Theorem limciccioolb 31486

Description: The limit of a function at the lower bound of a closed interval only depends on the values in the inner open interval (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
limciccioolb.1	|- ( ph -> A e. RR )
limciccioolb.2	|- ( ph -> B e. RR )
limciccioolb.3	|- ( ph -> A < B )
limciccioolb.4	|- ( ph -> F : ( A [,] B ) --> CC )

Assertion

Ref	Expression
limciccioolb	|- ( ph -> ( ( F |` ( A (,) B ) ) lim CC A ) = ( F lim CC A ) )

Proof of Theorem limciccioolb

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limciccioolb.4	. 2 |- ( ph -> F : ( A [,] B ) --> CC )
2		ioossicc 11622	. . 3 |- ( A (,) B ) C_ ( A [,] B )
3	2	a1i 11	. 2 |- ( ph -> ( A (,) B ) C_ ( A [,] B ) )
4		limciccioolb.1	. . . 4 |- ( ph -> A e. RR )
5		limciccioolb.2	. . . 4 |- ( ph -> B e. RR )
6		iccssre 11618	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
7	4, 5, 6	syl2anc 661	. . 3 |- ( ph -> ( A [,] B ) C_ RR )
8		ax-resscn 9561	. . . 4 |- RR C_ CC
9	8	a1i 11	. . 3 |- ( ph -> RR C_ CC )
10	7, 9	sstrd 3519	. 2 |- ( ph -> ( A [,] B ) C_ CC )
11		eqid 2467	. 2 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
12		eqid 2467	. 2 |- ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) ) = ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) )
13		mnflt 11345	. . . . . . . . . . . 12 |- ( A e. RR -> -oo < A )
14	4, 13	syl 16	. . . . . . . . . . 11 |- ( ph -> -oo < A )
15		limciccioolb.3	. . . . . . . . . . 11 |- ( ph -> A < B )
16	4, 14, 15	3jca 1176	. . . . . . . . . 10 |- ( ph -> ( A e. RR /\ -oo < A /\ A < B ) )
17		mnfxr 11335	. . . . . . . . . . . 12 |- -oo e. RR*
18	17	a1i 11	. . . . . . . . . . 11 |- ( ph -> -oo e. RR* )
19	5	rexrd 9655	. . . . . . . . . . 11 |- ( ph -> B e. RR* )
20		elioo2 11582	. . . . . . . . . . 11 |- ( ( -oo e. RR* /\ B e. RR* ) -> ( A e. ( -oo (,) B ) <-> ( A e. RR /\ -oo < A /\ A < B ) ) )
21	18, 19, 20	syl2anc 661	. . . . . . . . . 10 |- ( ph -> ( A e. ( -oo (,) B ) <-> ( A e. RR /\ -oo < A /\ A < B ) ) )
22	16, 21	mpbird 232	. . . . . . . . 9 |- ( ph -> A e. ( -oo (,) B ) )
23		retop 21136	. . . . . . . . . . . 12 |- ( topGen ` ran (,) ) e. Top
24	23	a1i 11	. . . . . . . . . . 11 |- ( ph -> ( topGen ` ran (,) ) e. Top )
25		iooretop 21141	. . . . . . . . . . . 12 |- ( -oo (,) B ) e. ( topGen ` ran (,) )
26	25	a1i 11	. . . . . . . . . . 11 |- ( ph -> ( -oo (,) B ) e. ( topGen ` ran (,) ) )
27		isopn3i 19451	. . . . . . . . . . 11 |- ( ( ( topGen ` ran (,) ) e. Top /\ ( -oo (,) B ) e. ( topGen ` ran (,) ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( -oo (,) B ) ) = ( -oo (,) B ) )
28	24, 26, 27	syl2anc 661	. . . . . . . . . 10 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( -oo (,) B ) ) = ( -oo (,) B ) )
29	28	eqcomd 2475	. . . . . . . . 9 |- ( ph -> ( -oo (,) B ) = ( ( int ` ( topGen ` ran (,) ) ) ` ( -oo (,) B ) ) )
30	22, 29	eleqtrd 2557	. . . . . . . 8 |- ( ph -> A e. ( ( int ` ( topGen ` ran (,) ) ) ` ( -oo (,) B ) ) )
31		icossre 11617	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. RR* ) -> ( A [,) B ) C_ RR )
32	4, 19, 31	syl2anc 661	. . . . . . . . . . . . 13 |- ( ph -> ( A [,) B ) C_ RR )
33		difssd 3637	. . . . . . . . . . . . 13 |- ( ph -> ( RR \ ( A [,] B ) ) C_ RR )
34		unss12 3681	. . . . . . . . . . . . 13 |- ( ( ( A [,) B ) C_ RR /\ ( RR \ ( A [,] B ) ) C_ RR ) -> ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) C_ ( RR u. RR ) )
35	32, 33, 34	syl2anc 661	. . . . . . . . . . . 12 |- ( ph -> ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) C_ ( RR u. RR ) )
36		unidm 3652	. . . . . . . . . . . 12 |- ( RR u. RR ) = RR
37	35, 36	syl6sseq 3555	. . . . . . . . . . 11 |- ( ph -> ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) C_ RR )
38		uniretop 21137	. . . . . . . . . . 11 |- RR = U. ( topGen ` ran (,) )
39	37, 38	syl6sseq 3555	. . . . . . . . . 10 |- ( ph -> ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) C_ U. ( topGen ` ran (,) ) )
40		elioore 11571	. . . . . . . . . . . . . . . . . 18 |- ( x e. ( -oo (,) B ) -> x e. RR )
41	40	ad2antlr 726	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> x e. RR )
42		simpr 461	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> A <_ x )
43		simpr 461	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> x e. ( -oo (,) B ) )
44	17	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> -oo e. RR* )
45	19	adantr 465	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> B e. RR* )
46		elioo2 11582	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( -oo e. RR* /\ B e. RR* ) -> ( x e. ( -oo (,) B ) <-> ( x e. RR /\ -oo < x /\ x < B ) ) )
47	44, 45, 46	syl2anc 661	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> ( x e. ( -oo (,) B ) <-> ( x e. RR /\ -oo < x /\ x < B ) ) )
48	43, 47	mpbid 210	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> ( x e. RR /\ -oo < x /\ x < B ) )
49	48	simp3d 1010	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> x < B )
50	49	adantr 465	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> x < B )
51	41, 42, 50	3jca 1176	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> ( x e. RR /\ A <_ x /\ x < B ) )
52	4	ad2antrr 725	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> A e. RR )
53	19	ad2antrr 725	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> B e. RR* )
54		elico2 11600	. . . . . . . . . . . . . . . . 17 |- ( ( A e. RR /\ B e. RR* ) -> ( x e. ( A [,) B ) <-> ( x e. RR /\ A <_ x /\ x < B ) ) )
55	52, 53, 54	syl2anc 661	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> ( x e. ( A [,) B ) <-> ( x e. RR /\ A <_ x /\ x < B ) ) )
56	51, 55	mpbird 232	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> x e. ( A [,) B ) )
57	56	orcd 392	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> ( x e. ( A [,) B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
58	40	ad2antlr 726	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> x e. RR )
59		simpr 461	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> -. A <_ x )
60	59	intnanrd 915	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> -. ( A <_ x /\ x <_ B ) )
61	4	rexrd 9655	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> A e. RR* )
62	61	ad2antrr 725	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> A e. RR* )
63	45	adantr 465	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> B e. RR* )
64	58	rexrd 9655	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> x e. RR* )
65		elicc4 11603	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) -> ( x e. ( A [,] B ) <-> ( A <_ x /\ x <_ B ) ) )
66	62, 63, 64, 65	syl3anc 1228	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> ( x e. ( A [,] B ) <-> ( A <_ x /\ x <_ B ) ) )
67	60, 66	mtbird 301	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> -. x e. ( A [,] B ) )
68	58, 67	jca 532	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> ( x e. RR /\ -. x e. ( A [,] B ) ) )
69		eldif 3491	. . . . . . . . . . . . . . . 16 |- ( x e. ( RR \ ( A [,] B ) ) <-> ( x e. RR /\ -. x e. ( A [,] B ) ) )
70	68, 69	sylibr 212	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> x e. ( RR \ ( A [,] B ) ) )
71	70	olcd 393	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> ( x e. ( A [,) B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
72	57, 71	pm2.61dan 789	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> ( x e. ( A [,) B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
73		elun 3650	. . . . . . . . . . . . 13 |- ( x e. ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) <-> ( x e. ( A [,) B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
74	72, 73	sylibr 212	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> x e. ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) )
75	74	ralrimiva 2881	. . . . . . . . . . 11 |- ( ph -> A. x e. ( -oo (,) B ) x e. ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) )
76		dfss3 3499	. . . . . . . . . . 11 |- ( ( -oo (,) B ) C_ ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) <-> A. x e. ( -oo (,) B ) x e. ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) )
77	75, 76	sylibr 212	. . . . . . . . . 10 |- ( ph -> ( -oo (,) B ) C_ ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) )
78		eqid 2467	. . . . . . . . . . 11 |- U. ( topGen ` ran (,) ) = U. ( topGen ` ran (,) )
79	78	ntrss 19424	. . . . . . . . . 10 |- ( ( ( topGen ` ran (,) ) e. Top /\ ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) C_ U. ( topGen ` ran (,) ) /\ ( -oo (,) B ) C_ ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( -oo (,) B ) ) C_ ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) )
80	24, 39, 77, 79	syl3anc 1228	. . . . . . . . 9 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( -oo (,) B ) ) C_ ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) )
81	80	sseld 3508	. . . . . . . 8 |- ( ph -> ( A e. ( ( int ` ( topGen ` ran (,) ) ) ` ( -oo (,) B ) ) -> A e. ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) ) )
82	30, 81	mpd 15	. . . . . . 7 |- ( ph -> A e. ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) )
83	4	leidd 10131	. . . . . . . . 9 |- ( ph -> A <_ A )
84	4, 5, 15	ltled 9744	. . . . . . . . 9 |- ( ph -> A <_ B )
85	4, 83, 84	3jca 1176	. . . . . . . 8 |- ( ph -> ( A e. RR /\ A <_ A /\ A <_ B ) )
86		elicc2 11601	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A e. ( A [,] B ) <-> ( A e. RR /\ A <_ A /\ A <_ B ) ) )
87	4, 5, 86	syl2anc 661	. . . . . . . 8 |- ( ph -> ( A e. ( A [,] B ) <-> ( A e. RR /\ A <_ A /\ A <_ B ) ) )
88	85, 87	mpbird 232	. . . . . . 7 |- ( ph -> A e. ( A [,] B ) )
89	82, 88	jca 532	. . . . . 6 |- ( ph -> ( A e. ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) /\ A e. ( A [,] B ) ) )
90		elin 3692	. . . . . 6 |- ( A e. ( ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) i^i ( A [,] B ) ) <-> ( A e. ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) /\ A e. ( A [,] B ) ) )
91	89, 90	sylibr 212	. . . . 5 |- ( ph -> A e. ( ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) i^i ( A [,] B ) ) )
92		icossicc 11623	. . . . . . . . 9 |- ( A [,) B ) C_ ( A [,] B )
93	92	a1i 11	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> ( A [,) B ) C_ ( A [,] B ) )
94	61, 19, 93	syl2anc 661	. . . . . . 7 |- ( ph -> ( A [,) B ) C_ ( A [,] B ) )
95		eqid 2467	. . . . . . . 8 |- ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) )
96	38, 95	restntr 19551	. . . . . . 7 |- ( ( ( topGen ` ran (,) ) e. Top /\ ( A [,] B ) C_ RR /\ ( A [,) B ) C_ ( A [,] B ) ) -> ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) = ( ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) i^i ( A [,] B ) ) )
97	24, 7, 94, 96	syl3anc 1228	. . . . . 6 |- ( ph -> ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) = ( ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) i^i ( A [,] B ) ) )
98	97	eqcomd 2475	. . . . 5 |- ( ph -> ( ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) i^i ( A [,] B ) ) = ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) )
99	91, 98	eleqtrd 2557	. . . 4 |- ( ph -> A e. ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) )
100		eqid 2467	. . . . . . . . 9 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
101	11, 100	rerest 21177	. . . . . . . 8 |- ( ( A [,] B ) C_ RR -> ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
102	7, 101	syl 16	. . . . . . 7 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
103	102	eqcomd 2475	. . . . . 6 |- ( ph -> ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) )
104	103	fveq2d 5876	. . . . 5 |- ( ph -> ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) = ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) )
105	104	fveq1d 5874	. . . 4 |- ( ph -> ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) = ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) )
106	99, 105	eleqtrd 2557	. . 3 |- ( ph -> A e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) )
107	88	snssd 4178	. . . . . . . 8 |- ( ph -> { A } C_ ( A [,] B ) )
108		ssequn2 3682	. . . . . . . 8 |- ( { A } C_ ( A [,] B ) <-> ( ( A [,] B ) u. { A } ) = ( A [,] B ) )
109	107, 108	sylib 196	. . . . . . 7 |- ( ph -> ( ( A [,] B ) u. { A } ) = ( A [,] B ) )
110	109	eqcomd 2475	. . . . . 6 |- ( ph -> ( A [,] B ) = ( ( A [,] B ) u. { A } ) )
111	110	oveq2d 6311	. . . . 5 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) ) )
112	111	fveq2d 5876	. . . 4 |- ( ph -> ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) = ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) ) ) )
113		uncom 3653	. . . . . . 7 |- ( ( A (,) B ) u. { A } ) = ( { A } u. ( A (,) B ) )
114	113	a1i 11	. . . . . 6 |- ( ph -> ( ( A (,) B ) u. { A } ) = ( { A } u. ( A (,) B ) ) )
115		snunioo 11658	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( { A } u. ( A (,) B ) ) = ( A [,) B ) )
116	61, 19, 15, 115	syl3anc 1228	. . . . . 6 |- ( ph -> ( { A } u. ( A (,) B ) ) = ( A [,) B ) )
117	114, 116	eqtrd 2508	. . . . 5 |- ( ph -> ( ( A (,) B ) u. { A } ) = ( A [,) B ) )
118	117	eqcomd 2475	. . . 4 |- ( ph -> ( A [,) B ) = ( ( A (,) B ) u. { A } ) )
119	112, 118	fveq12d 5878	. . 3 |- ( ph -> ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) = ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) ) ) ` ( ( A (,) B ) u. { A } ) ) )
120	106, 119	eleqtrd 2557	. 2 |- ( ph -> A e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) ) ) ` ( ( A (,) B ) u. { A } ) ) )
121	1, 3, 10, 11, 12, 120	limcres 22158	1 |- ( ph -> ( ( F |` ( A (,) B ) ) lim CC A ) = ( F lim CC A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   A.wral 2817   \ cdif 3478   u. cun 3479   i^i cin 3480   C_ wss 3481   {csn 4033   U.cuni 4251   class class class wbr 4453   ran crn 5006   |` cres 5007   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   -oocmnf 9638   RR*cxr 9639   < clt 9640   <_ cle 9641   (,)cioo 11541   [,)cico 11543   [,]cicc 11544   ↾_t crest 14693   TopOpenctopn 14694   topGenctg 14710  ℂ_fldccnfld 18290   Topctop 19263   intcnt 19386   lim CC climc 22134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fi 7883  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ico 11547  df-icc 11548  df-fz 11685  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-plusg 14585  df-mulr 14586  df-starv 14587  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-rest 14695  df-topn 14696  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-cnp 19597  df-xms 20691  df-ms 20692  df-limc 22138

This theorem is referenced by:  fourierdlem82  31812  fourierdlem93  31823  fourierdlem111  31841