Theorem fourierdlem33 38013

Description: Limit of a continuous function on an open subinterval. Upper bound version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem33.1	|- ( ph -> A e. RR )
fourierdlem33.2	|- ( ph -> B e. RR )
fourierdlem33.3	|- ( ph -> A < B )
fourierdlem33.4	|- ( ph -> F e. ( ( A (,) B ) -cn-> CC ) )
fourierdlem33.5	|- ( ph -> L e. ( F lim CC B ) )
fourierdlem33.6	|- ( ph -> C e. RR )
fourierdlem33.7	|- ( ph -> D e. RR )
fourierdlem33.8	|- ( ph -> C < D )
fourierdlem33.ss	|- ( ph -> ( C (,) D ) C_ ( A (,) B ) )
fourierdlem33.y	|- Y = if ( D = B , L , ( F ` D ) )
fourierdlem33.10	|- J = ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) )

Assertion

Ref	Expression
fourierdlem33	|- ( ph -> Y e. ( ( F |` ( C (,) D ) ) lim CC D ) )

Proof of Theorem fourierdlem33

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem33.5	. . . 4 |- ( ph -> L e. ( F lim CC B ) )
2	1	adantr 467	. . 3 |- ( ( ph /\ D = B ) -> L e. ( F lim CC B ) )
3		fourierdlem33.y	. . . . 5 |- Y = if ( D = B , L , ( F ` D ) )
4		iftrue 3889	. . . . 5 |- ( D = B -> if ( D = B , L , ( F ` D ) ) = L )
5	3, 4	syl5req 2500	. . . 4 |- ( D = B -> L = Y )
6	5	adantl 468	. . 3 |- ( ( ph /\ D = B ) -> L = Y )
7		oveq2 6303	. . . . 5 |- ( D = B -> ( ( F |` ( C (,) D ) ) lim CC D ) = ( ( F |` ( C (,) D ) ) lim CC B ) )
8	7	adantl 468	. . . 4 |- ( ( ph /\ D = B ) -> ( ( F |` ( C (,) D ) ) lim CC D ) = ( ( F |` ( C (,) D ) ) lim CC B ) )
9		fourierdlem33.4	. . . . . . 7 |- ( ph -> F e. ( ( A (,) B ) -cn-> CC ) )
10		cncff 21937	. . . . . . 7 |- ( F e. ( ( A (,) B ) -cn-> CC ) -> F : ( A (,) B ) --> CC )
11	9, 10	syl 17	. . . . . 6 |- ( ph -> F : ( A (,) B ) --> CC )
12	11	adantr 467	. . . . 5 |- ( ( ph /\ D = B ) -> F : ( A (,) B ) --> CC )
13		fourierdlem33.ss	. . . . . 6 |- ( ph -> ( C (,) D ) C_ ( A (,) B ) )
14	13	adantr 467	. . . . 5 |- ( ( ph /\ D = B ) -> ( C (,) D ) C_ ( A (,) B ) )
15		ioosscn 37601	. . . . . 6 |- ( A (,) B ) C_ CC
16	15	a1i 11	. . . . 5 |- ( ( ph /\ D = B ) -> ( A (,) B ) C_ CC )
17		eqid 2453	. . . . 5 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
18		fourierdlem33.10	. . . . 5 |- J = ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) )
19		fourierdlem33.7	. . . . . . . . 9 |- ( ph -> D e. RR )
20		fourierdlem33.8	. . . . . . . . 9 |- ( ph -> C < D )
21	19	leidd 10187	. . . . . . . . 9 |- ( ph -> D <_ D )
22		fourierdlem33.6	. . . . . . . . . . 11 |- ( ph -> C e. RR )
23	22	rexrd 9695	. . . . . . . . . 10 |- ( ph -> C e. RR* )
24		elioc2 11704	. . . . . . . . . 10 |- ( ( C e. RR* /\ D e. RR ) -> ( D e. ( C (,] D ) <-> ( D e. RR /\ C < D /\ D <_ D ) ) )
25	23, 19, 24	syl2anc 667	. . . . . . . . 9 |- ( ph -> ( D e. ( C (,] D ) <-> ( D e. RR /\ C < D /\ D <_ D ) ) )
26	19, 20, 21, 25	mpbir3and 1192	. . . . . . . 8 |- ( ph -> D e. ( C (,] D ) )
27	26	adantr 467	. . . . . . 7 |- ( ( ph /\ D = B ) -> D e. ( C (,] D ) )
28		eqcom 2460	. . . . . . . . 9 |- ( D = B <-> B = D )
29	28	biimpi 198	. . . . . . . 8 |- ( D = B -> B = D )
30	29	adantl 468	. . . . . . 7 |- ( ( ph /\ D = B ) -> B = D )
31	17	cnfldtop 21816	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) e. Top
32		fourierdlem33.1	. . . . . . . . . . . . . 14 |- ( ph -> A e. RR )
33	32	rexrd 9695	. . . . . . . . . . . . 13 |- ( ph -> A e. RR* )
34		fourierdlem33.2	. . . . . . . . . . . . . 14 |- ( ph -> B e. RR )
35	34	rexrd 9695	. . . . . . . . . . . . 13 |- ( ph -> B e. RR* )
36		fourierdlem33.3	. . . . . . . . . . . . 13 |- ( ph -> A < B )
37		snunioo2 37616	. . . . . . . . . . . . 13 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A (,) B ) u. { B } ) = ( A (,] B ) )
38	33, 35, 36, 37	syl3anc 1269	. . . . . . . . . . . 12 |- ( ph -> ( ( A (,) B ) u. { B } ) = ( A (,] B ) )
39		ovex 6323	. . . . . . . . . . . . 13 |- ( A (,] B ) e. _V
40	39	a1i 11	. . . . . . . . . . . 12 |- ( ph -> ( A (,] B ) e. _V )
41	38, 40	eqeltrd 2531	. . . . . . . . . . 11 |- ( ph -> ( ( A (,) B ) u. { B } ) e. _V )
42		resttop 20188	. . . . . . . . . . 11 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ ( ( A (,) B ) u. { B } ) e. _V ) -> ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) e. Top )
43	31, 41, 42	sylancr 670	. . . . . . . . . 10 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) e. Top )
44	18, 43	syl5eqel 2535	. . . . . . . . 9 |- ( ph -> J e. Top )
45	44	adantr 467	. . . . . . . 8 |- ( ( ph /\ D = B ) -> J e. Top )
46		oveq2 6303	. . . . . . . . . . 11 |- ( D = B -> ( C (,] D ) = ( C (,] B ) )
47	46	adantl 468	. . . . . . . . . 10 |- ( ( ph /\ D = B ) -> ( C (,] D ) = ( C (,] B ) )
48	23	adantr 467	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> C e. RR* )
49		pnfxr 11419	. . . . . . . . . . . . . . . . 17 |- +oo e. RR*
50	49	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> +oo e. RR* )
51		simpr 463	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( C (,] B ) ) -> x e. ( C (,] B ) )
52	34	adantr 467	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ x e. ( C (,] B ) ) -> B e. RR )
53		elioc2 11704	. . . . . . . . . . . . . . . . . . 19 |- ( ( C e. RR* /\ B e. RR ) -> ( x e. ( C (,] B ) <-> ( x e. RR /\ C < x /\ x <_ B ) ) )
54	48, 52, 53	syl2anc 667	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( C (,] B ) ) -> ( x e. ( C (,] B ) <-> ( x e. RR /\ C < x /\ x <_ B ) ) )
55	51, 54	mpbid 214	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( C (,] B ) ) -> ( x e. RR /\ C < x /\ x <_ B ) )
56	55	simp1d 1021	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> x e. RR )
57	55	simp2d 1022	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> C < x )
58	56	ltpnfd 11430	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> x < +oo )
59	48, 50, 56, 57, 58	eliood 37605	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( C (,] B ) ) -> x e. ( C (,) +oo ) )
60	32	adantr 467	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( C (,] B ) ) -> A e. RR )
61	22	adantr 467	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( C (,] B ) ) -> C e. RR )
62	32, 34, 22, 19, 20, 13	fourierdlem10 37989	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( A <_ C /\ D <_ B ) )
63	62	simpld 461	. . . . . . . . . . . . . . . . . 18 |- ( ph -> A <_ C )
64	63	adantr 467	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( C (,] B ) ) -> A <_ C )
65	60, 61, 56, 64, 57	lelttrd 9798	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> A < x )
66	55	simp3d 1023	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> x <_ B )
67	33	adantr 467	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( C (,] B ) ) -> A e. RR* )
68		elioc2 11704	. . . . . . . . . . . . . . . . 17 |- ( ( A e. RR* /\ B e. RR ) -> ( x e. ( A (,] B ) <-> ( x e. RR /\ A < x /\ x <_ B ) ) )
69	67, 52, 68	syl2anc 667	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> ( x e. ( A (,] B ) <-> ( x e. RR /\ A < x /\ x <_ B ) ) )
70	56, 65, 66, 69	mpbir3and 1192	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( C (,] B ) ) -> x e. ( A (,] B ) )
71	59, 70	elind 3620	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( C (,] B ) ) -> x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) )
72		elinel1 3621	. . . . . . . . . . . . . . . . 17 |- ( x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) -> x e. ( C (,) +oo ) )
73		elioore 11673	. . . . . . . . . . . . . . . . 17 |- ( x e. ( C (,) +oo ) -> x e. RR )
74	72, 73	syl 17	. . . . . . . . . . . . . . . 16 |- ( x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) -> x e. RR )
75	74	adantl 468	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> x e. RR )
76	23	adantr 467	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> C e. RR* )
77	49	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> +oo e. RR* )
78	72	adantl 468	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> x e. ( C (,) +oo ) )
79		ioogtlb 37602	. . . . . . . . . . . . . . . 16 |- ( ( C e. RR* /\ +oo e. RR* /\ x e. ( C (,) +oo ) ) -> C < x )
80	76, 77, 78, 79	syl3anc 1269	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> C < x )
81		elinel2 3622	. . . . . . . . . . . . . . . . . 18 |- ( x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) -> x e. ( A (,] B ) )
82	81	adantl 468	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> x e. ( A (,] B ) )
83	33	adantr 467	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> A e. RR* )
84	34	adantr 467	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> B e. RR )
85	83, 84, 68	syl2anc 667	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> ( x e. ( A (,] B ) <-> ( x e. RR /\ A < x /\ x <_ B ) ) )
86	82, 85	mpbid 214	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> ( x e. RR /\ A < x /\ x <_ B ) )
87	86	simp3d 1023	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> x <_ B )
88	76, 84, 53	syl2anc 667	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> ( x e. ( C (,] B ) <-> ( x e. RR /\ C < x /\ x <_ B ) ) )
89	75, 80, 87, 88	mpbir3and 1192	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> x e. ( C (,] B ) )
90	71, 89	impbida 844	. . . . . . . . . . . . 13 |- ( ph -> ( x e. ( C (,] B ) <-> x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) )
91	90	eqrdv 2451	. . . . . . . . . . . 12 |- ( ph -> ( C (,] B ) = ( ( C (,) +oo ) i^i ( A (,] B ) ) )
92		retop 21794	. . . . . . . . . . . . . 14 |- ( topGen ` ran (,) ) e. Top
93	92	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> ( topGen ` ran (,) ) e. Top )
94		iooretop 21798	. . . . . . . . . . . . . 14 |- ( C (,) +oo ) e. ( topGen ` ran (,) )
95	94	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> ( C (,) +oo ) e. ( topGen ` ran (,) ) )
96		elrestr 15339	. . . . . . . . . . . . 13 |- ( ( ( topGen ` ran (,) ) e. Top /\ ( A (,] B ) e. _V /\ ( C (,) +oo ) e. ( topGen ` ran (,) ) ) -> ( ( C (,) +oo ) i^i ( A (,] B ) ) e. ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) )
97	93, 40, 95, 96	syl3anc 1269	. . . . . . . . . . . 12 |- ( ph -> ( ( C (,) +oo ) i^i ( A (,] B ) ) e. ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) )
98	91, 97	eqeltrd 2531	. . . . . . . . . . 11 |- ( ph -> ( C (,] B ) e. ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) )
99	98	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ D = B ) -> ( C (,] B ) e. ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) )
100	47, 99	eqeltrd 2531	. . . . . . . . 9 |- ( ( ph /\ D = B ) -> ( C (,] D ) e. ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) )
101	18	a1i 11	. . . . . . . . . . 11 |- ( ph -> J = ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) )
102	38	oveq2d 6311	. . . . . . . . . . 11 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A (,] B ) ) )
103	17	tgioo2 21833	. . . . . . . . . . . . . 14 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
104	103	eqcomi 2462	. . . . . . . . . . . . 13 |- ( ( TopOpen ` ℂfld ) ↾t RR ) = ( topGen ` ran (,) )
105	104	oveq1i 6305	. . . . . . . . . . . 12 |- ( ( ( TopOpen ` ℂfld ) ↾t RR ) ↾t ( A (,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A (,] B ) )
106	31	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> ( TopOpen ` ℂfld ) e. Top )
107		iocssre 11721	. . . . . . . . . . . . . 14 |- ( ( A e. RR* /\ B e. RR ) -> ( A (,] B ) C_ RR )
108	33, 34, 107	syl2anc 667	. . . . . . . . . . . . 13 |- ( ph -> ( A (,] B ) C_ RR )
109		reex 9635	. . . . . . . . . . . . . 14 |- RR e. _V
110	109	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> RR e. _V )
111		restabs 20193	. . . . . . . . . . . . 13 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ ( A (,] B ) C_ RR /\ RR e. _V ) -> ( ( ( TopOpen ` ℂfld ) ↾t RR ) ↾t ( A (,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A (,] B ) ) )
112	106, 108, 110, 111	syl3anc 1269	. . . . . . . . . . . 12 |- ( ph -> ( ( ( TopOpen ` ℂfld ) ↾t RR ) ↾t ( A (,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A (,] B ) ) )
113	105, 112	syl5reqr 2502	. . . . . . . . . . 11 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A (,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) )
114	101, 102, 113	3eqtrrd 2492	. . . . . . . . . 10 |- ( ph -> ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) = J )
115	114	adantr 467	. . . . . . . . 9 |- ( ( ph /\ D = B ) -> ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) = J )
116	100, 115	eleqtrd 2533	. . . . . . . 8 |- ( ( ph /\ D = B ) -> ( C (,] D ) e. J )
117		isopn3i 20110	. . . . . . . 8 |- ( ( J e. Top /\ ( C (,] D ) e. J ) -> ( ( int ` J ) ` ( C (,] D ) ) = ( C (,] D ) )
118	45, 116, 117	syl2anc 667	. . . . . . 7 |- ( ( ph /\ D = B ) -> ( ( int ` J ) ` ( C (,] D ) ) = ( C (,] D ) )
119	27, 30, 118	3eltr4d 2546	. . . . . 6 |- ( ( ph /\ D = B ) -> B e. ( ( int ` J ) ` ( C (,] D ) ) )
120		sneq 3980	. . . . . . . . . . 11 |- ( D = B -> { D } = { B } )
121	120	eqcomd 2459	. . . . . . . . . 10 |- ( D = B -> { B } = { D } )
122	121	uneq2d 3590	. . . . . . . . 9 |- ( D = B -> ( ( C (,) D ) u. { B } ) = ( ( C (,) D ) u. { D } ) )
123	122	adantl 468	. . . . . . . 8 |- ( ( ph /\ D = B ) -> ( ( C (,) D ) u. { B } ) = ( ( C (,) D ) u. { D } ) )
124	19	rexrd 9695	. . . . . . . . . 10 |- ( ph -> D e. RR* )
125		snunioo2 37616	. . . . . . . . . 10 |- ( ( C e. RR* /\ D e. RR* /\ C < D ) -> ( ( C (,) D ) u. { D } ) = ( C (,] D ) )
126	23, 124, 20, 125	syl3anc 1269	. . . . . . . . 9 |- ( ph -> ( ( C (,) D ) u. { D } ) = ( C (,] D ) )
127	126	adantr 467	. . . . . . . 8 |- ( ( ph /\ D = B ) -> ( ( C (,) D ) u. { D } ) = ( C (,] D ) )
128	123, 127	eqtr2d 2488	. . . . . . 7 |- ( ( ph /\ D = B ) -> ( C (,] D ) = ( ( C (,) D ) u. { B } ) )
129	128	fveq2d 5874	. . . . . 6 |- ( ( ph /\ D = B ) -> ( ( int ` J ) ` ( C (,] D ) ) = ( ( int ` J ) ` ( ( C (,) D ) u. { B } ) ) )
130	119, 129	eleqtrd 2533	. . . . 5 |- ( ( ph /\ D = B ) -> B e. ( ( int ` J ) ` ( ( C (,) D ) u. { B } ) ) )
131	12, 14, 16, 17, 18, 130	limcres 22853	. . . 4 |- ( ( ph /\ D = B ) -> ( ( F |` ( C (,) D ) ) lim CC B ) = ( F lim CC B ) )
132	8, 131	eqtr2d 2488	. . 3 |- ( ( ph /\ D = B ) -> ( F lim CC B ) = ( ( F |` ( C (,) D ) ) lim CC D ) )
133	2, 6, 132	3eltr3d 2545	. 2 |- ( ( ph /\ D = B ) -> Y e. ( ( F |` ( C (,) D ) ) lim CC D ) )
134		limcresi 22852	. . 3 |- ( F lim CC D ) C_ ( ( F |` ( C (,) D ) ) lim CC D )
135		iffalse 3892	. . . . . 6 |- ( -. D = B -> if ( D = B , L , ( F ` D ) ) = ( F ` D ) )
136	3, 135	syl5eq 2499	. . . . 5 |- ( -. D = B -> Y = ( F ` D ) )
137	136	adantl 468	. . . 4 |- ( ( ph /\ -. D = B ) -> Y = ( F ` D ) )
138		ssid 3453	. . . . . . . . . . . . 13 |- CC C_ CC
139	138	a1i 11	. . . . . . . . . . . 12 |- ( ph -> CC C_ CC )
140		eqid 2453	. . . . . . . . . . . . 13 |- ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) )
141		unicntop 37381	. . . . . . . . . . . . . . . 16 |- CC = U. ( TopOpen ` ℂfld )
142	141	restid 15344	. . . . . . . . . . . . . . 15 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
143	31, 142	ax-mp 5	. . . . . . . . . . . . . 14 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
144	143	eqcomi 2462	. . . . . . . . . . . . 13 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
145	17, 140, 144	cncfcn 21953	. . . . . . . . . . . 12 |- ( ( ( A (,) B ) C_ CC /\ CC C_ CC ) -> ( ( A (,) B ) -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) Cn ( TopOpen ` ℂfld ) ) )
146	15, 139, 145	sylancr 670	. . . . . . . . . . 11 |- ( ph -> ( ( A (,) B ) -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) Cn ( TopOpen ` ℂfld ) ) )
147	9, 146	eleqtrd 2533	. . . . . . . . . 10 |- ( ph -> F e. ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) Cn ( TopOpen ` ℂfld ) ) )
148	17	cnfldtopon 21815	. . . . . . . . . . . 12 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
149	15	a1i 11	. . . . . . . . . . . 12 |- ( ph -> ( A (,) B ) C_ CC )
150		resttopon 20189	. . . . . . . . . . . 12 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ ( A (,) B ) C_ CC ) -> ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) e. (TopOn ` ( A (,) B ) ) )
151	148, 149, 150	sylancr 670	. . . . . . . . . . 11 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) e. (TopOn ` ( A (,) B ) ) )
152	148	a1i 11	. . . . . . . . . . 11 |- ( ph -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
153		cncnp 20308	. . . . . . . . . . 11 |- ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) e. (TopOn ` ( A (,) B ) ) /\ ( TopOpen ` ℂfld ) e. (TopOn ` CC ) ) -> ( F e. ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) Cn ( TopOpen ` ℂfld ) ) <-> ( F : ( A (,) B ) --> CC /\ A. x e. ( A (,) B ) F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) ) ) )
154	151, 152, 153	syl2anc 667	. . . . . . . . . 10 |- ( ph -> ( F e. ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) Cn ( TopOpen ` ℂfld ) ) <-> ( F : ( A (,) B ) --> CC /\ A. x e. ( A (,) B ) F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) ) ) )
155	147, 154	mpbid 214	. . . . . . . . 9 |- ( ph -> ( F : ( A (,) B ) --> CC /\ A. x e. ( A (,) B ) F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) ) )
156	155	simprd 465	. . . . . . . 8 |- ( ph -> A. x e. ( A (,) B ) F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) )
157	156	adantr 467	. . . . . . 7 |- ( ( ph /\ -. D = B ) -> A. x e. ( A (,) B ) F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) )
158	33	adantr 467	. . . . . . . 8 |- ( ( ph /\ -. D = B ) -> A e. RR* )
159	35	adantr 467	. . . . . . . 8 |- ( ( ph /\ -. D = B ) -> B e. RR* )
160	19	adantr 467	. . . . . . . 8 |- ( ( ph /\ -. D = B ) -> D e. RR )
161	32, 22, 19, 63, 20	lelttrd 9798	. . . . . . . . 9 |- ( ph -> A < D )
162	161	adantr 467	. . . . . . . 8 |- ( ( ph /\ -. D = B ) -> A < D )
163	34	adantr 467	. . . . . . . . 9 |- ( ( ph /\ -. D = B ) -> B e. RR )
164	62	simprd 465	. . . . . . . . . 10 |- ( ph -> D <_ B )
165	164	adantr 467	. . . . . . . . 9 |- ( ( ph /\ -. D = B ) -> D <_ B )
166		neqne 37384	. . . . . . . . . . 11 |- ( -. D = B -> D =/= B )
167	166	necomd 2681	. . . . . . . . . 10 |- ( -. D = B -> B =/= D )
168	167	adantl 468	. . . . . . . . 9 |- ( ( ph /\ -. D = B ) -> B =/= D )
169	160, 163, 165, 168	leneltd 9794	. . . . . . . 8 |- ( ( ph /\ -. D = B ) -> D < B )
170	158, 159, 160, 162, 169	eliood 37605	. . . . . . 7 |- ( ( ph /\ -. D = B ) -> D e. ( A (,) B ) )
171		fveq2 5870	. . . . . . . . 9 |- ( x = D -> ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) = ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` D ) )
172	171	eleq2d 2516	. . . . . . . 8 |- ( x = D -> ( F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) <-> F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` D ) ) )
173	172	rspccva 3151	. . . . . . 7 |- ( ( A. x e. ( A (,) B ) F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) /\ D e. ( A (,) B ) ) -> F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` D ) )
174	157, 170, 173	syl2anc 667	. . . . . 6 |- ( ( ph /\ -. D = B ) -> F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` D ) )
175	17, 140	cnplimc 22854	. . . . . . 7 |- ( ( ( A (,) B ) C_ CC /\ D e. ( A (,) B ) ) -> ( F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` D ) <-> ( F : ( A (,) B ) --> CC /\ ( F ` D ) e. ( F lim CC D ) ) ) )
176	15, 170, 175	sylancr 670	. . . . . 6 |- ( ( ph /\ -. D = B ) -> ( F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` D ) <-> ( F : ( A (,) B ) --> CC /\ ( F ` D ) e. ( F lim CC D ) ) ) )
177	174, 176	mpbid 214	. . . . 5 |- ( ( ph /\ -. D = B ) -> ( F : ( A (,) B ) --> CC /\ ( F ` D ) e. ( F lim CC D ) ) )
178	177	simprd 465	. . . 4 |- ( ( ph /\ -. D = B ) -> ( F ` D ) e. ( F lim CC D ) )
179	137, 178	eqeltrd 2531	. . 3 |- ( ( ph /\ -. D = B ) -> Y e. ( F lim CC D ) )
180	134, 179	sseldi 3432	. 2 |- ( ( ph /\ -. D = B ) -> Y e. ( ( F |` ( C (,) D ) ) lim CC D ) )
181	133, 180	pm2.61dan 801	1 |- ( ph -> Y e. ( ( F |` ( C (,) D ) ) lim CC D ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 986   = wceq 1446   e. wcel 1889   =/= wne 2624   A.wral 2739   _Vcvv 3047   u. cun 3404   i^i cin 3405   C_ wss 3406   ifcif 3883   {csn 3970   class class class wbr 4405   ran crn 4838   |` cres 4839   -->wf 5581   ` cfv 5585  (class class class)co 6295   CCcc 9542   RRcr 9543   +oocpnf 9677   RR*cxr 9679   < clt 9680   <_ cle 9681   (,)cioo 11642   (,]cioc 11643   ↾_t crest 15331   TopOpenctopn 15332   topGenctg 15348  ℂ_fldccnfld 18982   Topctop 19929  TopOnctopon 19930   intcnt 20044   Cn ccn 20252   CnP ccnp 20253   -cn->ccncf 21920   lim CC climc 22829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fi 7930  df-sup 7961  df-inf 7962  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ioo 11646  df-ioc 11647  df-icc 11649  df-fz 11792  df-seq 12221  df-exp 12280  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-plusg 15215  df-mulr 15216  df-starv 15217  df-tset 15221  df-ple 15222  df-ds 15224  df-unif 15225  df-rest 15333  df-topn 15334  df-topgen 15354  df-psmet 18974  df-xmet 18975  df-met 18976  df-bl 18977  df-mopn 18978  df-cnfld 18983  df-top 19933  df-bases 19934  df-topon 19935  df-topsp 19936  df-ntr 20047  df-cn 20255  df-cnp 20256  df-xms 21347  df-ms 21348  df-cncf 21922  df-limc 22833

This theorem is referenced by:  fourierdlem49  38029  fourierdlem76  38056  fourierdlem91  38071