396 |
386
|
cnfldtopon |
|- ( TopOpen ` CCfld ) e. ( TopOn ` CC ) |
397 |
396
|
a1i |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( TopOpen ` CCfld ) e. ( TopOn ` CC ) ) |
398 |
|
resttopon |
|- ( ( ( TopOpen ` CCfld ) e. ( TopOn ` CC ) /\ ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) C_ CC ) -> ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) e. ( TopOn ` ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) ) |
399 |
397 380 398
|
syl2anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) e. ( TopOn ` ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) ) |
400 |
|
cncnp |
|- ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) e. ( TopOn ` ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) /\ ( TopOpen ` CCfld ) e. ( TopOn ` CC ) ) -> ( ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) Cn ( TopOpen ` CCfld ) ) <-> ( ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) : ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) --> CC /\ A. t e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` t ) ) ) ) |
401 |
399 397 400
|
syl2anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) Cn ( TopOpen ` CCfld ) ) <-> ( ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) : ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) --> CC /\ A. t e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` t ) ) ) ) |
402 |
395 401
|
mpbid |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) : ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) --> CC /\ A. t e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` t ) ) ) |
403 |
402
|
simprd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> A. t e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` t ) ) |
404 |
|
ubioc1 |
|- ( ( ( H ` i ) e. RR* /\ ( H ` ( i + 1 ) ) e. RR* /\ ( H ` i ) < ( H ` ( i + 1 ) ) ) -> ( H ` ( i + 1 ) ) e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) |
405 |
376 251 180 404
|
syl3anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( H ` ( i + 1 ) ) e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) |
406 |
|
fveq2 |
|- ( t = ( H ` ( i + 1 ) ) -> ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` t ) = ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` ( H ` ( i + 1 ) ) ) ) |
407 |
406
|
eleq2d |
|- ( t = ( H ` ( i + 1 ) ) -> ( ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` t ) <-> ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` ( H ` ( i + 1 ) ) ) ) ) |
408 |
407
|
rspccva |
|- ( ( A. t e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` t ) /\ ( H ` ( i + 1 ) ) e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) -> ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` ( H ` ( i + 1 ) ) ) ) |
409 |
403 405 408
|
syl2anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` ( H ` ( i + 1 ) ) ) ) |
410 |
|
ioounsn |
|- ( ( ( H ` i ) e. RR* /\ ( H ` ( i + 1 ) ) e. RR* /\ ( H ` i ) < ( H ` ( i + 1 ) ) ) -> ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) = ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) |
411 |
376 251 180 410
|
syl3anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) = ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) |
412 |
265
|
eqcomd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` ( i + 1 ) ) = ( X + ( H ` ( i + 1 ) ) ) ) |
413 |
412
|
ad2antrr |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) /\ s = ( H ` ( i + 1 ) ) ) -> ( Q ` ( i + 1 ) ) = ( X + ( H ` ( i + 1 ) ) ) ) |
414 |
|
iftrue |
|- ( s = ( H ` ( i + 1 ) ) -> if ( s = ( H ` ( i + 1 ) ) , ( Q ` ( i + 1 ) ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) = ( Q ` ( i + 1 ) ) ) |
415 |
414
|
adantl |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) /\ s = ( H ` ( i + 1 ) ) ) -> if ( s = ( H ` ( i + 1 ) ) , ( Q ` ( i + 1 ) ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) = ( Q ` ( i + 1 ) ) ) |
416 |
|
oveq2 |
|- ( s = ( H ` ( i + 1 ) ) -> ( X + s ) = ( X + ( H ` ( i + 1 ) ) ) ) |
417 |
416
|
adantl |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) /\ s = ( H ` ( i + 1 ) ) ) -> ( X + s ) = ( X + ( H ` ( i + 1 ) ) ) ) |
418 |
413 415 417
|
3eqtr4d |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) /\ s = ( H ` ( i + 1 ) ) ) -> if ( s = ( H ` ( i + 1 ) ) , ( Q ` ( i + 1 ) ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) = ( X + s ) ) |
419 |
|
iffalse |
|- ( -. s = ( H ` ( i + 1 ) ) -> if ( s = ( H ` ( i + 1 ) ) , ( Q ` ( i + 1 ) ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) = ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) |
420 |
419
|
adantl |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) /\ -. s = ( H ` ( i + 1 ) ) ) -> if ( s = ( H ` ( i + 1 ) ) , ( Q ` ( i + 1 ) ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) = ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) |
421 |
|
eqidd |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) /\ -. s = ( H ` ( i + 1 ) ) ) -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) = ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) |
422 |
|
oveq2 |
|- ( t = s -> ( X + t ) = ( X + s ) ) |
423 |
422
|
adantl |
|- ( ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) /\ -. s = ( H ` ( i + 1 ) ) ) /\ t = s ) -> ( X + t ) = ( X + s ) ) |
424 |
|
velsn |
|- ( s e. { ( H ` ( i + 1 ) ) } <-> s = ( H ` ( i + 1 ) ) ) |
425 |
424
|
notbii |
|- ( -. s e. { ( H ` ( i + 1 ) ) } <-> -. s = ( H ` ( i + 1 ) ) ) |
426 |
|
elun |
|- ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) <-> ( s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) \/ s e. { ( H ` ( i + 1 ) ) } ) ) |
427 |
426
|
biimpi |
|- ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) -> ( s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) \/ s e. { ( H ` ( i + 1 ) ) } ) ) |
428 |
427
|
orcomd |
|- ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) -> ( s e. { ( H ` ( i + 1 ) ) } \/ s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) ) |
429 |
428
|
ord |
|- ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) -> ( -. s e. { ( H ` ( i + 1 ) ) } -> s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) ) |
430 |
425 429
|
syl5bir |
|- ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) -> ( -. s = ( H ` ( i + 1 ) ) -> s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) ) |
431 |
430
|
imp |
|- ( ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) /\ -. s = ( H ` ( i + 1 ) ) ) -> s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) |
432 |
431
|
adantll |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) /\ -. s = ( H ` ( i + 1 ) ) ) -> s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) |
433 |
5
|
ad2antrr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) -> X e. RR ) |
434 |
|
elioore |
|- ( s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) -> s e. RR ) |
435 |
434
|
adantl |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) -> s e. RR ) |
436 |
|
elsni |
|- ( s e. { ( H ` ( i + 1 ) ) } -> s = ( H ` ( i + 1 ) ) ) |
437 |
436
|
adantl |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. { ( H ` ( i + 1 ) ) } ) -> s = ( H ` ( i + 1 ) ) ) |
438 |
231
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. { ( H ` ( i + 1 ) ) } ) -> ( H ` ( i + 1 ) ) e. RR ) |
439 |
437 438
|
eqeltrd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. { ( H ` ( i + 1 ) ) } ) -> s e. RR ) |
440 |
435 439
|
jaodan |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ ( s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) \/ s e. { ( H ` ( i + 1 ) ) } ) ) -> s e. RR ) |
441 |
426 440
|
sylan2b |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) -> s e. RR ) |
442 |
433 441
|
readdcld |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) -> ( X + s ) e. RR ) |
443 |
442
|
adantr |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) /\ -. s = ( H ` ( i + 1 ) ) ) -> ( X + s ) e. RR ) |
444 |
421 423 432 443
|
fvmptd |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) /\ -. s = ( H ` ( i + 1 ) ) ) -> ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) = ( X + s ) ) |
445 |
420 444
|
eqtrd |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) /\ -. s = ( H ` ( i + 1 ) ) ) -> if ( s = ( H ` ( i + 1 ) ) , ( Q ` ( i + 1 ) ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) = ( X + s ) ) |
446 |
418 445
|
pm2.61dan |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) -> if ( s = ( H ` ( i + 1 ) ) , ( Q ` ( i + 1 ) ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) = ( X + s ) ) |
447 |
411 446
|
mpteq12dva |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) |-> if ( s = ( H ` ( i + 1 ) ) , ( Q ` ( i + 1 ) ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) ) = ( s e. ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) ) |
448 |
411
|
oveq2d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( TopOpen ` CCfld ) |`t ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) = ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) ) |
449 |
448
|
oveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( ( TopOpen ` CCfld ) |`t ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) CnP ( TopOpen ` CCfld ) ) = ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ) |
450 |
449
|
fveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( ( ( TopOpen ` CCfld ) |`t ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) CnP ( TopOpen ` CCfld ) ) ` ( H ` ( i + 1 ) ) ) = ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) (,] ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` ( H ` ( i + 1 ) ) ) ) |
451 |
409 447 450
|
3eltr4d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) |-> if ( s = ( H ` ( i + 1 ) ) , ( Q ` ( i + 1 ) ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) CnP ( TopOpen ` CCfld ) ) ` ( H ` ( i + 1 ) ) ) ) |
452 |
|
eqid |
|- ( ( TopOpen ` CCfld ) |`t ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) = ( ( TopOpen ` CCfld ) |`t ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) |
453 |
|
eqid |
|- ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) |-> if ( s = ( H ` ( i + 1 ) ) , ( Q ` ( i + 1 ) ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) ) = ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) |-> if ( s = ( H ` ( i + 1 ) ) , ( Q ` ( i + 1 ) ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) ) |
454 |
270 307
|
fssd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) : ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) --> CC ) |
455 |
231
|
recnd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( H ` ( i + 1 ) ) e. CC ) |
456 |
452 386 453 454 305 455
|
ellimc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( Q ` ( i + 1 ) ) e. ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) limCC ( H ` ( i + 1 ) ) ) <-> ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) |-> if ( s = ( H ` ( i + 1 ) ) , ( Q ` ( i + 1 ) ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` ( i + 1 ) ) } ) ) CnP ( TopOpen ` CCfld ) ) ` ( H ` ( i + 1 ) ) ) ) ) |
457 |
451 456
|
mpbird |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` ( i + 1 ) ) e. ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) limCC ( H ` ( i + 1 ) ) ) ) |
458 |
365 457 9
|
limccog |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> L e. ( ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) o. ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) limCC ( H ` ( i + 1 ) ) ) ) |
459 |
272 292
|
eqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) o. ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) = ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( F ` ( X + t ) ) ) ) |
460 |
459
|
oveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) o. ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) limCC ( H ` ( i + 1 ) ) ) = ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( F ` ( X + t ) ) ) limCC ( H ` ( i + 1 ) ) ) ) |
461 |
458 460
|
eleqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> L e. ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( F ` ( X + t ) ) ) limCC ( H ` ( i + 1 ) ) ) ) |
462 |
45
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> ( Q ` i ) e. RR ) |
463 |
462 340
|
gtned |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> r =/= ( Q ` i ) ) |
464 |
463
|
neneqd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> -. r = ( Q ` i ) ) |
465 |
|
velsn |
|- ( r e. { ( Q ` i ) } <-> r = ( Q ` i ) ) |
466 |
464 465
|
sylnibr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> -. r e. { ( Q ` i ) } ) |
467 |
357 466
|
eldifd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) -> r e. ( dom ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) \ { ( Q ` i ) } ) ) |
468 |
467
|
ralrimiva |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> A. r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) r e. ( dom ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) \ { ( Q ` i ) } ) ) |
469 |
|
dfss3 |
|- ( ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) C_ ( dom ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) \ { ( Q ` i ) } ) <-> A. r e. ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) r e. ( dom ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) \ { ( Q ` i ) } ) ) |
470 |
468 469
|
sylibr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ran ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) C_ ( dom ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) \ { ( Q ` i ) } ) ) |
471 |
|
icossre |
|- ( ( ( H ` i ) e. RR /\ ( H ` ( i + 1 ) ) e. RR* ) -> ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) C_ RR ) |
472 |
230 251 471
|
syl2anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) C_ RR ) |
473 |
472 379
|
sstrdi |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) C_ CC ) |
474 |
202
|
ad2antrr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) -> X e. CC ) |
475 |
473
|
sselda |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) -> s e. CC ) |
476 |
474 475
|
addcld |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) -> ( X + s ) e. CC ) |
477 |
366 375 473 381 476
|
cncfmptssg |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) -cn-> CC ) ) |
478 |
|
eqid |
|- ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) = ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) |
479 |
386 478 392
|
cncfcn |
|- ( ( ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) C_ CC /\ CC C_ CC ) -> ( ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) -cn-> CC ) = ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) Cn ( TopOpen ` CCfld ) ) ) |
480 |
473 381 479
|
syl2anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) -cn-> CC ) = ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) Cn ( TopOpen ` CCfld ) ) ) |
481 |
477 480
|
eleqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) Cn ( TopOpen ` CCfld ) ) ) |
482 |
|
resttopon |
|- ( ( ( TopOpen ` CCfld ) e. ( TopOn ` CC ) /\ ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) C_ CC ) -> ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) e. ( TopOn ` ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) ) |
483 |
397 473 482
|
syl2anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) e. ( TopOn ` ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) ) |
484 |
|
cncnp |
|- ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) e. ( TopOn ` ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) /\ ( TopOpen ` CCfld ) e. ( TopOn ` CC ) ) -> ( ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) Cn ( TopOpen ` CCfld ) ) <-> ( ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) : ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) --> CC /\ A. t e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` t ) ) ) ) |
485 |
483 397 484
|
syl2anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) Cn ( TopOpen ` CCfld ) ) <-> ( ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) : ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) --> CC /\ A. t e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` t ) ) ) ) |
486 |
481 485
|
mpbid |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) : ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) --> CC /\ A. t e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` t ) ) ) |
487 |
486
|
simprd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> A. t e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` t ) ) |
488 |
|
lbico1 |
|- ( ( ( H ` i ) e. RR* /\ ( H ` ( i + 1 ) ) e. RR* /\ ( H ` i ) < ( H ` ( i + 1 ) ) ) -> ( H ` i ) e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) |
489 |
376 251 180 488
|
syl3anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( H ` i ) e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) |
490 |
|
fveq2 |
|- ( t = ( H ` i ) -> ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` t ) = ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` ( H ` i ) ) ) |
491 |
490
|
eleq2d |
|- ( t = ( H ` i ) -> ( ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` t ) <-> ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` ( H ` i ) ) ) ) |
492 |
491
|
rspccva |
|- ( ( A. t e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` t ) /\ ( H ` i ) e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) -> ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` ( H ` i ) ) ) |
493 |
487 489 492
|
syl2anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` ( H ` i ) ) ) |
494 |
|
uncom |
|- ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) = ( { ( H ` i ) } u. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) |
495 |
|
snunioo |
|- ( ( ( H ` i ) e. RR* /\ ( H ` ( i + 1 ) ) e. RR* /\ ( H ` i ) < ( H ` ( i + 1 ) ) ) -> ( { ( H ` i ) } u. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) = ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) |
496 |
376 251 180 495
|
syl3anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( { ( H ` i ) } u. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) = ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) |
497 |
494 496
|
syl5eq |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) = ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) |
498 |
|
iftrue |
|- ( s = ( H ` i ) -> if ( s = ( H ` i ) , ( Q ` i ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) = ( Q ` i ) ) |
499 |
498
|
adantl |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s = ( H ` i ) ) -> if ( s = ( H ` i ) , ( Q ` i ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) = ( Q ` i ) ) |
500 |
246
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s = ( H ` i ) ) -> ( Q ` i ) = ( X + ( H ` i ) ) ) |
501 |
|
oveq2 |
|- ( s = ( H ` i ) -> ( X + s ) = ( X + ( H ` i ) ) ) |
502 |
501
|
eqcomd |
|- ( s = ( H ` i ) -> ( X + ( H ` i ) ) = ( X + s ) ) |
503 |
502
|
adantl |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s = ( H ` i ) ) -> ( X + ( H ` i ) ) = ( X + s ) ) |
504 |
499 500 503
|
3eqtrd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s = ( H ` i ) ) -> if ( s = ( H ` i ) , ( Q ` i ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) = ( X + s ) ) |
505 |
504
|
adantlr |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) /\ s = ( H ` i ) ) -> if ( s = ( H ` i ) , ( Q ` i ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) = ( X + s ) ) |
506 |
|
iffalse |
|- ( -. s = ( H ` i ) -> if ( s = ( H ` i ) , ( Q ` i ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) = ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) |
507 |
506
|
adantl |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) /\ -. s = ( H ` i ) ) -> if ( s = ( H ` i ) , ( Q ` i ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) = ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) |
508 |
|
eqidd |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) /\ -. s = ( H ` i ) ) -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) = ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) |
509 |
422
|
adantl |
|- ( ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) /\ -. s = ( H ` i ) ) /\ t = s ) -> ( X + t ) = ( X + s ) ) |
510 |
|
velsn |
|- ( s e. { ( H ` i ) } <-> s = ( H ` i ) ) |
511 |
510
|
notbii |
|- ( -. s e. { ( H ` i ) } <-> -. s = ( H ` i ) ) |
512 |
|
elun |
|- ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) <-> ( s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) \/ s e. { ( H ` i ) } ) ) |
513 |
512
|
biimpi |
|- ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) -> ( s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) \/ s e. { ( H ` i ) } ) ) |
514 |
513
|
orcomd |
|- ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) -> ( s e. { ( H ` i ) } \/ s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) ) |
515 |
514
|
ord |
|- ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) -> ( -. s e. { ( H ` i ) } -> s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) ) |
516 |
511 515
|
syl5bir |
|- ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) -> ( -. s = ( H ` i ) -> s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) ) |
517 |
516
|
imp |
|- ( ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) /\ -. s = ( H ` i ) ) -> s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) |
518 |
517
|
adantll |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) /\ -. s = ( H ` i ) ) -> s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) ) |
519 |
5
|
ad2antrr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) -> X e. RR ) |
520 |
|
elsni |
|- ( s e. { ( H ` i ) } -> s = ( H ` i ) ) |
521 |
520
|
adantl |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. { ( H ` i ) } ) -> s = ( H ` i ) ) |
522 |
230
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. { ( H ` i ) } ) -> ( H ` i ) e. RR ) |
523 |
521 522
|
eqeltrd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. { ( H ` i ) } ) -> s e. RR ) |
524 |
435 523
|
jaodan |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ ( s e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) \/ s e. { ( H ` i ) } ) ) -> s e. RR ) |
525 |
512 524
|
sylan2b |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) -> s e. RR ) |
526 |
519 525
|
readdcld |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) -> ( X + s ) e. RR ) |
527 |
526
|
adantr |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) /\ -. s = ( H ` i ) ) -> ( X + s ) e. RR ) |
528 |
508 509 518 527
|
fvmptd |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) /\ -. s = ( H ` i ) ) -> ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) = ( X + s ) ) |
529 |
507 528
|
eqtrd |
|- ( ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) /\ -. s = ( H ` i ) ) -> if ( s = ( H ` i ) , ( Q ` i ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) = ( X + s ) ) |
530 |
505 529
|
pm2.61dan |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) -> if ( s = ( H ` i ) , ( Q ` i ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) = ( X + s ) ) |
531 |
497 530
|
mpteq12dva |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) |-> if ( s = ( H ` i ) , ( Q ` i ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) ) = ( s e. ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) |-> ( X + s ) ) ) |
532 |
497
|
oveq2d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( TopOpen ` CCfld ) |`t ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) = ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) ) |
533 |
532
|
oveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( ( TopOpen ` CCfld ) |`t ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) CnP ( TopOpen ` CCfld ) ) = ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ) |
534 |
533
|
fveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( ( ( TopOpen ` CCfld ) |`t ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) CnP ( TopOpen ` CCfld ) ) ` ( H ` i ) ) = ( ( ( ( TopOpen ` CCfld ) |`t ( ( H ` i ) [,) ( H ` ( i + 1 ) ) ) ) CnP ( TopOpen ` CCfld ) ) ` ( H ` i ) ) ) |
535 |
493 531 534
|
3eltr4d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) |-> if ( s = ( H ` i ) , ( Q ` i ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) CnP ( TopOpen ` CCfld ) ) ` ( H ` i ) ) ) |
536 |
|
eqid |
|- ( ( TopOpen ` CCfld ) |`t ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) = ( ( TopOpen ` CCfld ) |`t ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) |
537 |
|
eqid |
|- ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) |-> if ( s = ( H ` i ) , ( Q ` i ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) ) = ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) |-> if ( s = ( H ` i ) , ( Q ` i ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) ) |
538 |
230
|
recnd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( H ` i ) e. CC ) |
539 |
536 386 537 454 305 538
|
ellimc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( Q ` i ) e. ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) limCC ( H ` i ) ) <-> ( s e. ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) |-> if ( s = ( H ` i ) , ( Q ` i ) , ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ` s ) ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) u. { ( H ` i ) } ) ) CnP ( TopOpen ` CCfld ) ) ` ( H ` i ) ) ) ) |
540 |
535 539
|
mpbird |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` i ) e. ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) limCC ( H ` i ) ) ) |
541 |
470 540 8
|
limccog |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> R e. ( ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) o. ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) limCC ( H ` i ) ) ) |
542 |
459
|
oveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) o. ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( X + t ) ) ) limCC ( H ` i ) ) = ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( F ` ( X + t ) ) ) limCC ( H ` i ) ) ) |
543 |
541 542
|
eleqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> R e. ( ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( F ` ( X + t ) ) ) limCC ( H ` i ) ) ) |
544 |
230 231 310 461 543
|
iblcncfioo |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( t e. ( ( H ` i ) (,) ( H ` ( i + 1 ) ) ) |-> ( F ` ( X + t ) ) ) e. L^1 ) |
545 |
6
|
ad2antrr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) ) -> F : ( -u _pi [,] _pi ) --> CC ) |
546 |
54
|
a1i |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) ) -> -u _pi e. RR* ) |
547 |
56
|
a1i |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) ) -> _pi e. RR* ) |
548 |
27
|
ad2antrr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) ) -> Q : ( 0 ... M ) --> ( -u _pi [,] _pi ) ) |
549 |
|
simplr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) ) -> i e. ( 0 ..^ M ) ) |
550 |
|
simpr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) ) -> t e. ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) ) |
551 |
169 179
|
oveq12d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) = ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) |
552 |
551
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) ) -> ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) = ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) |
553 |
550 552
|
eleqtrd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) ) -> t e. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ) |
554 |
553 117
|
syldan |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) ) -> ( X + t ) e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) |
555 |
546 547 548 549 554
|
fourierdlem1 |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) ) -> ( X + t ) e. ( -u _pi [,] _pi ) ) |
556 |
545 555
|
ffvelrnd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ t e. ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) ) -> ( F ` ( X + t ) ) e. CC ) |
557 |
230 231 544 556
|
ibliooicc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( t e. ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) |-> ( F ` ( X + t ) ) ) e. L^1 ) |
558 |
20 26 165 180 229 557
|
itgspltprt |
|- ( ph -> S. ( ( H ` 0 ) [,] ( H ` M ) ) ( F ` ( X + t ) ) _d t = sum_ i e. ( 0 ..^ M ) S. ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) ( F ` ( X + t ) ) _d t ) |
559 |
551
|
itgeq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> S. ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) ( F ` ( X + t ) ) _d t = S. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ( F ` ( X + t ) ) _d t ) |
560 |
559
|
sumeq2dv |
|- ( ph -> sum_ i e. ( 0 ..^ M ) S. ( ( H ` i ) [,] ( H ` ( i + 1 ) ) ) ( F ` ( X + t ) ) _d t = sum_ i e. ( 0 ..^ M ) S. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ( F ` ( X + t ) ) _d t ) |
561 |
558 560
|
eqtrd |
|- ( ph -> S. ( ( H ` 0 ) [,] ( H ` M ) ) ( F ` ( X + t ) ) _d t = sum_ i e. ( 0 ..^ M ) S. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ( F ` ( X + t ) ) _d t ) |
562 |
126 161 561
|
3eqtrd |
|- ( ph -> S. ( ( -u _pi - X ) [,] ( _pi - X ) ) ( F ` ( X + s ) ) _d s = sum_ i e. ( 0 ..^ M ) S. ( ( ( Q ` i ) - X ) [,] ( ( Q ` ( i + 1 ) ) - X ) ) ( F ` ( X + t ) ) _d t ) |
563 |
122 562
|
eqtr4d |
|- ( ph -> sum_ i e. ( 0 ..^ M ) S. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ( F ` t ) _d t = S. ( ( -u _pi - X ) [,] ( _pi - X ) ) ( F ` ( X + s ) ) _d s ) |
564 |
19 78 563
|
3eqtrd |
|- ( ph -> S. ( -u _pi [,] _pi ) ( F ` t ) _d t = S. ( ( -u _pi - X ) [,] ( _pi - X ) ) ( F ` ( X + s ) ) _d s ) |