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Metamath Proof Explorer

Theorem fourierdlem58

Description: The derivative of K is continuous on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses fourierdlem58.k 
|- K = ( s e. A |-> ( s / ( 2 x. ( sin ` ( s / 2 ) ) ) ) )
fourierdlem58.ass 
|- ( ph -> A C_ ( -u _pi [,] _pi ) )
fourierdlem58.0nA 
|- ( ph -> -. 0 e. A )
fourierdlem58.4 
|- ( ph -> A e. ( topGen ` ran (,) ) )
Assertion fourierdlem58 
|- ( ph -> ( RR _D K ) e. ( A -cn-> RR ) )

Proof

Step Hyp Ref Expression
1 fourierdlem58.k 
 |-  K = ( s e. A |-> ( s / ( 2 x. ( sin ` ( s / 2 ) ) ) ) )
2 fourierdlem58.ass 
 |-  ( ph -> A C_ ( -u _pi [,] _pi ) )
3 fourierdlem58.0nA 
 |-  ( ph -> -. 0 e. A )
4 fourierdlem58.4 
 |-  ( ph -> A e. ( topGen ` ran (,) ) )
5 pire 
 |-  _pi e. RR
6 5 a1i 
 |-  ( ( ph /\ s e. A ) -> _pi e. RR )
7 6 renegcld 
 |-  ( ( ph /\ s e. A ) -> -u _pi e. RR )
8 7 6 iccssred 
 |-  ( ( ph /\ s e. A ) -> ( -u _pi [,] _pi ) C_ RR )
9 2 sselda 
 |-  ( ( ph /\ s e. A ) -> s e. ( -u _pi [,] _pi ) )
10 8 9 sseldd 
 |-  ( ( ph /\ s e. A ) -> s e. RR )
11 2re 
 |-  2 e. RR
12 11 a1i 
 |-  ( ( ph /\ s e. A ) -> 2 e. RR )
13 10 rehalfcld 
 |-  ( ( ph /\ s e. A ) -> ( s / 2 ) e. RR )
14 13 resincld 
 |-  ( ( ph /\ s e. A ) -> ( sin ` ( s / 2 ) ) e. RR )
15 12 14 remulcld 
 |-  ( ( ph /\ s e. A ) -> ( 2 x. ( sin ` ( s / 2 ) ) ) e. RR )
16 2cnd 
 |-  ( ( ph /\ s e. A ) -> 2 e. CC )
17 10 recnd 
 |-  ( ( ph /\ s e. A ) -> s e. CC )
18 17 halfcld 
 |-  ( ( ph /\ s e. A ) -> ( s / 2 ) e. CC )
19 18 sincld 
 |-  ( ( ph /\ s e. A ) -> ( sin ` ( s / 2 ) ) e. CC )
20 2ne0 
 |-  2 =/= 0
21 20 a1i 
 |-  ( ( ph /\ s e. A ) -> 2 =/= 0 )
22 eqcom 
 |-  ( s = 0 <-> 0 = s )
23 22 biimpi 
 |-  ( s = 0 -> 0 = s )
24 23 adantl 
 |-  ( ( s e. A /\ s = 0 ) -> 0 = s )
25 simpl 
 |-  ( ( s e. A /\ s = 0 ) -> s e. A )
26 24 25 eqeltrd 
 |-  ( ( s e. A /\ s = 0 ) -> 0 e. A )
27 26 adantll 
 |-  ( ( ( ph /\ s e. A ) /\ s = 0 ) -> 0 e. A )
28 3 ad2antrr 
 |-  ( ( ( ph /\ s e. A ) /\ s = 0 ) -> -. 0 e. A )
29 27 28 pm2.65da 
 |-  ( ( ph /\ s e. A ) -> -. s = 0 )
30 29 neqned 
 |-  ( ( ph /\ s e. A ) -> s =/= 0 )
31 fourierdlem44 
 |-  ( ( s e. ( -u _pi [,] _pi ) /\ s =/= 0 ) -> ( sin ` ( s / 2 ) ) =/= 0 )
32 9 30 31 syl2anc 
 |-  ( ( ph /\ s e. A ) -> ( sin ` ( s / 2 ) ) =/= 0 )
33 16 19 21 32 mulne0d 
 |-  ( ( ph /\ s e. A ) -> ( 2 x. ( sin ` ( s / 2 ) ) ) =/= 0 )
34 10 15 33 redivcld 
 |-  ( ( ph /\ s e. A ) -> ( s / ( 2 x. ( sin ` ( s / 2 ) ) ) ) e. RR )
35 34 1 fmptd 
 |-  ( ph -> K : A --> RR )
36 5 a1i 
 |-  ( ph -> _pi e. RR )
37 36 renegcld 
 |-  ( ph -> -u _pi e. RR )
38 37 36 iccssred 
 |-  ( ph -> ( -u _pi [,] _pi ) C_ RR )
39 2 38 sstrd 
 |-  ( ph -> A C_ RR )
40 dvfre 
 |-  ( ( K : A --> RR /\ A C_ RR ) -> ( RR _D K ) : dom ( RR _D K ) --> RR )
41 35 39 40 syl2anc 
 |-  ( ph -> ( RR _D K ) : dom ( RR _D K ) --> RR )
42 eqidd 
 |-  ( ph -> ( s e. A |-> s ) = ( s e. A |-> s ) )
43 eqidd 
 |-  ( ph -> ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) = ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) )
44 4 10 15 42 43 offval2 
 |-  ( ph -> ( ( s e. A |-> s ) oF / ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) = ( s e. A |-> ( s / ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) )
45 1 44 eqtr4id 
 |-  ( ph -> K = ( ( s e. A |-> s ) oF / ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) )
46 45 oveq2d 
 |-  ( ph -> ( RR _D K ) = ( RR _D ( ( s e. A |-> s ) oF / ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) ) )
47 reelprrecn 
 |-  RR e. { RR , CC }
48 47 a1i 
 |-  ( ph -> RR e. { RR , CC } )
49 eqid 
 |-  ( s e. A |-> s ) = ( s e. A |-> s )
50 17 49 fmptd 
 |-  ( ph -> ( s e. A |-> s ) : A --> CC )
51 16 19 mulcld 
 |-  ( ( ph /\ s e. A ) -> ( 2 x. ( sin ` ( s / 2 ) ) ) e. CC )
52 33 neneqd 
 |-  ( ( ph /\ s e. A ) -> -. ( 2 x. ( sin ` ( s / 2 ) ) ) = 0 )
53 elsng 
 |-  ( ( 2 x. ( sin ` ( s / 2 ) ) ) e. RR -> ( ( 2 x. ( sin ` ( s / 2 ) ) ) e. { 0 } <-> ( 2 x. ( sin ` ( s / 2 ) ) ) = 0 ) )
54 15 53 syl 
 |-  ( ( ph /\ s e. A ) -> ( ( 2 x. ( sin ` ( s / 2 ) ) ) e. { 0 } <-> ( 2 x. ( sin ` ( s / 2 ) ) ) = 0 ) )
55 52 54 mtbird 
 |-  ( ( ph /\ s e. A ) -> -. ( 2 x. ( sin ` ( s / 2 ) ) ) e. { 0 } )
56 51 55 eldifd 
 |-  ( ( ph /\ s e. A ) -> ( 2 x. ( sin ` ( s / 2 ) ) ) e. ( CC \ { 0 } ) )
57 eqid 
 |-  ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) = ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) )
58 56 57 fmptd 
 |-  ( ph -> ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) : A --> ( CC \ { 0 } ) )
59 tgioo4 
 |-  ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR )
60 4 59 eleqtrdi 
 |-  ( ph -> A e. ( ( TopOpen ` CCfld ) |`t RR ) )
61 48 60 dvmptidg 
 |-  ( ph -> ( RR _D ( s e. A |-> s ) ) = ( s e. A |-> 1 ) )
62 ax-resscn 
 |-  RR C_ CC
63 62 a1i 
 |-  ( ph -> RR C_ CC )
64 39 63 sstrd 
 |-  ( ph -> A C_ CC )
65 1cnd 
 |-  ( ph -> 1 e. CC )
66 ssid 
 |-  CC C_ CC
67 66 a1i 
 |-  ( ph -> CC C_ CC )
68 64 65 67 constcncfg 
 |-  ( ph -> ( s e. A |-> 1 ) e. ( A -cn-> CC ) )
69 61 68 eqeltrd 
 |-  ( ph -> ( RR _D ( s e. A |-> s ) ) e. ( A -cn-> CC ) )
70 39 resmptd 
 |-  ( ph -> ( ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) |` A ) = ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) )
71 70 eqcomd 
 |-  ( ph -> ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) = ( ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) |` A ) )
72 71 oveq2d 
 |-  ( ph -> ( RR _D ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) = ( RR _D ( ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) |` A ) ) )
73 eqid 
 |-  ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) = ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) )
74 2cnd 
 |-  ( s e. RR -> 2 e. CC )
75 recn 
 |-  ( s e. RR -> s e. CC )
76 75 halfcld 
 |-  ( s e. RR -> ( s / 2 ) e. CC )
77 76 sincld 
 |-  ( s e. RR -> ( sin ` ( s / 2 ) ) e. CC )
78 74 77 mulcld 
 |-  ( s e. RR -> ( 2 x. ( sin ` ( s / 2 ) ) ) e. CC )
79 73 78 fmpti 
 |-  ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) : RR --> CC
80 79 a1i 
 |-  ( ph -> ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) : RR --> CC )
81 ssid 
 |-  RR C_ RR
82 81 a1i 
 |-  ( ph -> RR C_ RR )
83 eqid 
 |-  ( TopOpen ` CCfld ) = ( TopOpen ` CCfld )
84 83 59 dvres 
 |-  ( ( ( RR C_ CC /\ ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) : RR --> CC ) /\ ( RR C_ RR /\ A C_ RR ) ) -> ( RR _D ( ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) |` A ) ) = ( ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` A ) ) )
85 63 80 82 39 84 syl22anc 
 |-  ( ph -> ( RR _D ( ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) |` A ) ) = ( ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` A ) ) )
86 retop 
 |-  ( topGen ` ran (,) ) e. Top
87 86 a1i 
 |-  ( ph -> ( topGen ` ran (,) ) e. Top )
88 uniretop 
 |-  RR = U. ( topGen ` ran (,) )
89 88 isopn3 
 |-  ( ( ( topGen ` ran (,) ) e. Top /\ A C_ RR ) -> ( A e. ( topGen ` ran (,) ) <-> ( ( int ` ( topGen ` ran (,) ) ) ` A ) = A ) )
90 87 39 89 syl2anc 
 |-  ( ph -> ( A e. ( topGen ` ran (,) ) <-> ( ( int ` ( topGen ` ran (,) ) ) ` A ) = A ) )
91 4 90 mpbid 
 |-  ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` A ) = A )
92 91 reseq2d 
 |-  ( ph -> ( ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` A ) ) = ( ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |` A ) )
93 resmpt 
 |-  ( RR C_ CC -> ( ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) |` RR ) = ( s e. RR |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) )
94 62 93 ax-mp 
 |-  ( ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) |` RR ) = ( s e. RR |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) )
95 id 
 |-  ( s e. CC -> s e. CC )
96 2cnd 
 |-  ( s e. CC -> 2 e. CC )
97 20 a1i 
 |-  ( s e. CC -> 2 =/= 0 )
98 95 96 97 divrec2d 
 |-  ( s e. CC -> ( s / 2 ) = ( ( 1 / 2 ) x. s ) )
99 98 eqcomd 
 |-  ( s e. CC -> ( ( 1 / 2 ) x. s ) = ( s / 2 ) )
100 75 99 syl 
 |-  ( s e. RR -> ( ( 1 / 2 ) x. s ) = ( s / 2 ) )
101 100 fveq2d 
 |-  ( s e. RR -> ( sin ` ( ( 1 / 2 ) x. s ) ) = ( sin ` ( s / 2 ) ) )
102 101 oveq2d 
 |-  ( s e. RR -> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) = ( 2 x. ( sin ` ( s / 2 ) ) ) )
103 102 mpteq2ia 
 |-  ( s e. RR |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) = ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) )
104 94 103 eqtr2i 
 |-  ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) = ( ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) |` RR )
105 104 oveq2i 
 |-  ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) = ( RR _D ( ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) |` RR ) )
106 eqid 
 |-  ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) = ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) )
107 halfcn 
 |-  ( 1 / 2 ) e. CC
108 107 a1i 
 |-  ( s e. CC -> ( 1 / 2 ) e. CC )
109 108 95 mulcld 
 |-  ( s e. CC -> ( ( 1 / 2 ) x. s ) e. CC )
110 109 sincld 
 |-  ( s e. CC -> ( sin ` ( ( 1 / 2 ) x. s ) ) e. CC )
111 96 110 mulcld 
 |-  ( s e. CC -> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) e. CC )
112 106 111 fmpti 
 |-  ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) : CC --> CC
113 2cn 
 |-  2 e. CC
114 dvasinbx 
 |-  ( ( 2 e. CC /\ ( 1 / 2 ) e. CC ) -> ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) = ( s e. CC |-> ( ( 2 x. ( 1 / 2 ) ) x. ( cos ` ( ( 1 / 2 ) x. s ) ) ) ) )
115 113 107 114 mp2an 
 |-  ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) = ( s e. CC |-> ( ( 2 x. ( 1 / 2 ) ) x. ( cos ` ( ( 1 / 2 ) x. s ) ) ) )
116 113 20 recidi 
 |-  ( 2 x. ( 1 / 2 ) ) = 1
117 116 a1i 
 |-  ( s e. CC -> ( 2 x. ( 1 / 2 ) ) = 1 )
118 99 fveq2d 
 |-  ( s e. CC -> ( cos ` ( ( 1 / 2 ) x. s ) ) = ( cos ` ( s / 2 ) ) )
119 117 118 oveq12d 
 |-  ( s e. CC -> ( ( 2 x. ( 1 / 2 ) ) x. ( cos ` ( ( 1 / 2 ) x. s ) ) ) = ( 1 x. ( cos ` ( s / 2 ) ) ) )
120 halfcl 
 |-  ( s e. CC -> ( s / 2 ) e. CC )
121 120 coscld 
 |-  ( s e. CC -> ( cos ` ( s / 2 ) ) e. CC )
122 121 mullidd 
 |-  ( s e. CC -> ( 1 x. ( cos ` ( s / 2 ) ) ) = ( cos ` ( s / 2 ) ) )
123 119 122 eqtrd 
 |-  ( s e. CC -> ( ( 2 x. ( 1 / 2 ) ) x. ( cos ` ( ( 1 / 2 ) x. s ) ) ) = ( cos ` ( s / 2 ) ) )
124 123 mpteq2ia 
 |-  ( s e. CC |-> ( ( 2 x. ( 1 / 2 ) ) x. ( cos ` ( ( 1 / 2 ) x. s ) ) ) ) = ( s e. CC |-> ( cos ` ( s / 2 ) ) )
125 115 124 eqtri 
 |-  ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) = ( s e. CC |-> ( cos ` ( s / 2 ) ) )
126 125 dmeqi 
 |-  dom ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) = dom ( s e. CC |-> ( cos ` ( s / 2 ) ) )
127 dmmptg 
 |-  ( A. s e. CC ( cos ` ( s / 2 ) ) e. CC -> dom ( s e. CC |-> ( cos ` ( s / 2 ) ) ) = CC )
128 127 121 mprg 
 |-  dom ( s e. CC |-> ( cos ` ( s / 2 ) ) ) = CC
129 126 128 eqtri 
 |-  dom ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) = CC
130 62 129 sseqtrri 
 |-  RR C_ dom ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) )
131 dvres3 
 |-  ( ( ( RR e. { RR , CC } /\ ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) : CC --> CC ) /\ ( CC C_ CC /\ RR C_ dom ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) ) ) -> ( RR _D ( ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) |` RR ) ) = ( ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) |` RR ) )
132 47 112 66 130 131 mp4an 
 |-  ( RR _D ( ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) |` RR ) ) = ( ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) |` RR )
133 125 reseq1i 
 |-  ( ( CC _D ( s e. CC |-> ( 2 x. ( sin ` ( ( 1 / 2 ) x. s ) ) ) ) ) |` RR ) = ( ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |` RR )
134 105 132 133 3eqtri 
 |-  ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) = ( ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |` RR )
135 134 reseq1i 
 |-  ( ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |` A ) = ( ( ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |` RR ) |` A )
136 135 a1i 
 |-  ( ph -> ( ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |` A ) = ( ( ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |` RR ) |` A ) )
137 39 resabs1d 
 |-  ( ph -> ( ( ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |` RR ) |` A ) = ( ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |` A ) )
138 64 resmptd 
 |-  ( ph -> ( ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |` A ) = ( s e. A |-> ( cos ` ( s / 2 ) ) ) )
139 137 138 eqtrd 
 |-  ( ph -> ( ( ( s e. CC |-> ( cos ` ( s / 2 ) ) ) |` RR ) |` A ) = ( s e. A |-> ( cos ` ( s / 2 ) ) ) )
140 92 136 139 3eqtrd 
 |-  ( ph -> ( ( RR _D ( s e. RR |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` A ) ) = ( s e. A |-> ( cos ` ( s / 2 ) ) ) )
141 72 85 140 3eqtrd 
 |-  ( ph -> ( RR _D ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) = ( s e. A |-> ( cos ` ( s / 2 ) ) ) )
142 coscn 
 |-  cos e. ( CC -cn-> CC )
143 142 a1i 
 |-  ( ph -> cos e. ( CC -cn-> CC ) )
144 64 67 idcncfg 
 |-  ( ph -> ( s e. A |-> s ) e. ( A -cn-> CC ) )
145 2cnd 
 |-  ( ph -> 2 e. CC )
146 20 a1i 
 |-  ( ph -> 2 =/= 0 )
147 eldifsn 
 |-  ( 2 e. ( CC \ { 0 } ) <-> ( 2 e. CC /\ 2 =/= 0 ) )
148 145 146 147 sylanbrc 
 |-  ( ph -> 2 e. ( CC \ { 0 } ) )
149 difssd 
 |-  ( ph -> ( CC \ { 0 } ) C_ CC )
150 64 148 149 constcncfg 
 |-  ( ph -> ( s e. A |-> 2 ) e. ( A -cn-> ( CC \ { 0 } ) ) )
151 144 150 divcncf 
 |-  ( ph -> ( s e. A |-> ( s / 2 ) ) e. ( A -cn-> CC ) )
152 143 151 cncfmpt1f 
 |-  ( ph -> ( s e. A |-> ( cos ` ( s / 2 ) ) ) e. ( A -cn-> CC ) )
153 141 152 eqeltrd 
 |-  ( ph -> ( RR _D ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) e. ( A -cn-> CC ) )
154 48 50 58 69 153 dvdivcncf 
 |-  ( ph -> ( RR _D ( ( s e. A |-> s ) oF / ( s e. A |-> ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) ) e. ( A -cn-> CC ) )
155 46 154 eqeltrd 
 |-  ( ph -> ( RR _D K ) e. ( A -cn-> CC ) )
156 cncff 
 |-  ( ( RR _D K ) e. ( A -cn-> CC ) -> ( RR _D K ) : A --> CC )
157 fdm 
 |-  ( ( RR _D K ) : A --> CC -> dom ( RR _D K ) = A )
158 155 156 157 3syl 
 |-  ( ph -> dom ( RR _D K ) = A )
159 158 feq2d 
 |-  ( ph -> ( ( RR _D K ) : dom ( RR _D K ) --> RR <-> ( RR _D K ) : A --> RR ) )
160 41 159 mpbid 
 |-  ( ph -> ( RR _D K ) : A --> RR )
161 cncfcdm 
 |-  ( ( RR C_ CC /\ ( RR _D K ) e. ( A -cn-> CC ) ) -> ( ( RR _D K ) e. ( A -cn-> RR ) <-> ( RR _D K ) : A --> RR ) )
162 63 155 161 syl2anc 
 |-  ( ph -> ( ( RR _D K ) e. ( A -cn-> RR ) <-> ( RR _D K ) : A --> RR ) )
163 160 162 mpbird 
 |-  ( ph -> ( RR _D K ) e. ( A -cn-> RR ) )