Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem85.p |
|- P = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = ( -u _pi + X ) /\ ( p ` m ) = ( _pi + X ) ) /\ A. i e. ( 0 ..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } ) |
2 |
|
fourierdlem85.f |
|- ( ph -> F : RR --> RR ) |
3 |
|
fourierdlem85.x |
|- ( ph -> X e. ran V ) |
4 |
|
fourierdlem85.y |
|- ( ph -> Y e. ( ( F |` ( X (,) +oo ) ) limCC X ) ) |
5 |
|
fourierdlem85.w |
|- ( ph -> W e. RR ) |
6 |
|
fourierdlem85.h |
|- H = ( s e. ( -u _pi [,] _pi ) |-> if ( s = 0 , 0 , ( ( ( F ` ( X + s ) ) - if ( 0 < s , Y , W ) ) / s ) ) ) |
7 |
|
fourierdlem85.k |
|- K = ( s e. ( -u _pi [,] _pi ) |-> if ( s = 0 , 1 , ( s / ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) ) |
8 |
|
fourierdlem85.u |
|- U = ( s e. ( -u _pi [,] _pi ) |-> ( ( H ` s ) x. ( K ` s ) ) ) |
9 |
|
fourierdlem85.n |
|- ( ph -> N e. RR ) |
10 |
|
fourierdlem85.s |
|- S = ( s e. ( -u _pi [,] _pi ) |-> ( sin ` ( ( N + ( 1 / 2 ) ) x. s ) ) ) |
11 |
|
fourierdlem85.g |
|- G = ( s e. ( -u _pi [,] _pi ) |-> ( ( U ` s ) x. ( S ` s ) ) ) |
12 |
|
fourierdlem85.m |
|- ( ph -> M e. NN ) |
13 |
|
fourierdlem85.v |
|- ( ph -> V e. ( P ` M ) ) |
14 |
|
fourierdlem85.r |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> R e. ( ( F |` ( ( V ` i ) (,) ( V ` ( i + 1 ) ) ) ) limCC ( V ` i ) ) ) |
15 |
|
fourierdlem85.q |
|- Q = ( i e. ( 0 ... M ) |-> ( ( V ` i ) - X ) ) |
16 |
|
fourierdlem85.o |
|- O = ( 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 ) ) ) } ) |
17 |
|
fourierdlem85.i |
|- I = ( RR _D F ) |
18 |
|
fourierdlem85.ifn |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( I |` ( ( V ` i ) (,) ( V ` ( i + 1 ) ) ) ) : ( ( V ` i ) (,) ( V ` ( i + 1 ) ) ) --> CC ) |
19 |
|
fourierdlem85.e |
|- ( ph -> E e. ( ( I |` ( X (,) +oo ) ) limCC X ) ) |
20 |
|
fourierdlem85.a |
|- A = ( ( if ( ( V ` i ) = X , E , ( ( R - if ( ( V ` i ) < X , W , Y ) ) / ( Q ` i ) ) ) x. ( K ` ( Q ` i ) ) ) x. ( S ` ( Q ` i ) ) ) |
21 |
|
eqid |
|- ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( U ` s ) ) = ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( U ` s ) ) |
22 |
|
eqid |
|- ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( S ` s ) ) = ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( S ` s ) ) |
23 |
|
eqid |
|- ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( ( U ` s ) x. ( S ` s ) ) ) = ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( ( U ` s ) x. ( S ` s ) ) ) |
24 |
|
pire |
|- _pi e. RR |
25 |
24
|
renegcli |
|- -u _pi e. RR |
26 |
25
|
rexri |
|- -u _pi e. RR* |
27 |
26
|
a1i |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> -u _pi e. RR* ) |
28 |
24
|
rexri |
|- _pi e. RR* |
29 |
28
|
a1i |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> _pi e. RR* ) |
30 |
24
|
a1i |
|- ( ph -> _pi e. RR ) |
31 |
30
|
renegcld |
|- ( ph -> -u _pi e. RR ) |
32 |
1
|
fourierdlem2 |
|- ( M e. NN -> ( V e. ( P ` M ) <-> ( V e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( V ` 0 ) = ( -u _pi + X ) /\ ( V ` M ) = ( _pi + X ) ) /\ A. i e. ( 0 ..^ M ) ( V ` i ) < ( V ` ( i + 1 ) ) ) ) ) ) |
33 |
12 32
|
syl |
|- ( ph -> ( V e. ( P ` M ) <-> ( V e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( V ` 0 ) = ( -u _pi + X ) /\ ( V ` M ) = ( _pi + X ) ) /\ A. i e. ( 0 ..^ M ) ( V ` i ) < ( V ` ( i + 1 ) ) ) ) ) ) |
34 |
13 33
|
mpbid |
|- ( ph -> ( V e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( V ` 0 ) = ( -u _pi + X ) /\ ( V ` M ) = ( _pi + X ) ) /\ A. i e. ( 0 ..^ M ) ( V ` i ) < ( V ` ( i + 1 ) ) ) ) ) |
35 |
34
|
simpld |
|- ( ph -> V e. ( RR ^m ( 0 ... M ) ) ) |
36 |
|
elmapi |
|- ( V e. ( RR ^m ( 0 ... M ) ) -> V : ( 0 ... M ) --> RR ) |
37 |
|
frn |
|- ( V : ( 0 ... M ) --> RR -> ran V C_ RR ) |
38 |
35 36 37
|
3syl |
|- ( ph -> ran V C_ RR ) |
39 |
38 3
|
sseldd |
|- ( ph -> X e. RR ) |
40 |
31 30 39 1 16 12 13 15
|
fourierdlem14 |
|- ( ph -> Q e. ( O ` M ) ) |
41 |
16 12 40
|
fourierdlem15 |
|- ( ph -> Q : ( 0 ... M ) --> ( -u _pi [,] _pi ) ) |
42 |
41
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> Q : ( 0 ... M ) --> ( -u _pi [,] _pi ) ) |
43 |
42
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> Q : ( 0 ... M ) --> ( -u _pi [,] _pi ) ) |
44 |
|
simplr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> i e. ( 0 ..^ M ) ) |
45 |
27 29 43 44
|
fourierdlem8 |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) C_ ( -u _pi [,] _pi ) ) |
46 |
|
ioossicc |
|- ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) |
47 |
46
|
sseli |
|- ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -> s e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) |
48 |
47
|
adantl |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> s e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) |
49 |
45 48
|
sseldd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> s e. ( -u _pi [,] _pi ) ) |
50 |
|
ioossre |
|- ( X (,) +oo ) C_ RR |
51 |
50
|
a1i |
|- ( ph -> ( X (,) +oo ) C_ RR ) |
52 |
2 51
|
fssresd |
|- ( ph -> ( F |` ( X (,) +oo ) ) : ( X (,) +oo ) --> RR ) |
53 |
|
ax-resscn |
|- RR C_ CC |
54 |
51 53
|
sstrdi |
|- ( ph -> ( X (,) +oo ) C_ CC ) |
55 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
56 |
|
pnfxr |
|- +oo e. RR* |
57 |
56
|
a1i |
|- ( ph -> +oo e. RR* ) |
58 |
39
|
ltpnfd |
|- ( ph -> X < +oo ) |
59 |
55 57 39 58
|
lptioo1cn |
|- ( ph -> X e. ( ( limPt ` ( TopOpen ` CCfld ) ) ` ( X (,) +oo ) ) ) |
60 |
52 54 59 4
|
limcrecl |
|- ( ph -> Y e. RR ) |
61 |
2 39 60 5 6
|
fourierdlem9 |
|- ( ph -> H : ( -u _pi [,] _pi ) --> RR ) |
62 |
53
|
a1i |
|- ( ph -> RR C_ CC ) |
63 |
61 62
|
fssd |
|- ( ph -> H : ( -u _pi [,] _pi ) --> CC ) |
64 |
63
|
ad2antrr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> H : ( -u _pi [,] _pi ) --> CC ) |
65 |
64 49
|
ffvelrnd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( H ` s ) e. CC ) |
66 |
7
|
fourierdlem43 |
|- K : ( -u _pi [,] _pi ) --> RR |
67 |
66
|
a1i |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> K : ( -u _pi [,] _pi ) --> RR ) |
68 |
67 49
|
ffvelrnd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( K ` s ) e. RR ) |
69 |
68
|
recnd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( K ` s ) e. CC ) |
70 |
65 69
|
mulcld |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( ( H ` s ) x. ( K ` s ) ) e. CC ) |
71 |
8
|
fvmpt2 |
|- ( ( s e. ( -u _pi [,] _pi ) /\ ( ( H ` s ) x. ( K ` s ) ) e. CC ) -> ( U ` s ) = ( ( H ` s ) x. ( K ` s ) ) ) |
72 |
49 70 71
|
syl2anc |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( U ` s ) = ( ( H ` s ) x. ( K ` s ) ) ) |
73 |
72 70
|
eqeltrd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( U ` s ) e. CC ) |
74 |
9 10
|
fourierdlem18 |
|- ( ph -> S e. ( ( -u _pi [,] _pi ) -cn-> RR ) ) |
75 |
|
cncff |
|- ( S e. ( ( -u _pi [,] _pi ) -cn-> RR ) -> S : ( -u _pi [,] _pi ) --> RR ) |
76 |
74 75
|
syl |
|- ( ph -> S : ( -u _pi [,] _pi ) --> RR ) |
77 |
76
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> S : ( -u _pi [,] _pi ) --> RR ) |
78 |
77
|
adantr |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> S : ( -u _pi [,] _pi ) --> RR ) |
79 |
78 49
|
ffvelrnd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( S ` s ) e. RR ) |
80 |
79
|
recnd |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( S ` s ) e. CC ) |
81 |
|
eqid |
|- ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( H ` s ) ) = ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( H ` s ) ) |
82 |
|
eqid |
|- ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( K ` s ) ) = ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( K ` s ) ) |
83 |
|
eqid |
|- ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( ( H ` s ) x. ( K ` s ) ) ) = ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( ( H ` s ) x. ( K ` s ) ) ) |
84 |
|
eqid |
|- if ( ( V ` i ) = X , E , ( ( R - if ( ( V ` i ) < X , W , Y ) ) / ( Q ` i ) ) ) = if ( ( V ` i ) = X , E , ( ( R - if ( ( V ` i ) < X , W , Y ) ) / ( Q ` i ) ) ) |
85 |
39 1 2 3 4 5 6 12 13 14 15 16 17 18 19 84
|
fourierdlem75 |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> if ( ( V ` i ) = X , E , ( ( R - if ( ( V ` i ) < X , W , Y ) ) / ( Q ` i ) ) ) e. ( ( H |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) ) |
86 |
61
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> H : ( -u _pi [,] _pi ) --> RR ) |
87 |
26
|
a1i |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> -u _pi e. RR* ) |
88 |
28
|
a1i |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> _pi e. RR* ) |
89 |
|
simpr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> i e. ( 0 ..^ M ) ) |
90 |
87 88 42 89
|
fourierdlem8 |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) C_ ( -u _pi [,] _pi ) ) |
91 |
46 90
|
sstrid |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ ( -u _pi [,] _pi ) ) |
92 |
86 91
|
feqresmpt |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( H |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( H ` s ) ) ) |
93 |
92
|
oveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( H |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) = ( ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( H ` s ) ) limCC ( Q ` i ) ) ) |
94 |
85 93
|
eleqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> if ( ( V ` i ) = X , E , ( ( R - if ( ( V ` i ) < X , W , Y ) ) / ( Q ` i ) ) ) e. ( ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( H ` s ) ) limCC ( Q ` i ) ) ) |
95 |
|
limcresi |
|- ( K limCC ( Q ` i ) ) C_ ( ( K |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) |
96 |
|
ssid |
|- CC C_ CC |
97 |
|
cncfss |
|- ( ( RR C_ CC /\ CC C_ CC ) -> ( ( -u _pi [,] _pi ) -cn-> RR ) C_ ( ( -u _pi [,] _pi ) -cn-> CC ) ) |
98 |
53 96 97
|
mp2an |
|- ( ( -u _pi [,] _pi ) -cn-> RR ) C_ ( ( -u _pi [,] _pi ) -cn-> CC ) |
99 |
7
|
fourierdlem62 |
|- K e. ( ( -u _pi [,] _pi ) -cn-> RR ) |
100 |
98 99
|
sselii |
|- K e. ( ( -u _pi [,] _pi ) -cn-> CC ) |
101 |
100
|
a1i |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> K e. ( ( -u _pi [,] _pi ) -cn-> CC ) ) |
102 |
|
elfzofz |
|- ( i e. ( 0 ..^ M ) -> i e. ( 0 ... M ) ) |
103 |
102
|
adantl |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> i e. ( 0 ... M ) ) |
104 |
42 103
|
ffvelrnd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` i ) e. ( -u _pi [,] _pi ) ) |
105 |
101 104
|
cnlimci |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( K ` ( Q ` i ) ) e. ( K limCC ( Q ` i ) ) ) |
106 |
95 105
|
sseldi |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( K ` ( Q ` i ) ) e. ( ( K |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) ) |
107 |
|
cncff |
|- ( K e. ( ( -u _pi [,] _pi ) -cn-> CC ) -> K : ( -u _pi [,] _pi ) --> CC ) |
108 |
100 107
|
mp1i |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> K : ( -u _pi [,] _pi ) --> CC ) |
109 |
108 91
|
feqresmpt |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( K |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( K ` s ) ) ) |
110 |
109
|
oveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( K |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) = ( ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( K ` s ) ) limCC ( Q ` i ) ) ) |
111 |
106 110
|
eleqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( K ` ( Q ` i ) ) e. ( ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( K ` s ) ) limCC ( Q ` i ) ) ) |
112 |
81 82 83 65 69 94 111
|
mullimc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( if ( ( V ` i ) = X , E , ( ( R - if ( ( V ` i ) < X , W , Y ) ) / ( Q ` i ) ) ) x. ( K ` ( Q ` i ) ) ) e. ( ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( ( H ` s ) x. ( K ` s ) ) ) limCC ( Q ` i ) ) ) |
113 |
72
|
mpteq2dva |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( U ` s ) ) = ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( ( H ` s ) x. ( K ` s ) ) ) ) |
114 |
113
|
oveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( U ` s ) ) limCC ( Q ` i ) ) = ( ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( ( H ` s ) x. ( K ` s ) ) ) limCC ( Q ` i ) ) ) |
115 |
112 114
|
eleqtrrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( if ( ( V ` i ) = X , E , ( ( R - if ( ( V ` i ) < X , W , Y ) ) / ( Q ` i ) ) ) x. ( K ` ( Q ` i ) ) ) e. ( ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( U ` s ) ) limCC ( Q ` i ) ) ) |
116 |
|
limcresi |
|- ( S limCC ( Q ` i ) ) C_ ( ( S |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) |
117 |
74
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> S e. ( ( -u _pi [,] _pi ) -cn-> RR ) ) |
118 |
117 104
|
cnlimci |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( S ` ( Q ` i ) ) e. ( S limCC ( Q ` i ) ) ) |
119 |
116 118
|
sseldi |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( S ` ( Q ` i ) ) e. ( ( S |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) ) |
120 |
77 91
|
feqresmpt |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( S |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( S ` s ) ) ) |
121 |
120
|
oveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( S |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) = ( ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( S ` s ) ) limCC ( Q ` i ) ) ) |
122 |
119 121
|
eleqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( S ` ( Q ` i ) ) e. ( ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( S ` s ) ) limCC ( Q ` i ) ) ) |
123 |
21 22 23 73 80 115 122
|
mullimc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( if ( ( V ` i ) = X , E , ( ( R - if ( ( V ` i ) < X , W , Y ) ) / ( Q ` i ) ) ) x. ( K ` ( Q ` i ) ) ) x. ( S ` ( Q ` i ) ) ) e. ( ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( ( U ` s ) x. ( S ` s ) ) ) limCC ( Q ` i ) ) ) |
124 |
20 123
|
eqeltrid |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> A e. ( ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( ( U ` s ) x. ( S ` s ) ) ) limCC ( Q ` i ) ) ) |
125 |
11
|
reseq1i |
|- ( G |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( ( s e. ( -u _pi [,] _pi ) |-> ( ( U ` s ) x. ( S ` s ) ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) |
126 |
91
|
resmptd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( s e. ( -u _pi [,] _pi ) |-> ( ( U ` s ) x. ( S ` s ) ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( ( U ` s ) x. ( S ` s ) ) ) ) |
127 |
125 126
|
eqtr2id |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( ( U ` s ) x. ( S ` s ) ) ) = ( G |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) |
128 |
127
|
oveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( s e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( ( U ` s ) x. ( S ` s ) ) ) limCC ( Q ` i ) ) = ( ( G |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) ) |
129 |
124 128
|
eleqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> A e. ( ( G |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) ) |