Theorem fourierdlem33 40357

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 481	. . 3 |- ( ( ph /\ D = B ) -> L e. ( F lim CC B ) )
3		fourierdlem33.y	. . . . 5 |- Y = if ( D = B , L , ( F ` D ) )
4		iftrue 4092	. . . . 5 |- ( D = B -> if ( D = B , L , ( F ` D ) ) = L )
5	3, 4	syl5req 2669	. . . 4 |- ( D = B -> L = Y )
6	5	adantl 482	. . 3 |- ( ( ph /\ D = B ) -> L = Y )
7		oveq2 6658	. . . . 5 |- ( D = B -> ( ( F |` ( C (,) D ) ) lim CC D ) = ( ( F |` ( C (,) D ) ) lim CC B ) )
8	7	adantl 482	. . . 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 22696	. . . . . . 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 481	. . . . 5 |- ( ( ph /\ D = B ) -> F : ( A (,) B ) --> CC )
13		fourierdlem33.ss	. . . . . 6 |- ( ph -> ( C (,) D ) C_ ( A (,) B ) )
14	13	adantr 481	. . . . 5 |- ( ( ph /\ D = B ) -> ( C (,) D ) C_ ( A (,) B ) )
15		ioosscn 39716	. . . . . 6 |- ( A (,) B ) C_ CC
16	15	a1i 11	. . . . 5 |- ( ( ph /\ D = B ) -> ( A (,) B ) C_ CC )
17		eqid 2622	. . . . 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 10594	. . . . . . . . 9 |- ( ph -> D <_ D )
22		fourierdlem33.6	. . . . . . . . . . 11 |- ( ph -> C e. RR )
23	22	rexrd 10089	. . . . . . . . . 10 |- ( ph -> C e. RR* )
24		elioc2 12236	. . . . . . . . . 10 |- ( ( C e. RR* /\ D e. RR ) -> ( D e. ( C (,] D ) <-> ( D e. RR /\ C < D /\ D <_ D ) ) )
25	23, 19, 24	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( D e. ( C (,] D ) <-> ( D e. RR /\ C < D /\ D <_ D ) ) )
26	19, 20, 21, 25	mpbir3and 1245	. . . . . . . 8 |- ( ph -> D e. ( C (,] D ) )
27	26	adantr 481	. . . . . . 7 |- ( ( ph /\ D = B ) -> D e. ( C (,] D ) )
28		eqcom 2629	. . . . . . . . 9 |- ( D = B <-> B = D )
29	28	biimpi 206	. . . . . . . 8 |- ( D = B -> B = D )
30	29	adantl 482	. . . . . . 7 |- ( ( ph /\ D = B ) -> B = D )
31	17	cnfldtop 22587	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) e. Top
32		fourierdlem33.1	. . . . . . . . . . . . . 14 |- ( ph -> A e. RR )
33	32	rexrd 10089	. . . . . . . . . . . . 13 |- ( ph -> A e. RR* )
34		fourierdlem33.2	. . . . . . . . . . . . . 14 |- ( ph -> B e. RR )
35	34	rexrd 10089	. . . . . . . . . . . . 13 |- ( ph -> B e. RR* )
36		fourierdlem33.3	. . . . . . . . . . . . 13 |- ( ph -> A < B )
37		snunioo2 39731	. . . . . . . . . . . . 13 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A (,) B ) u. { B } ) = ( A (,] B ) )
38	33, 35, 36, 37	syl3anc 1326	. . . . . . . . . . . 12 |- ( ph -> ( ( A (,) B ) u. { B } ) = ( A (,] B ) )
39		ovex 6678	. . . . . . . . . . . . 13 |- ( A (,] B ) e. _V
40	39	a1i 11	. . . . . . . . . . . 12 |- ( ph -> ( A (,] B ) e. _V )
41	38, 40	eqeltrd 2701	. . . . . . . . . . 11 |- ( ph -> ( ( A (,) B ) u. { B } ) e. _V )
42		resttop 20964	. . . . . . . . . . 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 695	. . . . . . . . . 10 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) e. Top )
44	18, 43	syl5eqel 2705	. . . . . . . . 9 |- ( ph -> J e. Top )
45	44	adantr 481	. . . . . . . 8 |- ( ( ph /\ D = B ) -> J e. Top )
46		oveq2 6658	. . . . . . . . . . 11 |- ( D = B -> ( C (,] D ) = ( C (,] B ) )
47	46	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ D = B ) -> ( C (,] D ) = ( C (,] B ) )
48	23	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> C e. RR* )
49		pnfxr 10092	. . . . . . . . . . . . . . . . 17 |- +oo e. RR*
50	49	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> +oo e. RR* )
51		simpr 477	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( C (,] B ) ) -> x e. ( C (,] B ) )
52	34	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ x e. ( C (,] B ) ) -> B e. RR )
53		elioc2 12236	. . . . . . . . . . . . . . . . . . 19 |- ( ( C e. RR* /\ B e. RR ) -> ( x e. ( C (,] B ) <-> ( x e. RR /\ C < x /\ x <_ B ) ) )
54	48, 52, 53	syl2anc 693	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( C (,] B ) ) -> ( x e. ( C (,] B ) <-> ( x e. RR /\ C < x /\ x <_ B ) ) )
55	51, 54	mpbid 222	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( C (,] B ) ) -> ( x e. RR /\ C < x /\ x <_ B ) )
56	55	simp1d 1073	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> x e. RR )
57	55	simp2d 1074	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> C < x )
58	56	ltpnfd 11955	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> x < +oo )
59	48, 50, 56, 57, 58	eliood 39720	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( C (,] B ) ) -> x e. ( C (,) +oo ) )
60	32	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( C (,] B ) ) -> A e. RR )
61	22	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( C (,] B ) ) -> C e. RR )
62	32, 34, 22, 19, 20, 13	fourierdlem10 40334	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( A <_ C /\ D <_ B ) )
63	62	simpld 475	. . . . . . . . . . . . . . . . . 18 |- ( ph -> A <_ C )
64	63	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( C (,] B ) ) -> A <_ C )
65	60, 61, 56, 64, 57	lelttrd 10195	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> A < x )
66	55	simp3d 1075	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> x <_ B )
67	33	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( C (,] B ) ) -> A e. RR* )
68		elioc2 12236	. . . . . . . . . . . . . . . . 17 |- ( ( A e. RR* /\ B e. RR ) -> ( x e. ( A (,] B ) <-> ( x e. RR /\ A < x /\ x <_ B ) ) )
69	67, 52, 68	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> ( x e. ( A (,] B ) <-> ( x e. RR /\ A < x /\ x <_ B ) ) )
70	56, 65, 66, 69	mpbir3and 1245	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( C (,] B ) ) -> x e. ( A (,] B ) )
71	59, 70	elind 3798	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( C (,] B ) ) -> x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) )
72		elinel1 3799	. . . . . . . . . . . . . . . . 17 |- ( x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) -> x e. ( C (,) +oo ) )
73		elioore 12205	. . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> x e. RR )
76	23	adantr 481	. . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> x e. ( C (,) +oo ) )
79		ioogtlb 39717	. . . . . . . . . . . . . . . 16 |- ( ( C e. RR* /\ +oo e. RR* /\ x e. ( C (,) +oo ) ) -> C < x )
80	76, 77, 78, 79	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> C < x )
81		elinel2 3800	. . . . . . . . . . . . . . . . . 18 |- ( x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) -> x e. ( A (,] B ) )
82	81	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> x e. ( A (,] B ) )
83	33	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> A e. RR* )
84	34	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> B e. RR )
85	83, 84, 68	syl2anc 693	. . . . . . . . . . . . . . . . 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 222	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> ( x e. RR /\ A < x /\ x <_ B ) )
87	86	simp3d 1075	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> x <_ B )
88	76, 84, 53	syl2anc 693	. . . . . . . . . . . . . . 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 1245	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> x e. ( C (,] B ) )
90	71, 89	impbida 877	. . . . . . . . . . . . 13 |- ( ph -> ( x e. ( C (,] B ) <-> x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) )
91	90	eqrdv 2620	. . . . . . . . . . . 12 |- ( ph -> ( C (,] B ) = ( ( C (,) +oo ) i^i ( A (,] B ) ) )
92		retop 22565	. . . . . . . . . . . . . 14 |- ( topGen ` ran (,) ) e. Top
93	92	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> ( topGen ` ran (,) ) e. Top )
94		iooretop 22569	. . . . . . . . . . . . . 14 |- ( C (,) +oo ) e. ( topGen ` ran (,) )
95	94	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> ( C (,) +oo ) e. ( topGen ` ran (,) ) )
96		elrestr 16089	. . . . . . . . . . . . 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 1326	. . . . . . . . . . . 12 |- ( ph -> ( ( C (,) +oo ) i^i ( A (,] B ) ) e. ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) )
98	91, 97	eqeltrd 2701	. . . . . . . . . . 11 |- ( ph -> ( C (,] B ) e. ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) )
99	98	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ D = B ) -> ( C (,] B ) e. ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) )
100	47, 99	eqeltrd 2701	. . . . . . . . 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 6666	. . . . . . . . . . 11 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A (,] B ) ) )
103	17	tgioo2 22606	. . . . . . . . . . . . . 14 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
104	103	eqcomi 2631	. . . . . . . . . . . . 13 |- ( ( TopOpen ` ℂfld ) ↾t RR ) = ( topGen ` ran (,) )
105	104	oveq1i 6660	. . . . . . . . . . . 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 12253	. . . . . . . . . . . . . 14 |- ( ( A e. RR* /\ B e. RR ) -> ( A (,] B ) C_ RR )
108	33, 34, 107	syl2anc 693	. . . . . . . . . . . . 13 |- ( ph -> ( A (,] B ) C_ RR )
109		reex 10027	. . . . . . . . . . . . . 14 |- RR e. _V
110	109	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> RR e. _V )
111		restabs 20969	. . . . . . . . . . . . 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 1326	. . . . . . . . . . . 12 |- ( ph -> ( ( ( TopOpen ` ℂfld ) ↾t RR ) ↾t ( A (,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A (,] B ) ) )
113	105, 112	syl5reqr 2671	. . . . . . . . . . 11 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A (,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) )
114	101, 102, 113	3eqtrrd 2661	. . . . . . . . . 10 |- ( ph -> ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) = J )
115	114	adantr 481	. . . . . . . . 9 |- ( ( ph /\ D = B ) -> ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) = J )
116	100, 115	eleqtrd 2703	. . . . . . . 8 |- ( ( ph /\ D = B ) -> ( C (,] D ) e. J )
117		isopn3i 20886	. . . . . . . 8 |- ( ( J e. Top /\ ( C (,] D ) e. J ) -> ( ( int ` J ) ` ( C (,] D ) ) = ( C (,] D ) )
118	45, 116, 117	syl2anc 693	. . . . . . 7 |- ( ( ph /\ D = B ) -> ( ( int ` J ) ` ( C (,] D ) ) = ( C (,] D ) )
119	27, 30, 118	3eltr4d 2716	. . . . . 6 |- ( ( ph /\ D = B ) -> B e. ( ( int ` J ) ` ( C (,] D ) ) )
120		sneq 4187	. . . . . . . . . . 11 |- ( D = B -> { D } = { B } )
121	120	eqcomd 2628	. . . . . . . . . 10 |- ( D = B -> { B } = { D } )
122	121	uneq2d 3767	. . . . . . . . 9 |- ( D = B -> ( ( C (,) D ) u. { B } ) = ( ( C (,) D ) u. { D } ) )
123	122	adantl 482	. . . . . . . 8 |- ( ( ph /\ D = B ) -> ( ( C (,) D ) u. { B } ) = ( ( C (,) D ) u. { D } ) )
124	19	rexrd 10089	. . . . . . . . . 10 |- ( ph -> D e. RR* )
125		snunioo2 39731	. . . . . . . . . 10 |- ( ( C e. RR* /\ D e. RR* /\ C < D ) -> ( ( C (,) D ) u. { D } ) = ( C (,] D ) )
126	23, 124, 20, 125	syl3anc 1326	. . . . . . . . 9 |- ( ph -> ( ( C (,) D ) u. { D } ) = ( C (,] D ) )
127	126	adantr 481	. . . . . . . 8 |- ( ( ph /\ D = B ) -> ( ( C (,) D ) u. { D } ) = ( C (,] D ) )
128	123, 127	eqtr2d 2657	. . . . . . 7 |- ( ( ph /\ D = B ) -> ( C (,] D ) = ( ( C (,) D ) u. { B } ) )
129	128	fveq2d 6195	. . . . . 6 |- ( ( ph /\ D = B ) -> ( ( int ` J ) ` ( C (,] D ) ) = ( ( int ` J ) ` ( ( C (,) D ) u. { B } ) ) )
130	119, 129	eleqtrd 2703	. . . . 5 |- ( ( ph /\ D = B ) -> B e. ( ( int ` J ) ` ( ( C (,) D ) u. { B } ) ) )
131	12, 14, 16, 17, 18, 130	limcres 23650	. . . 4 |- ( ( ph /\ D = B ) -> ( ( F |` ( C (,) D ) ) lim CC B ) = ( F lim CC B ) )
132	8, 131	eqtr2d 2657	. . 3 |- ( ( ph /\ D = B ) -> ( F lim CC B ) = ( ( F |` ( C (,) D ) ) lim CC D ) )
133	2, 6, 132	3eltr3d 2715	. 2 |- ( ( ph /\ D = B ) -> Y e. ( ( F |` ( C (,) D ) ) lim CC D ) )
134		limcresi 23649	. . 3 |- ( F lim CC D ) C_ ( ( F |` ( C (,) D ) ) lim CC D )
135		iffalse 4095	. . . . . 6 |- ( -. D = B -> if ( D = B , L , ( F ` D ) ) = ( F ` D ) )
136	3, 135	syl5eq 2668	. . . . 5 |- ( -. D = B -> Y = ( F ` D ) )
137	136	adantl 482	. . . 4 |- ( ( ph /\ -. D = B ) -> Y = ( F ` D ) )
138		ssid 3624	. . . . . . . . . . . . 13 |- CC C_ CC
139	138	a1i 11	. . . . . . . . . . . 12 |- ( ph -> CC C_ CC )
140		eqid 2622	. . . . . . . . . . . . 13 |- ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) )
141		unicntop 22589	. . . . . . . . . . . . . . . 16 |- CC = U. ( TopOpen ` ℂfld )
142	141	restid 16094	. . . . . . . . . . . . . . 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 2631	. . . . . . . . . . . . 13 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
145	17, 140, 144	cncfcn 22712	. . . . . . . . . . . 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 695	. . . . . . . . . . 11 |- ( ph -> ( ( A (,) B ) -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) Cn ( TopOpen ` ℂfld ) ) )
147	9, 146	eleqtrd 2703	. . . . . . . . . 10 |- ( ph -> F e. ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) Cn ( TopOpen ` ℂfld ) ) )
148	17	cnfldtopon 22586	. . . . . . . . . . . 12 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
149	15	a1i 11	. . . . . . . . . . . 12 |- ( ph -> ( A (,) B ) C_ CC )
150		resttopon 20965	. . . . . . . . . . . 12 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ ( A (,) B ) C_ CC ) -> ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) e. (TopOn ` ( A (,) B ) ) )
151	148, 149, 150	sylancr 695	. . . . . . . . . . 11 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) e. (TopOn ` ( A (,) B ) ) )
152	148	a1i 11	. . . . . . . . . . 11 |- ( ph -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
153		cncnp 21084	. . . . . . . . . . 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 693	. . . . . . . . . 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 222	. . . . . . . . 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 479	. . . . . . . 8 |- ( ph -> A. x e. ( A (,) B ) F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) )
157	156	adantr 481	. . . . . . 7 |- ( ( ph /\ -. D = B ) -> A. x e. ( A (,) B ) F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) )
158	33	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. D = B ) -> A e. RR* )
159	35	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. D = B ) -> B e. RR* )
160	19	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. D = B ) -> D e. RR )
161	32, 22, 19, 63, 20	lelttrd 10195	. . . . . . . . 9 |- ( ph -> A < D )
162	161	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. D = B ) -> A < D )
163	34	adantr 481	. . . . . . . . 9 |- ( ( ph /\ -. D = B ) -> B e. RR )
164	62	simprd 479	. . . . . . . . . 10 |- ( ph -> D <_ B )
165	164	adantr 481	. . . . . . . . 9 |- ( ( ph /\ -. D = B ) -> D <_ B )
166		neqne 2802	. . . . . . . . . . 11 |- ( -. D = B -> D =/= B )
167	166	necomd 2849	. . . . . . . . . 10 |- ( -. D = B -> B =/= D )
168	167	adantl 482	. . . . . . . . 9 |- ( ( ph /\ -. D = B ) -> B =/= D )
169	160, 163, 165, 168	leneltd 10191	. . . . . . . 8 |- ( ( ph /\ -. D = B ) -> D < B )
170	158, 159, 160, 162, 169	eliood 39720	. . . . . . 7 |- ( ( ph /\ -. D = B ) -> D e. ( A (,) B ) )
171		fveq2 6191	. . . . . . . . 9 |- ( x = D -> ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) = ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` D ) )
172	171	eleq2d 2687	. . . . . . . 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 3308	. . . . . . 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 693	. . . . . 6 |- ( ( ph /\ -. D = B ) -> F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` D ) )
175	17, 140	cnplimc 23651	. . . . . . 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 695	. . . . . 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 222	. . . . 5 |- ( ( ph /\ -. D = B ) -> ( F : ( A (,) B ) --> CC /\ ( F ` D ) e. ( F lim CC D ) ) )
178	177	simprd 479	. . . 4 |- ( ( ph /\ -. D = B ) -> ( F ` D ) e. ( F lim CC D ) )
179	137, 178	eqeltrd 2701	. . 3 |- ( ( ph /\ -. D = B ) -> Y e. ( F lim CC D ) )
180	134, 179	sseldi 3601	. 2 |- ( ( ph /\ -. D = B ) -> Y e. ( ( F |` ( C (,) D ) ) lim CC D ) )
181	133, 180	pm2.61dan 832	1 |- ( ph -> Y e. ( ( F |` ( C (,) D ) ) lim CC D ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   ifcif 4086   {csn 4177   class class class wbr 4653   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175   (,]cioc 12176   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   Topctop 20698  TopOnctopon 20715   intcnt 20821   Cn ccn 21028   CnP ccnp 21029   -cn->ccncf 22679   lim CC climc 23626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-icc 12182  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-ntr 20824  df-cn 21031  df-cnp 21032  df-xms 22125  df-ms 22126  df-cncf 22681  df-limc 23630

This theorem is referenced by:  fourierdlem49  40372  fourierdlem76  40399  fourierdlem91  40414