Theorem fourierdlem102 40425

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
fourierdlem102.f	|- ( ph -> F : RR --> RR )
fourierdlem102.t	|- T = ( 2 x. pi )
fourierdlem102.per	|- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
fourierdlem102.g	|- G = ( ( RR _D F ) |` ( -u pi (,) pi ) )
fourierdlem102.dmdv	|- ( ph -> ( ( -u pi (,) pi ) \ dom G ) e. Fin )
fourierdlem102.gcn	|- ( ph -> G e. ( dom G -cn-> CC ) )
fourierdlem102.rlim	|- ( ( ph /\ x e. ( ( -u pi [,) pi ) \ dom G ) ) -> ( ( G |` ( x (,) +oo ) ) lim CC x ) =/= (/) )
fourierdlem102.llim	|- ( ( ph /\ x e. ( ( -u pi (,] pi ) \ dom G ) ) -> ( ( G |` ( -oo (,) x ) ) lim CC x ) =/= (/) )
fourierdlem102.x	|- ( ph -> X e. RR )
fourierdlem102.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 ) ) ) } )
fourierdlem102.e	|- E = ( x e. RR |-> ( x + ( ( |_ ` ( ( pi - x ) / T ) ) x. T ) ) )
fourierdlem102.h	|- H = ( { -u pi , pi , ( E ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) )
fourierdlem102.m	|- M = ( ( # ` H ) - 1 )
fourierdlem102.q	|- Q = ( iota g g Isom < , < ( ( 0 ... M ) , H ) )

Assertion

Ref	Expression
fourierdlem102	|- ( ph -> ( ( ( F |` ( -oo (,) X ) ) lim CC X ) =/= (/) /\ ( ( F |` ( X (,) +oo ) ) lim CC X ) =/= (/) ) )

Distinct variable groups:   x, E   i, F, n, x   i, G, x   g, H   g, M   i, M, n, p   x, M   Q, g   Q, i, n, p   x, Q   T, i, n, p   x, T   i, X, n, p   x, X   ph, g   ph, i, n, x

Allowed substitution hints:   ph( p)   P( x, g, i, n, p)   T( g)   E( g, i, n, p)   F( g, p)   G( g, n, p)   H( x, i, n, p)   X( g)

Proof of Theorem fourierdlem102

Step	Hyp	Ref	Expression
1		fourierdlem102.f	. 2 |- ( ph -> F : RR --> RR )
2		fourierdlem102.t	. 2 |- T = ( 2 x. pi )
3		fourierdlem102.per	. 2 |- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
4		fourierdlem102.x	. 2 |- ( ph -> X e. RR )
5		fourierdlem102.p	. 2 |- 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 ) ) ) } )
6		fourierdlem102.m	. . 3 |- M = ( ( # ` H ) - 1 )
7		2z 11409	. . . . . 6 |- 2 e. ZZ
8	7	a1i 11	. . . . 5 |- ( ph -> 2 e. ZZ )
9		fourierdlem102.h	. . . . . . . 8 |- H = ( { -u pi , pi , ( E ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) )
10		tpfi 8236	. . . . . . . . . 10 |- { -u pi , pi , ( E ` X ) } e. Fin
11	10	a1i 11	. . . . . . . . 9 |- ( ph -> { -u pi , pi , ( E ` X ) } e. Fin )
12		pire 24210	. . . . . . . . . . . . . . 15 |- pi e. RR
13	12	renegcli 10342	. . . . . . . . . . . . . 14 |- -u pi e. RR
14	13	rexri 10097	. . . . . . . . . . . . 13 |- -u pi e. RR*
15	12	rexri 10097	. . . . . . . . . . . . 13 |- pi e. RR*
16		negpilt0 39492	. . . . . . . . . . . . . . 15 |- -u pi < 0
17		pipos 24212	. . . . . . . . . . . . . . 15 |- 0 < pi
18		0re 10040	. . . . . . . . . . . . . . . 16 |- 0 e. RR
19	13, 18, 12	lttri 10163	. . . . . . . . . . . . . . 15 |- ( ( -u pi < 0 /\ 0 < pi ) -> -u pi < pi )
20	16, 17, 19	mp2an 708	. . . . . . . . . . . . . 14 |- -u pi < pi
21	13, 12, 20	ltleii 10160	. . . . . . . . . . . . 13 |- -u pi <_ pi
22		prunioo 12301	. . . . . . . . . . . . 13 |- ( ( -u pi e. RR* /\ pi e. RR* /\ -u pi <_ pi ) -> ( ( -u pi (,) pi ) u. { -u pi , pi } ) = ( -u pi [,] pi ) )
23	14, 15, 21, 22	mp3an 1424	. . . . . . . . . . . 12 |- ( ( -u pi (,) pi ) u. { -u pi , pi } ) = ( -u pi [,] pi )
24	23	difeq1i 3724	. . . . . . . . . . 11 |- ( ( ( -u pi (,) pi ) u. { -u pi , pi } ) \ dom G ) = ( ( -u pi [,] pi ) \ dom G )
25		difundir 3880	. . . . . . . . . . 11 |- ( ( ( -u pi (,) pi ) u. { -u pi , pi } ) \ dom G ) = ( ( ( -u pi (,) pi ) \ dom G ) u. ( { -u pi , pi } \ dom G ) )
26	24, 25	eqtr3i 2646	. . . . . . . . . 10 |- ( ( -u pi [,] pi ) \ dom G ) = ( ( ( -u pi (,) pi ) \ dom G ) u. ( { -u pi , pi } \ dom G ) )
27		fourierdlem102.dmdv	. . . . . . . . . . 11 |- ( ph -> ( ( -u pi (,) pi ) \ dom G ) e. Fin )
28		prfi 8235	. . . . . . . . . . . 12 |- { -u pi , pi } e. Fin
29		diffi 8192	. . . . . . . . . . . 12 |- ( { -u pi , pi } e. Fin -> ( { -u pi , pi } \ dom G ) e. Fin )
30	28, 29	mp1i 13	. . . . . . . . . . 11 |- ( ph -> ( { -u pi , pi } \ dom G ) e. Fin )
31		unfi 8227	. . . . . . . . . . 11 |- ( ( ( ( -u pi (,) pi ) \ dom G ) e. Fin /\ ( { -u pi , pi } \ dom G ) e. Fin ) -> ( ( ( -u pi (,) pi ) \ dom G ) u. ( { -u pi , pi } \ dom G ) ) e. Fin )
32	27, 30, 31	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( ( ( -u pi (,) pi ) \ dom G ) u. ( { -u pi , pi } \ dom G ) ) e. Fin )
33	26, 32	syl5eqel 2705	. . . . . . . . 9 |- ( ph -> ( ( -u pi [,] pi ) \ dom G ) e. Fin )
34		unfi 8227	. . . . . . . . 9 |- ( ( { -u pi , pi , ( E ` X ) } e. Fin /\ ( ( -u pi [,] pi ) \ dom G ) e. Fin ) -> ( { -u pi , pi , ( E ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) ) e. Fin )
35	11, 33, 34	syl2anc 693	. . . . . . . 8 |- ( ph -> ( { -u pi , pi , ( E ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) ) e. Fin )
36	9, 35	syl5eqel 2705	. . . . . . 7 |- ( ph -> H e. Fin )
37		hashcl 13147	. . . . . . 7 |- ( H e. Fin -> ( # ` H ) e. NN0 )
38	36, 37	syl 17	. . . . . 6 |- ( ph -> ( # ` H ) e. NN0 )
39	38	nn0zd 11480	. . . . 5 |- ( ph -> ( # ` H ) e. ZZ )
40	13, 20	ltneii 10150	. . . . . . 7 |- -u pi =/= pi
41		hashprg 13182	. . . . . . . 8 |- ( ( -u pi e. RR /\ pi e. RR ) -> ( -u pi =/= pi <-> ( # ` { -u pi , pi } ) = 2 ) )
42	13, 12, 41	mp2an 708	. . . . . . 7 |- ( -u pi =/= pi <-> ( # ` { -u pi , pi } ) = 2 )
43	40, 42	mpbi 220	. . . . . 6 |- ( # ` { -u pi , pi } ) = 2
44	10	elexi 3213	. . . . . . . . . 10 |- { -u pi , pi , ( E ` X ) } e. _V
45		ovex 6678	. . . . . . . . . . 11 |- ( -u pi [,] pi ) e. _V
46		difexg 4808	. . . . . . . . . . 11 |- ( ( -u pi [,] pi ) e. _V -> ( ( -u pi [,] pi ) \ dom G ) e. _V )
47	45, 46	ax-mp 5	. . . . . . . . . 10 |- ( ( -u pi [,] pi ) \ dom G ) e. _V
48	44, 47	unex 6956	. . . . . . . . 9 |- ( { -u pi , pi , ( E ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) ) e. _V
49	9, 48	eqeltri 2697	. . . . . . . 8 |- H e. _V
50		negex 10279	. . . . . . . . . . 11 |- -u pi e. _V
51	50	tpid1 4303	. . . . . . . . . 10 |- -u pi e. { -u pi , pi , ( E ` X ) }
52	12	elexi 3213	. . . . . . . . . . 11 |- pi e. _V
53	52	tpid2 4304	. . . . . . . . . 10 |- pi e. { -u pi , pi , ( E ` X ) }
54		prssi 4353	. . . . . . . . . 10 |- ( ( -u pi e. { -u pi , pi , ( E ` X ) } /\ pi e. { -u pi , pi , ( E ` X ) } ) -> { -u pi , pi } C_ { -u pi , pi , ( E ` X ) } )
55	51, 53, 54	mp2an 708	. . . . . . . . 9 |- { -u pi , pi } C_ { -u pi , pi , ( E ` X ) }
56		ssun1 3776	. . . . . . . . . 10 |- { -u pi , pi , ( E ` X ) } C_ ( { -u pi , pi , ( E ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) )
57	56, 9	sseqtr4i 3638	. . . . . . . . 9 |- { -u pi , pi , ( E ` X ) } C_ H
58	55, 57	sstri 3612	. . . . . . . 8 |- { -u pi , pi } C_ H
59		hashss 13197	. . . . . . . 8 |- ( ( H e. _V /\ { -u pi , pi } C_ H ) -> ( # ` { -u pi , pi } ) <_ ( # ` H ) )
60	49, 58, 59	mp2an 708	. . . . . . 7 |- ( # ` { -u pi , pi } ) <_ ( # ` H )
61	60	a1i 11	. . . . . 6 |- ( ph -> ( # ` { -u pi , pi } ) <_ ( # ` H ) )
62	43, 61	syl5eqbrr 4689	. . . . 5 |- ( ph -> 2 <_ ( # ` H ) )
63		eluz2 11693	. . . . 5 |- ( ( # ` H ) e. ( ZZ>= ` 2 ) <-> ( 2 e. ZZ /\ ( # ` H ) e. ZZ /\ 2 <_ ( # ` H ) ) )
64	8, 39, 62, 63	syl3anbrc 1246	. . . 4 |- ( ph -> ( # ` H ) e. ( ZZ>= ` 2 ) )
65		uz2m1nn 11763	. . . 4 |- ( ( # ` H ) e. ( ZZ>= ` 2 ) -> ( ( # ` H ) - 1 ) e. NN )
66	64, 65	syl 17	. . 3 |- ( ph -> ( ( # ` H ) - 1 ) e. NN )
67	6, 66	syl5eqel 2705	. 2 |- ( ph -> M e. NN )
68	13	a1i 11	. . . . . . . . . . 11 |- ( ph -> -u pi e. RR )
69	12	a1i 11	. . . . . . . . . . 11 |- ( ph -> pi e. RR )
70		negpitopissre 24286	. . . . . . . . . . . 12 |- ( -u pi (,] pi ) C_ RR
71	20	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> -u pi < pi )
72		picn 24211	. . . . . . . . . . . . . . . 16 |- pi e. CC
73	72	2timesi 11147	. . . . . . . . . . . . . . 15 |- ( 2 x. pi ) = ( pi + pi )
74	72, 72	subnegi 10360	. . . . . . . . . . . . . . 15 |- ( pi - -u pi ) = ( pi + pi )
75	73, 2, 74	3eqtr4i 2654	. . . . . . . . . . . . . 14 |- T = ( pi - -u pi )
76		fourierdlem102.e	. . . . . . . . . . . . . 14 |- E = ( x e. RR |-> ( x + ( ( |_ ` ( ( pi - x ) / T ) ) x. T ) ) )
77	68, 69, 71, 75, 76	fourierdlem4 40328	. . . . . . . . . . . . 13 |- ( ph -> E : RR --> ( -u pi (,] pi ) )
78	77, 4	ffvelrnd 6360	. . . . . . . . . . . 12 |- ( ph -> ( E ` X ) e. ( -u pi (,] pi ) )
79	70, 78	sseldi 3601	. . . . . . . . . . 11 |- ( ph -> ( E ` X ) e. RR )
80	68, 69, 79	3jca 1242	. . . . . . . . . 10 |- ( ph -> ( -u pi e. RR /\ pi e. RR /\ ( E ` X ) e. RR ) )
81		fvex 6201	. . . . . . . . . . 11 |- ( E ` X ) e. _V
82	50, 52, 81	tpss 4368	. . . . . . . . . 10 |- ( ( -u pi e. RR /\ pi e. RR /\ ( E ` X ) e. RR ) <-> { -u pi , pi , ( E ` X ) } C_ RR )
83	80, 82	sylib 208	. . . . . . . . 9 |- ( ph -> { -u pi , pi , ( E ` X ) } C_ RR )
84		iccssre 12255	. . . . . . . . . . 11 |- ( ( -u pi e. RR /\ pi e. RR ) -> ( -u pi [,] pi ) C_ RR )
85	13, 12, 84	mp2an 708	. . . . . . . . . 10 |- ( -u pi [,] pi ) C_ RR
86		ssdifss 3741	. . . . . . . . . 10 |- ( ( -u pi [,] pi ) C_ RR -> ( ( -u pi [,] pi ) \ dom G ) C_ RR )
87	85, 86	mp1i 13	. . . . . . . . 9 |- ( ph -> ( ( -u pi [,] pi ) \ dom G ) C_ RR )
88	83, 87	unssd 3789	. . . . . . . 8 |- ( ph -> ( { -u pi , pi , ( E ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) ) C_ RR )
89	9, 88	syl5eqss 3649	. . . . . . 7 |- ( ph -> H C_ RR )
90		fourierdlem102.q	. . . . . . 7 |- Q = ( iota g g Isom < , < ( ( 0 ... M ) , H ) )
91	36, 89, 90, 6	fourierdlem36 40360	. . . . . 6 |- ( ph -> Q Isom < , < ( ( 0 ... M ) , H ) )
92		isof1o 6573	. . . . . 6 |- ( Q Isom < , < ( ( 0 ... M ) , H ) -> Q : ( 0 ... M ) -1-1-onto-> H )
93		f1of 6137	. . . . . 6 |- ( Q : ( 0 ... M ) -1-1-onto-> H -> Q : ( 0 ... M ) --> H )
94	91, 92, 93	3syl 18	. . . . 5 |- ( ph -> Q : ( 0 ... M ) --> H )
95	94, 89	fssd 6057	. . . 4 |- ( ph -> Q : ( 0 ... M ) --> RR )
96		reex 10027	. . . . 5 |- RR e. _V
97		ovex 6678	. . . . 5 |- ( 0 ... M ) e. _V
98	96, 97	elmap 7886	. . . 4 |- ( Q e. ( RR ^m ( 0 ... M ) ) <-> Q : ( 0 ... M ) --> RR )
99	95, 98	sylibr 224	. . 3 |- ( ph -> Q e. ( RR ^m ( 0 ... M ) ) )
100		fveq2 6191	. . . . . . . . . . 11 |- ( 0 = i -> ( Q ` 0 ) = ( Q ` i ) )
101	100	adantl 482	. . . . . . . . . 10 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ 0 = i ) -> ( Q ` 0 ) = ( Q ` i ) )
102	95	ffvelrnda 6359	. . . . . . . . . . . 12 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( Q ` i ) e. RR )
103	102	leidd 10594	. . . . . . . . . . 11 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( Q ` i ) <_ ( Q ` i ) )
104	103	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ 0 = i ) -> ( Q ` i ) <_ ( Q ` i ) )
105	101, 104	eqbrtrd 4675	. . . . . . . . 9 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ 0 = i ) -> ( Q ` 0 ) <_ ( Q ` i ) )
106		elfzelz 12342	. . . . . . . . . . . . 13 |- ( i e. ( 0 ... M ) -> i e. ZZ )
107	106	zred 11482	. . . . . . . . . . . 12 |- ( i e. ( 0 ... M ) -> i e. RR )
108	107	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ -. 0 = i ) -> i e. RR )
109		elfzle1 12344	. . . . . . . . . . . 12 |- ( i e. ( 0 ... M ) -> 0 <_ i )
110	109	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ -. 0 = i ) -> 0 <_ i )
111		neqne 2802	. . . . . . . . . . . . 13 |- ( -. 0 = i -> 0 =/= i )
112	111	necomd 2849	. . . . . . . . . . . 12 |- ( -. 0 = i -> i =/= 0 )
113	112	adantl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ -. 0 = i ) -> i =/= 0 )
114	108, 110, 113	ne0gt0d 10174	. . . . . . . . . 10 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ -. 0 = i ) -> 0 < i )
115		nnssnn0 11295	. . . . . . . . . . . . . . . . 17 |- NN C_ NN0
116		nn0uz 11722	. . . . . . . . . . . . . . . . 17 |- NN0 = ( ZZ>= ` 0 )
117	115, 116	sseqtri 3637	. . . . . . . . . . . . . . . 16 |- NN C_ ( ZZ>= ` 0 )
118	117, 67	sseldi 3601	. . . . . . . . . . . . . . 15 |- ( ph -> M e. ( ZZ>= ` 0 ) )
119		eluzfz1 12348	. . . . . . . . . . . . . . 15 |- ( M e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... M ) )
120	118, 119	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> 0 e. ( 0 ... M ) )
121	94, 120	ffvelrnd 6360	. . . . . . . . . . . . 13 |- ( ph -> ( Q ` 0 ) e. H )
122	89, 121	sseldd 3604	. . . . . . . . . . . 12 |- ( ph -> ( Q ` 0 ) e. RR )
123	122	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ 0 < i ) -> ( Q ` 0 ) e. RR )
124	102	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ 0 < i ) -> ( Q ` i ) e. RR )
125		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ 0 < i ) -> 0 < i )
126	91	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ 0 < i ) -> Q Isom < , < ( ( 0 ... M ) , H ) )
127	120	anim1i 592	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( 0 e. ( 0 ... M ) /\ i e. ( 0 ... M ) ) )
128	127	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ 0 < i ) -> ( 0 e. ( 0 ... M ) /\ i e. ( 0 ... M ) ) )
129		isorel 6576	. . . . . . . . . . . . 13 |- ( ( Q Isom < , < ( ( 0 ... M ) , H ) /\ ( 0 e. ( 0 ... M ) /\ i e. ( 0 ... M ) ) ) -> ( 0 < i <-> ( Q ` 0 ) < ( Q ` i ) ) )
130	126, 128, 129	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ 0 < i ) -> ( 0 < i <-> ( Q ` 0 ) < ( Q ` i ) ) )
131	125, 130	mpbid 222	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ 0 < i ) -> ( Q ` 0 ) < ( Q ` i ) )
132	123, 124, 131	ltled 10185	. . . . . . . . . 10 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ 0 < i ) -> ( Q ` 0 ) <_ ( Q ` i ) )
133	114, 132	syldan 487	. . . . . . . . 9 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ -. 0 = i ) -> ( Q ` 0 ) <_ ( Q ` i ) )
134	105, 133	pm2.61dan 832	. . . . . . . 8 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( Q ` 0 ) <_ ( Q ` i ) )
135	134	adantr 481	. . . . . . 7 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ ( Q ` i ) = -u pi ) -> ( Q ` 0 ) <_ ( Q ` i ) )
136		simpr 477	. . . . . . 7 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ ( Q ` i ) = -u pi ) -> ( Q ` i ) = -u pi )
137	135, 136	breqtrd 4679	. . . . . 6 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ ( Q ` i ) = -u pi ) -> ( Q ` 0 ) <_ -u pi )
138	68	rexrd 10089	. . . . . . . 8 |- ( ph -> -u pi e. RR* )
139	69	rexrd 10089	. . . . . . . 8 |- ( ph -> pi e. RR* )
140		lbicc2 12288	. . . . . . . . . . . . . 14 |- ( ( -u pi e. RR* /\ pi e. RR* /\ -u pi <_ pi ) -> -u pi e. ( -u pi [,] pi ) )
141	14, 15, 21, 140	mp3an 1424	. . . . . . . . . . . . 13 |- -u pi e. ( -u pi [,] pi )
142	141	a1i 11	. . . . . . . . . . . 12 |- ( ph -> -u pi e. ( -u pi [,] pi ) )
143		ubicc2 12289	. . . . . . . . . . . . . 14 |- ( ( -u pi e. RR* /\ pi e. RR* /\ -u pi <_ pi ) -> pi e. ( -u pi [,] pi ) )
144	14, 15, 21, 143	mp3an 1424	. . . . . . . . . . . . 13 |- pi e. ( -u pi [,] pi )
145	144	a1i 11	. . . . . . . . . . . 12 |- ( ph -> pi e. ( -u pi [,] pi ) )
146		iocssicc 12261	. . . . . . . . . . . . 13 |- ( -u pi (,] pi ) C_ ( -u pi [,] pi )
147	146, 78	sseldi 3601	. . . . . . . . . . . 12 |- ( ph -> ( E ` X ) e. ( -u pi [,] pi ) )
148		tpssi 4369	. . . . . . . . . . . 12 |- ( ( -u pi e. ( -u pi [,] pi ) /\ pi e. ( -u pi [,] pi ) /\ ( E ` X ) e. ( -u pi [,] pi ) ) -> { -u pi , pi , ( E ` X ) } C_ ( -u pi [,] pi ) )
149	142, 145, 147, 148	syl3anc 1326	. . . . . . . . . . 11 |- ( ph -> { -u pi , pi , ( E ` X ) } C_ ( -u pi [,] pi ) )
150		difssd 3738	. . . . . . . . . . 11 |- ( ph -> ( ( -u pi [,] pi ) \ dom G ) C_ ( -u pi [,] pi ) )
151	149, 150	unssd 3789	. . . . . . . . . 10 |- ( ph -> ( { -u pi , pi , ( E ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) ) C_ ( -u pi [,] pi ) )
152	9, 151	syl5eqss 3649	. . . . . . . . 9 |- ( ph -> H C_ ( -u pi [,] pi ) )
153	152, 121	sseldd 3604	. . . . . . . 8 |- ( ph -> ( Q ` 0 ) e. ( -u pi [,] pi ) )
154		iccgelb 12230	. . . . . . . 8 |- ( ( -u pi e. RR* /\ pi e. RR* /\ ( Q ` 0 ) e. ( -u pi [,] pi ) ) -> -u pi <_ ( Q ` 0 ) )
155	138, 139, 153, 154	syl3anc 1326	. . . . . . 7 |- ( ph -> -u pi <_ ( Q ` 0 ) )
156	155	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ ( Q ` i ) = -u pi ) -> -u pi <_ ( Q ` 0 ) )
157	122	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ ( Q ` i ) = -u pi ) -> ( Q ` 0 ) e. RR )
158	13	a1i 11	. . . . . . 7 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ ( Q ` i ) = -u pi ) -> -u pi e. RR )
159	157, 158	letri3d 10179	. . . . . 6 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ ( Q ` i ) = -u pi ) -> ( ( Q ` 0 ) = -u pi <-> ( ( Q ` 0 ) <_ -u pi /\ -u pi <_ ( Q ` 0 ) ) ) )
160	137, 156, 159	mpbir2and 957	. . . . 5 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ ( Q ` i ) = -u pi ) -> ( Q ` 0 ) = -u pi )
161	57, 51	sselii 3600	. . . . . . 7 |- -u pi e. H
162		f1ofo 6144	. . . . . . . . 9 |- ( Q : ( 0 ... M ) -1-1-onto-> H -> Q : ( 0 ... M ) -onto-> H )
163	92, 162	syl 17	. . . . . . . 8 |- ( Q Isom < , < ( ( 0 ... M ) , H ) -> Q : ( 0 ... M ) -onto-> H )
164		forn 6118	. . . . . . . 8 |- ( Q : ( 0 ... M ) -onto-> H -> ran Q = H )
165	91, 163, 164	3syl 18	. . . . . . 7 |- ( ph -> ran Q = H )
166	161, 165	syl5eleqr 2708	. . . . . 6 |- ( ph -> -u pi e. ran Q )
167		ffn 6045	. . . . . . 7 |- ( Q : ( 0 ... M ) --> H -> Q Fn ( 0 ... M ) )
168		fvelrnb 6243	. . . . . . 7 |- ( Q Fn ( 0 ... M ) -> ( -u pi e. ran Q <-> E. i e. ( 0 ... M ) ( Q ` i ) = -u pi ) )
169	94, 167, 168	3syl 18	. . . . . 6 |- ( ph -> ( -u pi e. ran Q <-> E. i e. ( 0 ... M ) ( Q ` i ) = -u pi ) )
170	166, 169	mpbid 222	. . . . 5 |- ( ph -> E. i e. ( 0 ... M ) ( Q ` i ) = -u pi )
171	160, 170	r19.29a 3078	. . . 4 |- ( ph -> ( Q ` 0 ) = -u pi )
172	57, 53	sselii 3600	. . . . . . 7 |- pi e. H
173	172, 165	syl5eleqr 2708	. . . . . 6 |- ( ph -> pi e. ran Q )
174		fvelrnb 6243	. . . . . . 7 |- ( Q Fn ( 0 ... M ) -> ( pi e. ran Q <-> E. i e. ( 0 ... M ) ( Q ` i ) = pi ) )
175	94, 167, 174	3syl 18	. . . . . 6 |- ( ph -> ( pi e. ran Q <-> E. i e. ( 0 ... M ) ( Q ` i ) = pi ) )
176	173, 175	mpbid 222	. . . . 5 |- ( ph -> E. i e. ( 0 ... M ) ( Q ` i ) = pi )
177	94, 152	fssd 6057	. . . . . . . . . 10 |- ( ph -> Q : ( 0 ... M ) --> ( -u pi [,] pi ) )
178		eluzfz2 12349	. . . . . . . . . . 11 |- ( M e. ( ZZ>= ` 0 ) -> M e. ( 0 ... M ) )
179	118, 178	syl 17	. . . . . . . . . 10 |- ( ph -> M e. ( 0 ... M ) )
180	177, 179	ffvelrnd 6360	. . . . . . . . 9 |- ( ph -> ( Q ` M ) e. ( -u pi [,] pi ) )
181		iccleub 12229	. . . . . . . . 9 |- ( ( -u pi e. RR* /\ pi e. RR* /\ ( Q ` M ) e. ( -u pi [,] pi ) ) -> ( Q ` M ) <_ pi )
182	138, 139, 180, 181	syl3anc 1326	. . . . . . . 8 |- ( ph -> ( Q ` M ) <_ pi )
183	182	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ i e. ( 0 ... M ) /\ ( Q ` i ) = pi ) -> ( Q ` M ) <_ pi )
184		id 22	. . . . . . . . . 10 |- ( ( Q ` i ) = pi -> ( Q ` i ) = pi )
185	184	eqcomd 2628	. . . . . . . . 9 |- ( ( Q ` i ) = pi -> pi = ( Q ` i ) )
186	185	3ad2ant3 1084	. . . . . . . 8 |- ( ( ph /\ i e. ( 0 ... M ) /\ ( Q ` i ) = pi ) -> pi = ( Q ` i ) )
187	103	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ i = M ) -> ( Q ` i ) <_ ( Q ` i ) )
188		fveq2 6191	. . . . . . . . . . . 12 |- ( i = M -> ( Q ` i ) = ( Q ` M ) )
189	188	adantl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ i = M ) -> ( Q ` i ) = ( Q ` M ) )
190	187, 189	breqtrd 4679	. . . . . . . . . 10 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ i = M ) -> ( Q ` i ) <_ ( Q ` M ) )
191	107	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ -. i = M ) -> i e. RR )
192		elfzel2 12340	. . . . . . . . . . . . . 14 |- ( i e. ( 0 ... M ) -> M e. ZZ )
193	192	zred 11482	. . . . . . . . . . . . 13 |- ( i e. ( 0 ... M ) -> M e. RR )
194	193	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ -. i = M ) -> M e. RR )
195		elfzle2 12345	. . . . . . . . . . . . 13 |- ( i e. ( 0 ... M ) -> i <_ M )
196	195	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ -. i = M ) -> i <_ M )
197		neqne 2802	. . . . . . . . . . . . . 14 |- ( -. i = M -> i =/= M )
198	197	necomd 2849	. . . . . . . . . . . . 13 |- ( -. i = M -> M =/= i )
199	198	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ -. i = M ) -> M =/= i )
200	191, 194, 196, 199	leneltd 10191	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ -. i = M ) -> i < M )
201	102	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ i < M ) -> ( Q ` i ) e. RR )
202	85, 180	sseldi 3601	. . . . . . . . . . . . 13 |- ( ph -> ( Q ` M ) e. RR )
203	202	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ i < M ) -> ( Q ` M ) e. RR )
204		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ i < M ) -> i < M )
205	91	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ i < M ) -> Q Isom < , < ( ( 0 ... M ) , H ) )
206		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ i e. ( 0 ... M ) ) -> i e. ( 0 ... M ) )
207	179	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ i e. ( 0 ... M ) ) -> M e. ( 0 ... M ) )
208	206, 207	jca 554	. . . . . . . . . . . . . . 15 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( i e. ( 0 ... M ) /\ M e. ( 0 ... M ) ) )
209	208	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ i < M ) -> ( i e. ( 0 ... M ) /\ M e. ( 0 ... M ) ) )
210		isorel 6576	. . . . . . . . . . . . . 14 |- ( ( Q Isom < , < ( ( 0 ... M ) , H ) /\ ( i e. ( 0 ... M ) /\ M e. ( 0 ... M ) ) ) -> ( i < M <-> ( Q ` i ) < ( Q ` M ) ) )
211	205, 209, 210	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ i < M ) -> ( i < M <-> ( Q ` i ) < ( Q ` M ) ) )
212	204, 211	mpbid 222	. . . . . . . . . . . 12 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ i < M ) -> ( Q ` i ) < ( Q ` M ) )
213	201, 203, 212	ltled 10185	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ i < M ) -> ( Q ` i ) <_ ( Q ` M ) )
214	200, 213	syldan 487	. . . . . . . . . 10 |- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ -. i = M ) -> ( Q ` i ) <_ ( Q ` M ) )
215	190, 214	pm2.61dan 832	. . . . . . . . 9 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( Q ` i ) <_ ( Q ` M ) )
216	215	3adant3 1081	. . . . . . . 8 |- ( ( ph /\ i e. ( 0 ... M ) /\ ( Q ` i ) = pi ) -> ( Q ` i ) <_ ( Q ` M ) )
217	186, 216	eqbrtrd 4675	. . . . . . 7 |- ( ( ph /\ i e. ( 0 ... M ) /\ ( Q ` i ) = pi ) -> pi <_ ( Q ` M ) )
218	202	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ i e. ( 0 ... M ) /\ ( Q ` i ) = pi ) -> ( Q ` M ) e. RR )
219	12	a1i 11	. . . . . . . 8 |- ( ( ph /\ i e. ( 0 ... M ) /\ ( Q ` i ) = pi ) -> pi e. RR )
220	218, 219	letri3d 10179	. . . . . . 7 |- ( ( ph /\ i e. ( 0 ... M ) /\ ( Q ` i ) = pi ) -> ( ( Q ` M ) = pi <-> ( ( Q ` M ) <_ pi /\ pi <_ ( Q ` M ) ) ) )
221	183, 217, 220	mpbir2and 957	. . . . . 6 |- ( ( ph /\ i e. ( 0 ... M ) /\ ( Q ` i ) = pi ) -> ( Q ` M ) = pi )
222	221	rexlimdv3a 3033	. . . . 5 |- ( ph -> ( E. i e. ( 0 ... M ) ( Q ` i ) = pi -> ( Q ` M ) = pi ) )
223	176, 222	mpd 15	. . . 4 |- ( ph -> ( Q ` M ) = pi )
224		elfzoelz 12470	. . . . . . . . 9 |- ( i e. ( 0..^ M ) -> i e. ZZ )
225	224	zred 11482	. . . . . . . 8 |- ( i e. ( 0..^ M ) -> i e. RR )
226	225	ltp1d 10954	. . . . . . 7 |- ( i e. ( 0..^ M ) -> i < ( i + 1 ) )
227	226	adantl 482	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> i < ( i + 1 ) )
228		elfzofz 12485	. . . . . . . 8 |- ( i e. ( 0..^ M ) -> i e. ( 0 ... M ) )
229		fzofzp1 12565	. . . . . . . 8 |- ( i e. ( 0..^ M ) -> ( i + 1 ) e. ( 0 ... M ) )
230	228, 229	jca 554	. . . . . . 7 |- ( i e. ( 0..^ M ) -> ( i e. ( 0 ... M ) /\ ( i + 1 ) e. ( 0 ... M ) ) )
231		isorel 6576	. . . . . . 7 |- ( ( Q Isom < , < ( ( 0 ... M ) , H ) /\ ( i e. ( 0 ... M ) /\ ( i + 1 ) e. ( 0 ... M ) ) ) -> ( i < ( i + 1 ) <-> ( Q ` i ) < ( Q ` ( i + 1 ) ) ) )
232	91, 230, 231	syl2an 494	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( i < ( i + 1 ) <-> ( Q ` i ) < ( Q ` ( i + 1 ) ) ) )
233	227, 232	mpbid 222	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` i ) < ( Q ` ( i + 1 ) ) )
234	233	ralrimiva 2966	. . . 4 |- ( ph -> A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) )
235	171, 223, 234	jca31 557	. . 3 |- ( ph -> ( ( ( Q ` 0 ) = -u pi /\ ( Q ` M ) = pi ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) )
236	5	fourierdlem2 40326	. . . 4 |- ( 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 ) ) ) ) ) )
237	67, 236	syl 17	. . 3 |- ( 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 ) ) ) ) ) )
238	99, 235, 237	mpbir2and 957	. 2 |- ( ph -> Q e. ( P ` M ) )
239		fourierdlem102.g	. . . . 5 |- G = ( ( RR _D F ) |` ( -u pi (,) pi ) )
240	239	reseq1i 5392	. . . 4 |- ( G |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( ( ( RR _D F ) |` ( -u pi (,) pi ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) )
241	14	a1i 11	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> -u pi e. RR* )
242	15	a1i 11	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> pi e. RR* )
243	177	adantr 481	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> Q : ( 0 ... M ) --> ( -u pi [,] pi ) )
244		simpr 477	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> i e. ( 0..^ M ) )
245	241, 242, 243, 244	fourierdlem27 40351	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ ( -u pi (,) pi ) )
246	245	resabs1d 5428	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( ( RR _D F ) |` ( -u pi (,) pi ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) )
247	240, 246	syl5req 2669	. . 3 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( G |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) )
248		fourierdlem102.gcn	. . . 4 |- ( ph -> G e. ( dom G -cn-> CC ) )
249	248, 5, 67, 238, 9, 165	fourierdlem38 40362	. . 3 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( G |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
250	247, 249	eqeltrd 2701	. 2 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
251	247	oveq1d 6665	. . 3 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) = ( ( G |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) )
252	248	adantr 481	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> G e. ( dom G -cn-> CC ) )
253		fourierdlem102.rlim	. . . . . 6 |- ( ( ph /\ x e. ( ( -u pi [,) pi ) \ dom G ) ) -> ( ( G |` ( x (,) +oo ) ) lim CC x ) =/= (/) )
254	253	adantlr 751	. . . . 5 |- ( ( ( ph /\ i e. ( 0..^ M ) ) /\ x e. ( ( -u pi [,) pi ) \ dom G ) ) -> ( ( G |` ( x (,) +oo ) ) lim CC x ) =/= (/) )
255		fourierdlem102.llim	. . . . . 6 |- ( ( ph /\ x e. ( ( -u pi (,] pi ) \ dom G ) ) -> ( ( G |` ( -oo (,) x ) ) lim CC x ) =/= (/) )
256	255	adantlr 751	. . . . 5 |- ( ( ( ph /\ i e. ( 0..^ M ) ) /\ x e. ( ( -u pi (,] pi ) \ dom G ) ) -> ( ( G |` ( -oo (,) x ) ) lim CC x ) =/= (/) )
257	91	adantr 481	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> Q Isom < , < ( ( 0 ... M ) , H ) )
258	257, 92, 93	3syl 18	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> Q : ( 0 ... M ) --> H )
259	79	adantr 481	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( E ` X ) e. RR )
260	257, 163, 164	3syl 18	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ran Q = H )
261	252, 254, 256, 257, 258, 244, 233, 245, 259, 9, 260	fourierdlem46 40369	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( ( G |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) =/= (/) /\ ( ( G |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) =/= (/) ) )
262	261	simpld 475	. . 3 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( G |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) =/= (/) )
263	251, 262	eqnetrd 2861	. 2 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) =/= (/) )
264	247	oveq1d 6665	. . 3 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) = ( ( G |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) )
265	261	simprd 479	. . 3 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( G |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) =/= (/) )
266	264, 265	eqnetrd 2861	. 2 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( ( RR _D F ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) =/= (/) )
267	1, 2, 3, 4, 5, 67, 238, 250, 263, 266	fourierdlem94 40417	1 |- ( ph -> ( ( ( F |` ( -oo (,) X ) ) lim CC X ) =/= (/) /\ ( ( F |` ( X (,) +oo ) ) lim CC X ) =/= (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {cpr 4179   {ctp 4181   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   iotacio 5849   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   (,)cioo 12175   (,]cioc 12176   [,)cico 12177   [,]cicc 12178   ...cfz 12326  ..^cfzo 12465   |_cfl 12591   #chash 13117   picpi 14797   -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-xnn0 11364  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:  fourierdlem106  40429