Theorem fourierdlem94 40417

Description: For a piecewise smooth function, the left and the right limits exist at any point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem94.f	|- ( ph -> F : RR --> RR )
fourierdlem94.t	|- T = ( 2 x. pi )
fourierdlem94.per	|- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
fourierdlem94.x	|- ( ph -> X e. RR )
fourierdlem94.p	|- P = ( n e. NN |-> { p e. ( RR ^m ( 0 ... n ) ) | ( ( ( p ` 0 ) = -u pi /\ ( p ` n ) = pi ) /\ A. i e. ( 0..^ n ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
fourierdlem94.m	|- ( ph -> M e. NN )
fourierdlem94.q	|- ( ph -> Q e. ( P ` M ) )
fourierdlem94.dvcn	|- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
fourierdlem94.dvlb	|- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) =/= (/) )
fourierdlem94.dvub	|- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) =/= (/) )

Assertion

Ref	Expression
fourierdlem94	|- ( ph -> ( ( ( F |` ( -oo (,) X ) ) lim CC X ) =/= (/) /\ ( ( F |` ( X (,) +oo ) ) lim CC X ) =/= (/) ) )

Distinct variable groups:   i, F, n, x   i, M, x, n   M, p, i, n   Q, i, x, n   Q, p   T, i, x, n   T, p   i, X, x, n   X, p   ph, i, x, n

Allowed substitution hints:   ph( p)   P( x, i, n, p)   F( p)

Proof of Theorem fourierdlem94

Dummy variables k j w y t z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pire 24210	. . . . 5 |- pi e. RR
2	1	renegcli 10342	. . . 4 |- -u pi e. RR
3	2	a1i 11	. . 3 |- ( ph -> -u pi e. RR )
4	1	a1i 11	. . 3 |- ( ph -> pi e. RR )
5		negpilt0 39492	. . . . 5 |- -u pi < 0
6		pipos 24212	. . . . 5 |- 0 < pi
7		0re 10040	. . . . . 6 |- 0 e. RR
8	2, 7, 1	lttri 10163	. . . . 5 |- ( ( -u pi < 0 /\ 0 < pi ) -> -u pi < pi )
9	5, 6, 8	mp2an 708	. . . 4 |- -u pi < pi
10	9	a1i 11	. . 3 |- ( ph -> -u pi < pi )
11		fourierdlem94.p	. . 3 |- P = ( n e. NN |-> { p e. ( RR ^m ( 0 ... n ) ) | ( ( ( p ` 0 ) = -u pi /\ ( p ` n ) = pi ) /\ A. i e. ( 0..^ n ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
12		picn 24211	. . . . 5 |- pi e. CC
13	12	2timesi 11147	. . . 4 |- ( 2 x. pi ) = ( pi + pi )
14		fourierdlem94.t	. . . 4 |- T = ( 2 x. pi )
15	12, 12	subnegi 10360	. . . 4 |- ( pi - -u pi ) = ( pi + pi )
16	13, 14, 15	3eqtr4i 2654	. . 3 |- T = ( pi - -u pi )
17		fourierdlem94.m	. . 3 |- ( ph -> M e. NN )
18		fourierdlem94.q	. . 3 |- ( ph -> Q e. ( P ` M ) )
19		ssid 3624	. . . 4 |- RR C_ RR
20	19	a1i 11	. . 3 |- ( ph -> RR C_ RR )
21		fourierdlem94.f	. . 3 |- ( ph -> F : RR --> RR )
22		simp2 1062	. . . 4 |- ( ( ph /\ x e. RR /\ k e. ZZ ) -> x e. RR )
23		zre 11381	. . . . . 6 |- ( k e. ZZ -> k e. RR )
24	23	3ad2ant3 1084	. . . . 5 |- ( ( ph /\ x e. RR /\ k e. ZZ ) -> k e. RR )
25		2re 11090	. . . . . . . . . 10 |- 2 e. RR
26	25, 1	remulcli 10054	. . . . . . . . 9 |- ( 2 x. pi ) e. RR
27	26	a1i 11	. . . . . . . 8 |- ( ph -> ( 2 x. pi ) e. RR )
28	14, 27	syl5eqel 2705	. . . . . . 7 |- ( ph -> T e. RR )
29	28	adantr 481	. . . . . 6 |- ( ( ph /\ k e. ZZ ) -> T e. RR )
30	29	3adant2 1080	. . . . 5 |- ( ( ph /\ x e. RR /\ k e. ZZ ) -> T e. RR )
31	24, 30	remulcld 10070	. . . 4 |- ( ( ph /\ x e. RR /\ k e. ZZ ) -> ( k x. T ) e. RR )
32	22, 31	readdcld 10069	. . 3 |- ( ( ph /\ x e. RR /\ k e. ZZ ) -> ( x + ( k x. T ) ) e. RR )
33		simp1 1061	. . . 4 |- ( ( ph /\ x e. RR /\ k e. ZZ ) -> ph )
34		simp3 1063	. . . 4 |- ( ( ph /\ x e. RR /\ k e. ZZ ) -> k e. ZZ )
35		ax-resscn 9993	. . . . . . . . 9 |- RR C_ CC
36	35	a1i 11	. . . . . . . 8 |- ( ph -> RR C_ CC )
37	21, 36	fssd 6057	. . . . . . 7 |- ( ph -> F : RR --> CC )
38	37	adantr 481	. . . . . 6 |- ( ( ph /\ k e. ZZ ) -> F : RR --> CC )
39	38	adantr 481	. . . . 5 |- ( ( ( ph /\ k e. ZZ ) /\ x e. RR ) -> F : RR --> CC )
40	29	adantr 481	. . . . 5 |- ( ( ( ph /\ k e. ZZ ) /\ x e. RR ) -> T e. RR )
41		simplr 792	. . . . 5 |- ( ( ( ph /\ k e. ZZ ) /\ x e. RR ) -> k e. ZZ )
42		simpr 477	. . . . 5 |- ( ( ( ph /\ k e. ZZ ) /\ x e. RR ) -> x e. RR )
43		eleq1 2689	. . . . . . . . 9 |- ( x = y -> ( x e. RR <-> y e. RR ) )
44	43	anbi2d 740	. . . . . . . 8 |- ( x = y -> ( ( ph /\ x e. RR ) <-> ( ph /\ y e. RR ) ) )
45		oveq1 6657	. . . . . . . . . 10 |- ( x = y -> ( x + T ) = ( y + T ) )
46	45	fveq2d 6195	. . . . . . . . 9 |- ( x = y -> ( F ` ( x + T ) ) = ( F ` ( y + T ) ) )
47		fveq2 6191	. . . . . . . . 9 |- ( x = y -> ( F ` x ) = ( F ` y ) )
48	46, 47	eqeq12d 2637	. . . . . . . 8 |- ( x = y -> ( ( F ` ( x + T ) ) = ( F ` x ) <-> ( F ` ( y + T ) ) = ( F ` y ) ) )
49	44, 48	imbi12d 334	. . . . . . 7 |- ( x = y -> ( ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) ) <-> ( ( ph /\ y e. RR ) -> ( F ` ( y + T ) ) = ( F ` y ) ) ) )
50		fourierdlem94.per	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
51	49, 50	chvarv 2263	. . . . . 6 |- ( ( ph /\ y e. RR ) -> ( F ` ( y + T ) ) = ( F ` y ) )
52	51	ad4ant14 1293	. . . . 5 |- ( ( ( ( ph /\ k e. ZZ ) /\ x e. RR ) /\ y e. RR ) -> ( F ` ( y + T ) ) = ( F ` y ) )
53	39, 40, 41, 42, 52	fperiodmul 39518	. . . 4 |- ( ( ( ph /\ k e. ZZ ) /\ x e. RR ) -> ( F ` ( x + ( k x. T ) ) ) = ( F ` x ) )
54	33, 34, 22, 53	syl21anc 1325	. . 3 |- ( ( ph /\ x e. RR /\ k e. ZZ ) -> ( F ` ( x + ( k x. T ) ) ) = ( F ` x ) )
55	35	a1i 11	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> RR C_ CC )
56		ioossre 12235	. . . . . . . 8 |- ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ RR
57	56	a1i 11	. . . . . . 7 |- ( ph -> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ RR )
58	21, 57	fssresd 6071	. . . . . 6 |- ( ph -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) : ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) --> RR )
59	58, 36	fssd 6057	. . . . 5 |- ( ph -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) : ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) --> CC )
60	59	adantr 481	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) : ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) --> CC )
61	56	a1i 11	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ RR )
62	37	adantr 481	. . . . . . 7 |- ( ( ph /\ i e. ( 0..^ M ) ) -> F : RR --> CC )
63	19	a1i 11	. . . . . . 7 |- ( ( ph /\ i e. ( 0..^ M ) ) -> RR C_ RR )
64		eqid 2622	. . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
65	64	tgioo2 22606	. . . . . . . 8 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
66	64, 65	dvres 23675	. . . . . . 7 |- ( ( ( RR C_ CC /\ F : RR --> CC ) /\ ( RR C_ RR /\ ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ RR ) ) -> ( RR _D ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) )
67	55, 62, 63, 61, 66	syl22anc 1327	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( RR _D ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) )
68	67	dmeqd 5326	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> dom ( RR _D ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) = dom ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) )
69		ioontr 39736	. . . . . . . 8 |- ( ( int ` ( topGen ` ran (,) ) ) ` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) )
70	69	reseq2i 5393	. . . . . . 7 |- ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) = ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) )
71	70	dmeqi 5325	. . . . . 6 |- dom ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) = dom ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) )
72	71	a1i 11	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> dom ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) = dom ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) )
73		fourierdlem94.dvcn	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
74		cncff 22696	. . . . . 6 |- ( ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) -> ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) : ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) --> CC )
75		fdm 6051	. . . . . 6 |- ( ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) : ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) --> CC -> dom ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) )
76	73, 74, 75	3syl 18	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> dom ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) )
77	68, 72, 76	3eqtrd 2660	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> dom ( RR _D ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) = ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) )
78		dvcn 23684	. . . 4 |- ( ( ( RR C_ CC /\ ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) : ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) --> CC /\ ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ RR ) /\ dom ( RR _D ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) = ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
79	55, 60, 61, 77, 78	syl31anc 1329	. . 3 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
80	61, 35	syl6ss 3615	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ CC )
81	11	fourierdlem2 40326	. . . . . . . . . . . 12 |- ( M e. NN -> ( Q e. ( P ` M ) <-> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = -u pi /\ ( Q ` M ) = pi ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) )
82	17, 81	syl 17	. . . . . . . . . . 11 |- ( ph -> ( Q e. ( P ` M ) <-> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = -u pi /\ ( Q ` M ) = pi ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) )
83	18, 82	mpbid 222	. . . . . . . . . 10 |- ( ph -> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = -u pi /\ ( Q ` M ) = pi ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) )
84	83	simpld 475	. . . . . . . . 9 |- ( ph -> Q e. ( RR ^m ( 0 ... M ) ) )
85		elmapi 7879	. . . . . . . . 9 |- ( Q e. ( RR ^m ( 0 ... M ) ) -> Q : ( 0 ... M ) --> RR )
86	84, 85	syl 17	. . . . . . . 8 |- ( ph -> Q : ( 0 ... M ) --> RR )
87	86	adantr 481	. . . . . . 7 |- ( ( ph /\ i e. ( 0..^ M ) ) -> Q : ( 0 ... M ) --> RR )
88		elfzofz 12485	. . . . . . . 8 |- ( i e. ( 0..^ M ) -> i e. ( 0 ... M ) )
89	88	adantl 482	. . . . . . 7 |- ( ( ph /\ i e. ( 0..^ M ) ) -> i e. ( 0 ... M ) )
90	87, 89	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` i ) e. RR )
91	90	rexrd 10089	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` i ) e. RR* )
92		fzofzp1 12565	. . . . . . 7 |- ( i e. ( 0..^ M ) -> ( i + 1 ) e. ( 0 ... M ) )
93	92	adantl 482	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( i + 1 ) e. ( 0 ... M ) )
94	87, 93	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` ( i + 1 ) ) e. RR )
95	83	simprrd 797	. . . . . 6 |- ( ph -> A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) )
96	95	r19.21bi 2932	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` i ) < ( Q ` ( i + 1 ) ) )
97	64, 91, 94, 96	lptioo2cn 39877	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` ( i + 1 ) ) e. ( ( limPt ` ( TopOpen ` ℂfld ) ) ` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) )
98	58	adantr 481	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) : ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) --> RR )
99	36, 37, 20	dvbss 23665	. . . . . . . 8 |- ( ph -> dom ( RR _D F ) C_ RR )
100		dvfre 23714	. . . . . . . . 9 |- ( ( F : RR --> RR /\ RR C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )
101	21, 20, 100	syl2anc 693	. . . . . . . 8 |- ( ph -> ( RR _D F ) : dom ( RR _D F ) --> RR )
102	83	simprd 479	. . . . . . . . 9 |- ( ph -> ( ( ( Q ` 0 ) = -u pi /\ ( Q ` M ) = pi ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) )
103	102	simplld 791	. . . . . . . 8 |- ( ph -> ( Q ` 0 ) = -u pi )
104	102	simplrd 793	. . . . . . . 8 |- ( ph -> ( Q ` M ) = pi )
105	73, 74	syl 17	. . . . . . . . 9 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) : ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) --> CC )
106	94	rexrd 10089	. . . . . . . . . 10 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` ( i + 1 ) ) e. RR* )
107	64, 106, 90, 96	lptioo1cn 39878	. . . . . . . . 9 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` i ) e. ( ( limPt ` ( TopOpen ` ℂfld ) ) ` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) )
108		fourierdlem94.dvlb	. . . . . . . . 9 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) =/= (/) )
109	105, 80, 107, 108, 64	ellimciota 39846	. . . . . . . 8 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( iota x x e. ( ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) ) e. ( ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) )
110		fourierdlem94.dvub	. . . . . . . . 9 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) =/= (/) )
111	105, 80, 97, 110, 64	ellimciota 39846	. . . . . . . 8 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( iota x x e. ( ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) ) e. ( ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) )
112	23	adantl 482	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ZZ ) -> k e. RR )
113	112, 29	remulcld 10070	. . . . . . . . . . 11 |- ( ( ph /\ k e. ZZ ) -> ( k x. T ) e. RR )
114	38	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ZZ ) /\ t e. RR ) -> F : RR --> CC )
115	29	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ZZ ) /\ t e. RR ) -> T e. RR )
116		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ZZ ) /\ t e. RR ) -> k e. ZZ )
117		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ZZ ) /\ t e. RR ) -> t e. RR )
118	50	ad4ant14 1293	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ k e. ZZ ) /\ t e. RR ) /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
119	114, 115, 116, 117, 118	fperiodmul 39518	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ZZ ) /\ t e. RR ) -> ( F ` ( t + ( k x. T ) ) ) = ( F ` t ) )
120		eqid 2622	. . . . . . . . . . 11 |- ( RR _D F ) = ( RR _D F )
121	38, 113, 119, 120	fperdvper 40133	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ZZ ) /\ t e. dom ( RR _D F ) ) -> ( ( t + ( k x. T ) ) e. dom ( RR _D F ) /\ ( ( RR _D F ) ` ( t + ( k x. T ) ) ) = ( ( RR _D F ) ` t ) ) )
122	121	an32s 846	. . . . . . . . 9 |- ( ( ( ph /\ t e. dom ( RR _D F ) ) /\ k e. ZZ ) -> ( ( t + ( k x. T ) ) e. dom ( RR _D F ) /\ ( ( RR _D F ) ` ( t + ( k x. T ) ) ) = ( ( RR _D F ) ` t ) ) )
123	122	simpld 475	. . . . . . . 8 |- ( ( ( ph /\ t e. dom ( RR _D F ) ) /\ k e. ZZ ) -> ( t + ( k x. T ) ) e. dom ( RR _D F ) )
124	122	simprd 479	. . . . . . . 8 |- ( ( ( ph /\ t e. dom ( RR _D F ) ) /\ k e. ZZ ) -> ( ( RR _D F ) ` ( t + ( k x. T ) ) ) = ( ( RR _D F ) ` t ) )
125		fveq2 6191	. . . . . . . . . 10 |- ( j = i -> ( Q ` j ) = ( Q ` i ) )
126		oveq1 6657	. . . . . . . . . . 11 |- ( j = i -> ( j + 1 ) = ( i + 1 ) )
127	126	fveq2d 6195	. . . . . . . . . 10 |- ( j = i -> ( Q ` ( j + 1 ) ) = ( Q ` ( i + 1 ) ) )
128	125, 127	oveq12d 6668	. . . . . . . . 9 |- ( j = i -> ( ( Q ` j ) (,) ( Q ` ( j + 1 ) ) ) = ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) )
129	128	cbvmptv 4750	. . . . . . . 8 |- ( j e. ( 0..^ M ) |-> ( ( Q ` j ) (,) ( Q ` ( j + 1 ) ) ) ) = ( i e. ( 0..^ M ) |-> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) )
130		eqid 2622	. . . . . . . 8 |- ( t e. RR |-> ( t + ( ( |_ ` ( ( pi - t ) / T ) ) x. T ) ) ) = ( t e. RR |-> ( t + ( ( |_ ` ( ( pi - t ) / T ) ) x. T ) ) )
131	99, 101, 3, 4, 10, 16, 17, 86, 103, 104, 73, 109, 111, 123, 124, 129, 130	fourierdlem71 40394	. . . . . . 7 |- ( ph -> E. z e. RR A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z )
132	131	adantr 481	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> E. z e. RR A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z )
133		nfv 1843	. . . . . . . . . 10 |- F/ t ( ph /\ i e. ( 0..^ M ) )
134		nfra1 2941	. . . . . . . . . 10 |- F/ t A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z
135	133, 134	nfan 1828	. . . . . . . . 9 |- F/ t ( ( ph /\ i e. ( 0..^ M ) ) /\ A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z )
136	67, 70	syl6eq 2672	. . . . . . . . . . . . . . 15 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( RR _D ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) = ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) )
137	136	fveq1d 6193	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( RR _D ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) ` t ) = ( ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` t ) )
138		fvres 6207	. . . . . . . . . . . . . 14 |- ( t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -> ( ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ` t ) = ( ( RR _D F ) ` t ) )
139	137, 138	sylan9eq 2676	. . . . . . . . . . . . 13 |- ( ( ( ph /\ i e. ( 0..^ M ) ) /\ t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( ( RR _D ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) ` t ) = ( ( RR _D F ) ` t ) )
140	139	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ph /\ i e. ( 0..^ M ) ) /\ t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( abs ` ( ( RR _D ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) ` t ) ) = ( abs ` ( ( RR _D F ) ` t ) ) )
141	140	adantlr 751	. . . . . . . . . . 11 |- ( ( ( ( ph /\ i e. ( 0..^ M ) ) /\ A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z ) /\ t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( abs ` ( ( RR _D ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) ` t ) ) = ( abs ` ( ( RR _D F ) ` t ) ) )
142		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 0..^ M ) ) /\ A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z ) /\ t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z )
143		ssdmres 5420	. . . . . . . . . . . . . . 15 |- ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ dom ( RR _D F ) <-> dom ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) )
144	76, 143	sylibr 224	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ dom ( RR _D F ) )
145	144	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ i e. ( 0..^ M ) ) /\ A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z ) /\ t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ dom ( RR _D F ) )
146		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ i e. ( 0..^ M ) ) /\ A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z ) /\ t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) )
147	145, 146	sseldd 3604	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ i e. ( 0..^ M ) ) /\ A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z ) /\ t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> t e. dom ( RR _D F ) )
148		rspa 2930	. . . . . . . . . . . 12 |- ( ( A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z /\ t e. dom ( RR _D F ) ) -> ( abs ` ( ( RR _D F ) ` t ) ) <_ z )
149	142, 147, 148	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ( ph /\ i e. ( 0..^ M ) ) /\ A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z ) /\ t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( abs ` ( ( RR _D F ) ` t ) ) <_ z )
150	141, 149	eqbrtrd 4675	. . . . . . . . . 10 |- ( ( ( ( ph /\ i e. ( 0..^ M ) ) /\ A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z ) /\ t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) -> ( abs ` ( ( RR _D ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) ` t ) ) <_ z )
151	150	ex 450	. . . . . . . . 9 |- ( ( ( ph /\ i e. ( 0..^ M ) ) /\ A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z ) -> ( t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -> ( abs ` ( ( RR _D ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) ` t ) ) <_ z ) )
152	135, 151	ralrimi 2957	. . . . . . . 8 |- ( ( ( ph /\ i e. ( 0..^ M ) ) /\ A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z ) -> A. t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ( abs ` ( ( RR _D ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) ` t ) ) <_ z )
153	152	ex 450	. . . . . . 7 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z -> A. t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ( abs ` ( ( RR _D ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) ` t ) ) <_ z ) )
154	153	reximdv 3016	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( E. z e. RR A. t e. dom ( RR _D F ) ( abs ` ( ( RR _D F ) ` t ) ) <_ z -> E. z e. RR A. t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ( abs ` ( ( RR _D ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) ` t ) ) <_ z ) )
155	132, 154	mpd 15	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> E. z e. RR A. t e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ( abs ` ( ( RR _D ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) ` t ) ) <_ z )
156	90, 94, 98, 77, 155	ioodvbdlimc2 40150	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) =/= (/) )
157	60, 80, 97, 156, 64	ellimciota 39846	. . 3 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( iota y y e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) ) e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) )
158		fourierdlem94.x	. . 3 |- ( ph -> X e. RR )
159		oveq2 6658	. . . . . . 7 |- ( y = x -> ( pi - y ) = ( pi - x ) )
160	159	oveq1d 6665	. . . . . 6 |- ( y = x -> ( ( pi - y ) / T ) = ( ( pi - x ) / T ) )
161	160	fveq2d 6195	. . . . 5 |- ( y = x -> ( |_ ` ( ( pi - y ) / T ) ) = ( |_ ` ( ( pi - x ) / T ) ) )
162	161	oveq1d 6665	. . . 4 |- ( y = x -> ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) = ( ( |_ ` ( ( pi - x ) / T ) ) x. T ) )
163	162	cbvmptv 4750	. . 3 |- ( y e. RR |-> ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) = ( x e. RR |-> ( ( |_ ` ( ( pi - x ) / T ) ) x. T ) )
164		id 22	. . . . 5 |- ( z = x -> z = x )
165		fveq2 6191	. . . . 5 |- ( z = x -> ( ( y e. RR |-> ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ` z ) = ( ( y e. RR |-> ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ` x ) )
166	164, 165	oveq12d 6668	. . . 4 |- ( z = x -> ( z + ( ( y e. RR |-> ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ` z ) ) = ( x + ( ( y e. RR |-> ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ` x ) ) )
167	166	cbvmptv 4750	. . 3 |- ( z e. RR |-> ( z + ( ( y e. RR |-> ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ` z ) ) ) = ( x e. RR |-> ( x + ( ( y e. RR |-> ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ` x ) ) )
168	3, 4, 10, 11, 16, 17, 18, 20, 21, 32, 54, 79, 157, 158, 163, 167	fourierdlem49 40372	. 2 |- ( ph -> ( ( F |` ( -oo (,) X ) ) lim CC X ) =/= (/) )
169	90, 94, 98, 77, 155	ioodvbdlimc1 40148	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) =/= (/) )
170	60, 80, 107, 169, 64	ellimciota 39846	. . 3 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( iota y y e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) ) e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) )
171		biid 251	. . 3 |- ( ( ( ( ( ph /\ i e. ( 0..^ M ) ) /\ w e. ( ( Q ` i ) [,) ( Q ` ( i + 1 ) ) ) ) /\ k e. ZZ ) /\ w = ( X + ( k x. T ) ) ) <-> ( ( ( ( ph /\ i e. ( 0..^ M ) ) /\ w e. ( ( Q ` i ) [,) ( Q ` ( i + 1 ) ) ) ) /\ k e. ZZ ) /\ w = ( X + ( k x. T ) ) ) )
172	3, 4, 10, 11, 16, 17, 18, 21, 32, 54, 79, 170, 158, 163, 167, 171	fourierdlem48 40371	. 2 |- ( ph -> ( ( F |` ( X (,) +oo ) ) lim CC X ) =/= (/) )
173	168, 172	jca 554	1 |- ( ph -> ( ( ( F |` ( -oo (,) X ) ) lim CC X ) =/= (/) /\ ( ( F |` ( X (,) +oo ) ) lim CC X ) =/= (/) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   iotacio 5849   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   -oocmnf 10072   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   ZZcz 11377   (,)cioo 12175   [,)cico 12177   ...cfz 12326  ..^cfzo 12465   |_cfl 12591   abscabs 13974   picpi 14797   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   intcnt 20821   -cn->ccncf 22679   lim CC climc 23626   _D cdv 23627

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-inf2 8538  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  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-iin 4523  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  fourierdlem102  40425