Theorem fourierdlem33 38115

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 472	. . 3 |- ( ( ph /\ D = B ) -> L e. ( F lim CC B ) )
3		fourierdlem33.y	. . . . 5 |- Y = if ( D = B , L , ( F ` D ) )
4		iftrue 3878	. . . . 5 |- ( D = B -> if ( D = B , L , ( F ` D ) ) = L )
5	3, 4	syl5req 2518	. . . 4 |- ( D = B -> L = Y )
6	5	adantl 473	. . 3 |- ( ( ph /\ D = B ) -> L = Y )
7		oveq2 6316	. . . . 5 |- ( D = B -> ( ( F |` ( C (,) D ) ) lim CC D ) = ( ( F |` ( C (,) D ) ) lim CC B ) )
8	7	adantl 473	. . . 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 22003	. . . . . . 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 472	. . . . 5 |- ( ( ph /\ D = B ) -> F : ( A (,) B ) --> CC )
13		fourierdlem33.ss	. . . . . 6 |- ( ph -> ( C (,) D ) C_ ( A (,) B ) )
14	13	adantr 472	. . . . 5 |- ( ( ph /\ D = B ) -> ( C (,) D ) C_ ( A (,) B ) )
15		ioosscn 37687	. . . . . 6 |- ( A (,) B ) C_ CC
16	15	a1i 11	. . . . 5 |- ( ( ph /\ D = B ) -> ( A (,) B ) C_ CC )
17		eqid 2471	. . . . 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 10201	. . . . . . . . 9 |- ( ph -> D <_ D )
22		fourierdlem33.6	. . . . . . . . . . 11 |- ( ph -> C e. RR )
23	22	rexrd 9708	. . . . . . . . . 10 |- ( ph -> C e. RR* )
24		elioc2 11722	. . . . . . . . . 10 |- ( ( C e. RR* /\ D e. RR ) -> ( D e. ( C (,] D ) <-> ( D e. RR /\ C < D /\ D <_ D ) ) )
25	23, 19, 24	syl2anc 673	. . . . . . . . 9 |- ( ph -> ( D e. ( C (,] D ) <-> ( D e. RR /\ C < D /\ D <_ D ) ) )
26	19, 20, 21, 25	mpbir3and 1213	. . . . . . . 8 |- ( ph -> D e. ( C (,] D ) )
27	26	adantr 472	. . . . . . 7 |- ( ( ph /\ D = B ) -> D e. ( C (,] D ) )
28		eqcom 2478	. . . . . . . . 9 |- ( D = B <-> B = D )
29	28	biimpi 199	. . . . . . . 8 |- ( D = B -> B = D )
30	29	adantl 473	. . . . . . 7 |- ( ( ph /\ D = B ) -> B = D )
31	17	cnfldtop 21882	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) e. Top
32		fourierdlem33.1	. . . . . . . . . . . . . 14 |- ( ph -> A e. RR )
33	32	rexrd 9708	. . . . . . . . . . . . 13 |- ( ph -> A e. RR* )
34		fourierdlem33.2	. . . . . . . . . . . . . 14 |- ( ph -> B e. RR )
35	34	rexrd 9708	. . . . . . . . . . . . 13 |- ( ph -> B e. RR* )
36		fourierdlem33.3	. . . . . . . . . . . . 13 |- ( ph -> A < B )
37		snunioo2 37702	. . . . . . . . . . . . 13 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A (,) B ) u. { B } ) = ( A (,] B ) )
38	33, 35, 36, 37	syl3anc 1292	. . . . . . . . . . . 12 |- ( ph -> ( ( A (,) B ) u. { B } ) = ( A (,] B ) )
39		ovex 6336	. . . . . . . . . . . . 13 |- ( A (,] B ) e. _V
40	39	a1i 11	. . . . . . . . . . . 12 |- ( ph -> ( A (,] B ) e. _V )
41	38, 40	eqeltrd 2549	. . . . . . . . . . 11 |- ( ph -> ( ( A (,) B ) u. { B } ) e. _V )
42		resttop 20253	. . . . . . . . . . 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 676	. . . . . . . . . 10 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) e. Top )
44	18, 43	syl5eqel 2553	. . . . . . . . 9 |- ( ph -> J e. Top )
45	44	adantr 472	. . . . . . . 8 |- ( ( ph /\ D = B ) -> J e. Top )
46		oveq2 6316	. . . . . . . . . . 11 |- ( D = B -> ( C (,] D ) = ( C (,] B ) )
47	46	adantl 473	. . . . . . . . . 10 |- ( ( ph /\ D = B ) -> ( C (,] D ) = ( C (,] B ) )
48	23	adantr 472	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> C e. RR* )
49		pnfxr 11435	. . . . . . . . . . . . . . . . 17 |- +oo e. RR*
50	49	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> +oo e. RR* )
51		simpr 468	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( C (,] B ) ) -> x e. ( C (,] B ) )
52	34	adantr 472	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ x e. ( C (,] B ) ) -> B e. RR )
53		elioc2 11722	. . . . . . . . . . . . . . . . . . 19 |- ( ( C e. RR* /\ B e. RR ) -> ( x e. ( C (,] B ) <-> ( x e. RR /\ C < x /\ x <_ B ) ) )
54	48, 52, 53	syl2anc 673	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( C (,] B ) ) -> ( x e. ( C (,] B ) <-> ( x e. RR /\ C < x /\ x <_ B ) ) )
55	51, 54	mpbid 215	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( C (,] B ) ) -> ( x e. RR /\ C < x /\ x <_ B ) )
56	55	simp1d 1042	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> x e. RR )
57	55	simp2d 1043	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> C < x )
58	56	ltpnfd 11446	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> x < +oo )
59	48, 50, 56, 57, 58	eliood 37691	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( C (,] B ) ) -> x e. ( C (,) +oo ) )
60	32	adantr 472	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( C (,] B ) ) -> A e. RR )
61	22	adantr 472	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( C (,] B ) ) -> C e. RR )
62	32, 34, 22, 19, 20, 13	fourierdlem10 38091	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( A <_ C /\ D <_ B ) )
63	62	simpld 466	. . . . . . . . . . . . . . . . . 18 |- ( ph -> A <_ C )
64	63	adantr 472	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( C (,] B ) ) -> A <_ C )
65	60, 61, 56, 64, 57	lelttrd 9810	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> A < x )
66	55	simp3d 1044	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> x <_ B )
67	33	adantr 472	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( C (,] B ) ) -> A e. RR* )
68		elioc2 11722	. . . . . . . . . . . . . . . . 17 |- ( ( A e. RR* /\ B e. RR ) -> ( x e. ( A (,] B ) <-> ( x e. RR /\ A < x /\ x <_ B ) ) )
69	67, 52, 68	syl2anc 673	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( C (,] B ) ) -> ( x e. ( A (,] B ) <-> ( x e. RR /\ A < x /\ x <_ B ) ) )
70	56, 65, 66, 69	mpbir3and 1213	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( C (,] B ) ) -> x e. ( A (,] B ) )
71	59, 70	elind 3609	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( C (,] B ) ) -> x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) )
72		elinel1 3610	. . . . . . . . . . . . . . . . 17 |- ( x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) -> x e. ( C (,) +oo ) )
73		elioore 11691	. . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> x e. RR )
76	23	adantr 472	. . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> x e. ( C (,) +oo ) )
79		ioogtlb 37688	. . . . . . . . . . . . . . . 16 |- ( ( C e. RR* /\ +oo e. RR* /\ x e. ( C (,) +oo ) ) -> C < x )
80	76, 77, 78, 79	syl3anc 1292	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> C < x )
81		elinel2 3611	. . . . . . . . . . . . . . . . . 18 |- ( x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) -> x e. ( A (,] B ) )
82	81	adantl 473	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> x e. ( A (,] B ) )
83	33	adantr 472	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> A e. RR* )
84	34	adantr 472	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> B e. RR )
85	83, 84, 68	syl2anc 673	. . . . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> ( x e. RR /\ A < x /\ x <_ B ) )
87	86	simp3d 1044	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> x <_ B )
88	76, 84, 53	syl2anc 673	. . . . . . . . . . . . . . 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 1213	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) -> x e. ( C (,] B ) )
90	71, 89	impbida 850	. . . . . . . . . . . . 13 |- ( ph -> ( x e. ( C (,] B ) <-> x e. ( ( C (,) +oo ) i^i ( A (,] B ) ) ) )
91	90	eqrdv 2469	. . . . . . . . . . . 12 |- ( ph -> ( C (,] B ) = ( ( C (,) +oo ) i^i ( A (,] B ) ) )
92		retop 21860	. . . . . . . . . . . . . 14 |- ( topGen ` ran (,) ) e. Top
93	92	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> ( topGen ` ran (,) ) e. Top )
94		iooretop 21864	. . . . . . . . . . . . . 14 |- ( C (,) +oo ) e. ( topGen ` ran (,) )
95	94	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> ( C (,) +oo ) e. ( topGen ` ran (,) ) )
96		elrestr 15405	. . . . . . . . . . . . 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 1292	. . . . . . . . . . . 12 |- ( ph -> ( ( C (,) +oo ) i^i ( A (,] B ) ) e. ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) )
98	91, 97	eqeltrd 2549	. . . . . . . . . . 11 |- ( ph -> ( C (,] B ) e. ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) )
99	98	adantr 472	. . . . . . . . . 10 |- ( ( ph /\ D = B ) -> ( C (,] B ) e. ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) )
100	47, 99	eqeltrd 2549	. . . . . . . . 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 6324	. . . . . . . . . . 11 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( ( A (,) B ) u. { B } ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A (,] B ) ) )
103	17	tgioo2 21899	. . . . . . . . . . . . . 14 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
104	103	eqcomi 2480	. . . . . . . . . . . . 13 |- ( ( TopOpen ` ℂfld ) ↾t RR ) = ( topGen ` ran (,) )
105	104	oveq1i 6318	. . . . . . . . . . . 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 11739	. . . . . . . . . . . . . 14 |- ( ( A e. RR* /\ B e. RR ) -> ( A (,] B ) C_ RR )
108	33, 34, 107	syl2anc 673	. . . . . . . . . . . . 13 |- ( ph -> ( A (,] B ) C_ RR )
109		reex 9648	. . . . . . . . . . . . . 14 |- RR e. _V
110	109	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> RR e. _V )
111		restabs 20258	. . . . . . . . . . . . 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 1292	. . . . . . . . . . . 12 |- ( ph -> ( ( ( TopOpen ` ℂfld ) ↾t RR ) ↾t ( A (,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A (,] B ) ) )
113	105, 112	syl5reqr 2520	. . . . . . . . . . 11 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A (,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) )
114	101, 102, 113	3eqtrrd 2510	. . . . . . . . . 10 |- ( ph -> ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) = J )
115	114	adantr 472	. . . . . . . . 9 |- ( ( ph /\ D = B ) -> ( ( topGen ` ran (,) ) ↾t ( A (,] B ) ) = J )
116	100, 115	eleqtrd 2551	. . . . . . . 8 |- ( ( ph /\ D = B ) -> ( C (,] D ) e. J )
117		isopn3i 20175	. . . . . . . 8 |- ( ( J e. Top /\ ( C (,] D ) e. J ) -> ( ( int ` J ) ` ( C (,] D ) ) = ( C (,] D ) )
118	45, 116, 117	syl2anc 673	. . . . . . 7 |- ( ( ph /\ D = B ) -> ( ( int ` J ) ` ( C (,] D ) ) = ( C (,] D ) )
119	27, 30, 118	3eltr4d 2564	. . . . . 6 |- ( ( ph /\ D = B ) -> B e. ( ( int ` J ) ` ( C (,] D ) ) )
120		sneq 3969	. . . . . . . . . . 11 |- ( D = B -> { D } = { B } )
121	120	eqcomd 2477	. . . . . . . . . 10 |- ( D = B -> { B } = { D } )
122	121	uneq2d 3579	. . . . . . . . 9 |- ( D = B -> ( ( C (,) D ) u. { B } ) = ( ( C (,) D ) u. { D } ) )
123	122	adantl 473	. . . . . . . 8 |- ( ( ph /\ D = B ) -> ( ( C (,) D ) u. { B } ) = ( ( C (,) D ) u. { D } ) )
124	19	rexrd 9708	. . . . . . . . . 10 |- ( ph -> D e. RR* )
125		snunioo2 37702	. . . . . . . . . 10 |- ( ( C e. RR* /\ D e. RR* /\ C < D ) -> ( ( C (,) D ) u. { D } ) = ( C (,] D ) )
126	23, 124, 20, 125	syl3anc 1292	. . . . . . . . 9 |- ( ph -> ( ( C (,) D ) u. { D } ) = ( C (,] D ) )
127	126	adantr 472	. . . . . . . 8 |- ( ( ph /\ D = B ) -> ( ( C (,) D ) u. { D } ) = ( C (,] D ) )
128	123, 127	eqtr2d 2506	. . . . . . 7 |- ( ( ph /\ D = B ) -> ( C (,] D ) = ( ( C (,) D ) u. { B } ) )
129	128	fveq2d 5883	. . . . . 6 |- ( ( ph /\ D = B ) -> ( ( int ` J ) ` ( C (,] D ) ) = ( ( int ` J ) ` ( ( C (,) D ) u. { B } ) ) )
130	119, 129	eleqtrd 2551	. . . . 5 |- ( ( ph /\ D = B ) -> B e. ( ( int ` J ) ` ( ( C (,) D ) u. { B } ) ) )
131	12, 14, 16, 17, 18, 130	limcres 22920	. . . 4 |- ( ( ph /\ D = B ) -> ( ( F |` ( C (,) D ) ) lim CC B ) = ( F lim CC B ) )
132	8, 131	eqtr2d 2506	. . 3 |- ( ( ph /\ D = B ) -> ( F lim CC B ) = ( ( F |` ( C (,) D ) ) lim CC D ) )
133	2, 6, 132	3eltr3d 2563	. 2 |- ( ( ph /\ D = B ) -> Y e. ( ( F |` ( C (,) D ) ) lim CC D ) )
134		limcresi 22919	. . 3 |- ( F lim CC D ) C_ ( ( F |` ( C (,) D ) ) lim CC D )
135		iffalse 3881	. . . . . 6 |- ( -. D = B -> if ( D = B , L , ( F ` D ) ) = ( F ` D ) )
136	3, 135	syl5eq 2517	. . . . 5 |- ( -. D = B -> Y = ( F ` D ) )
137	136	adantl 473	. . . 4 |- ( ( ph /\ -. D = B ) -> Y = ( F ` D ) )
138		ssid 3437	. . . . . . . . . . . . 13 |- CC C_ CC
139	138	a1i 11	. . . . . . . . . . . 12 |- ( ph -> CC C_ CC )
140		eqid 2471	. . . . . . . . . . . . 13 |- ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) )
141		unicntop 37433	. . . . . . . . . . . . . . . 16 |- CC = U. ( TopOpen ` ℂfld )
142	141	restid 15410	. . . . . . . . . . . . . . 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 2480	. . . . . . . . . . . . 13 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
145	17, 140, 144	cncfcn 22019	. . . . . . . . . . . 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 676	. . . . . . . . . . 11 |- ( ph -> ( ( A (,) B ) -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) Cn ( TopOpen ` ℂfld ) ) )
147	9, 146	eleqtrd 2551	. . . . . . . . . 10 |- ( ph -> F e. ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) Cn ( TopOpen ` ℂfld ) ) )
148	17	cnfldtopon 21881	. . . . . . . . . . . 12 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
149	15	a1i 11	. . . . . . . . . . . 12 |- ( ph -> ( A (,) B ) C_ CC )
150		resttopon 20254	. . . . . . . . . . . 12 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ ( A (,) B ) C_ CC ) -> ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) e. (TopOn ` ( A (,) B ) ) )
151	148, 149, 150	sylancr 676	. . . . . . . . . . 11 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) e. (TopOn ` ( A (,) B ) ) )
152	148	a1i 11	. . . . . . . . . . 11 |- ( ph -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
153		cncnp 20373	. . . . . . . . . . 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 673	. . . . . . . . . 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 215	. . . . . . . . 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 470	. . . . . . . 8 |- ( ph -> A. x e. ( A (,) B ) F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) )
157	156	adantr 472	. . . . . . 7 |- ( ( ph /\ -. D = B ) -> A. x e. ( A (,) B ) F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) )
158	33	adantr 472	. . . . . . . 8 |- ( ( ph /\ -. D = B ) -> A e. RR* )
159	35	adantr 472	. . . . . . . 8 |- ( ( ph /\ -. D = B ) -> B e. RR* )
160	19	adantr 472	. . . . . . . 8 |- ( ( ph /\ -. D = B ) -> D e. RR )
161	32, 22, 19, 63, 20	lelttrd 9810	. . . . . . . . 9 |- ( ph -> A < D )
162	161	adantr 472	. . . . . . . 8 |- ( ( ph /\ -. D = B ) -> A < D )
163	34	adantr 472	. . . . . . . . 9 |- ( ( ph /\ -. D = B ) -> B e. RR )
164	62	simprd 470	. . . . . . . . . 10 |- ( ph -> D <_ B )
165	164	adantr 472	. . . . . . . . 9 |- ( ( ph /\ -. D = B ) -> D <_ B )
166		neqne 2651	. . . . . . . . . . 11 |- ( -. D = B -> D =/= B )
167	166	necomd 2698	. . . . . . . . . 10 |- ( -. D = B -> B =/= D )
168	167	adantl 473	. . . . . . . . 9 |- ( ( ph /\ -. D = B ) -> B =/= D )
169	160, 163, 165, 168	leneltd 9806	. . . . . . . 8 |- ( ( ph /\ -. D = B ) -> D < B )
170	158, 159, 160, 162, 169	eliood 37691	. . . . . . 7 |- ( ( ph /\ -. D = B ) -> D e. ( A (,) B ) )
171		fveq2 5879	. . . . . . . . 9 |- ( x = D -> ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` x ) = ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` D ) )
172	171	eleq2d 2534	. . . . . . . 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 3135	. . . . . . 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 673	. . . . . 6 |- ( ( ph /\ -. D = B ) -> F e. ( ( ( ( TopOpen ` ℂfld ) ↾t ( A (,) B ) ) CnP ( TopOpen ` ℂfld ) ) ` D ) )
175	17, 140	cnplimc 22921	. . . . . . 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 676	. . . . . 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 215	. . . . 5 |- ( ( ph /\ -. D = B ) -> ( F : ( A (,) B ) --> CC /\ ( F ` D ) e. ( F lim CC D ) ) )
178	177	simprd 470	. . . 4 |- ( ( ph /\ -. D = B ) -> ( F ` D ) e. ( F lim CC D ) )
179	137, 178	eqeltrd 2549	. . 3 |- ( ( ph /\ -. D = B ) -> Y e. ( F lim CC D ) )
180	134, 179	sseldi 3416	. 2 |- ( ( ph /\ -. D = B ) -> Y e. ( ( F |` ( C (,) D ) ) lim CC D ) )
181	133, 180	pm2.61dan 808	1 |- ( ph -> Y e. ( ( F |` ( C (,) D ) ) lim CC D ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   _Vcvv 3031   u. cun 3388   i^i cin 3389   C_ wss 3390   ifcif 3872   {csn 3959   class class class wbr 4395   ran crn 4840   |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   +oocpnf 9690   RR*cxr 9692   < clt 9693   <_ cle 9694   (,)cioo 11660   (,]cioc 11661   ↾_t crest 15397   TopOpenctopn 15398   topGenctg 15414  ℂ_fldccnfld 19047   Topctop 19994  TopOnctopon 19995   intcnt 20109   Cn ccn 20317   CnP ccnp 20318   -cn->ccncf 21986   lim CC climc 22896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-icc 11667  df-fz 11811  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-ntr 20112  df-cn 20320  df-cnp 20321  df-xms 21413  df-ms 21414  df-cncf 21988  df-limc 22900

This theorem is referenced by:  fourierdlem49  38131  fourierdlem76  38158  fourierdlem91  38173