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