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Theorem fourierdlem35 38117
Description: There is a single point in  ( A (,] B ) that's distant from  X a multiple integer of  T. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem35.a  |-  ( ph  ->  A  e.  RR )
fourierdlem35.b  |-  ( ph  ->  B  e.  RR )
fourierdlem35.altb  |-  ( ph  ->  A  <  B )
fourierdlem35.t  |-  T  =  ( B  -  A
)
fourierdlem35.5  |-  ( ph  ->  X  e.  RR )
fourierdlem35.i  |-  ( ph  ->  I  e.  ZZ )
fourierdlem35.j  |-  ( ph  ->  J  e.  ZZ )
fourierdlem35.iel  |-  ( ph  ->  ( X  +  ( I  x.  T ) )  e.  ( A (,] B ) )
fourierdlem35.jel  |-  ( ph  ->  ( X  +  ( J  x.  T ) )  e.  ( A (,] B ) )
Assertion
Ref Expression
fourierdlem35  |-  ( ph  ->  I  =  J )

Proof of Theorem fourierdlem35
StepHypRef Expression
1 neqne 2651 . . 3  |-  ( -.  I  =  J  ->  I  =/=  J )
2 fourierdlem35.a . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
32adantr 472 . . . . . . 7  |-  ( (
ph  /\  I  <  J )  ->  A  e.  RR )
4 fourierdlem35.b . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
54adantr 472 . . . . . . 7  |-  ( (
ph  /\  I  <  J )  ->  B  e.  RR )
6 fourierdlem35.altb . . . . . . . 8  |-  ( ph  ->  A  <  B )
76adantr 472 . . . . . . 7  |-  ( (
ph  /\  I  <  J )  ->  A  <  B )
8 fourierdlem35.t . . . . . . 7  |-  T  =  ( B  -  A
)
9 fourierdlem35.5 . . . . . . . 8  |-  ( ph  ->  X  e.  RR )
109adantr 472 . . . . . . 7  |-  ( (
ph  /\  I  <  J )  ->  X  e.  RR )
11 fourierdlem35.i . . . . . . . 8  |-  ( ph  ->  I  e.  ZZ )
1211adantr 472 . . . . . . 7  |-  ( (
ph  /\  I  <  J )  ->  I  e.  ZZ )
13 fourierdlem35.j . . . . . . . 8  |-  ( ph  ->  J  e.  ZZ )
1413adantr 472 . . . . . . 7  |-  ( (
ph  /\  I  <  J )  ->  J  e.  ZZ )
15 simpr 468 . . . . . . 7  |-  ( (
ph  /\  I  <  J )  ->  I  <  J )
16 iocssicc 11747 . . . . . . . . 9  |-  ( A (,] B )  C_  ( A [,] B )
17 fourierdlem35.iel . . . . . . . . 9  |-  ( ph  ->  ( X  +  ( I  x.  T ) )  e.  ( A (,] B ) )
1816, 17sseldi 3416 . . . . . . . 8  |-  ( ph  ->  ( X  +  ( I  x.  T ) )  e.  ( A [,] B ) )
1918adantr 472 . . . . . . 7  |-  ( (
ph  /\  I  <  J )  ->  ( X  +  ( I  x.  T ) )  e.  ( A [,] B
) )
20 fourierdlem35.jel . . . . . . . . 9  |-  ( ph  ->  ( X  +  ( J  x.  T ) )  e.  ( A (,] B ) )
2116, 20sseldi 3416 . . . . . . . 8  |-  ( ph  ->  ( X  +  ( J  x.  T ) )  e.  ( A [,] B ) )
2221adantr 472 . . . . . . 7  |-  ( (
ph  /\  I  <  J )  ->  ( X  +  ( J  x.  T ) )  e.  ( A [,] B
) )
233, 5, 7, 8, 10, 12, 14, 15, 19, 22fourierdlem6 38087 . . . . . 6  |-  ( (
ph  /\  I  <  J )  ->  J  =  ( I  +  1
) )
2423orcd 399 . . . . 5  |-  ( (
ph  /\  I  <  J )  ->  ( J  =  ( I  + 
1 )  \/  I  =  ( J  + 
1 ) ) )
2524adantlr 729 . . . 4  |-  ( ( ( ph  /\  I  =/=  J )  /\  I  <  J )  ->  ( J  =  ( I  +  1 )  \/  I  =  ( J  +  1 ) ) )
26 simpll 768 . . . . 5  |-  ( ( ( ph  /\  I  =/=  J )  /\  -.  I  <  J )  ->  ph )
2713zred 11063 . . . . . . 7  |-  ( ph  ->  J  e.  RR )
2826, 27syl 17 . . . . . 6  |-  ( ( ( ph  /\  I  =/=  J )  /\  -.  I  <  J )  ->  J  e.  RR )
2911zred 11063 . . . . . . 7  |-  ( ph  ->  I  e.  RR )
3026, 29syl 17 . . . . . 6  |-  ( ( ( ph  /\  I  =/=  J )  /\  -.  I  <  J )  ->  I  e.  RR )
31 id 22 . . . . . . . 8  |-  ( I  =/=  J  ->  I  =/=  J )
3231necomd 2698 . . . . . . 7  |-  ( I  =/=  J  ->  J  =/=  I )
3332ad2antlr 741 . . . . . 6  |-  ( ( ( ph  /\  I  =/=  J )  /\  -.  I  <  J )  ->  J  =/=  I )
34 simpr 468 . . . . . 6  |-  ( ( ( ph  /\  I  =/=  J )  /\  -.  I  <  J )  ->  -.  I  <  J )
3528, 30, 33, 34lttri5d 37605 . . . . 5  |-  ( ( ( ph  /\  I  =/=  J )  /\  -.  I  <  J )  ->  J  <  I )
362adantr 472 . . . . . . 7  |-  ( (
ph  /\  J  <  I )  ->  A  e.  RR )
374adantr 472 . . . . . . 7  |-  ( (
ph  /\  J  <  I )  ->  B  e.  RR )
386adantr 472 . . . . . . 7  |-  ( (
ph  /\  J  <  I )  ->  A  <  B )
399adantr 472 . . . . . . 7  |-  ( (
ph  /\  J  <  I )  ->  X  e.  RR )
4013adantr 472 . . . . . . 7  |-  ( (
ph  /\  J  <  I )  ->  J  e.  ZZ )
4111adantr 472 . . . . . . 7  |-  ( (
ph  /\  J  <  I )  ->  I  e.  ZZ )
42 simpr 468 . . . . . . 7  |-  ( (
ph  /\  J  <  I )  ->  J  <  I )
4321adantr 472 . . . . . . 7  |-  ( (
ph  /\  J  <  I )  ->  ( X  +  ( J  x.  T ) )  e.  ( A [,] B
) )
4418adantr 472 . . . . . . 7  |-  ( (
ph  /\  J  <  I )  ->  ( X  +  ( I  x.  T ) )  e.  ( A [,] B
) )
4536, 37, 38, 8, 39, 40, 41, 42, 43, 44fourierdlem6 38087 . . . . . 6  |-  ( (
ph  /\  J  <  I )  ->  I  =  ( J  +  1
) )
4645olcd 400 . . . . 5  |-  ( (
ph  /\  J  <  I )  ->  ( J  =  ( I  + 
1 )  \/  I  =  ( J  + 
1 ) ) )
4726, 35, 46syl2anc 673 . . . 4  |-  ( ( ( ph  /\  I  =/=  J )  /\  -.  I  <  J )  -> 
( J  =  ( I  +  1 )  \/  I  =  ( J  +  1 ) ) )
4825, 47pm2.61dan 808 . . 3  |-  ( (
ph  /\  I  =/=  J )  ->  ( J  =  ( I  + 
1 )  \/  I  =  ( J  + 
1 ) ) )
491, 48sylan2 482 . 2  |-  ( (
ph  /\  -.  I  =  J )  ->  ( J  =  ( I  +  1 )  \/  I  =  ( J  +  1 ) ) )
502rexrd 9708 . . . . . . . 8  |-  ( ph  ->  A  e.  RR* )
514rexrd 9708 . . . . . . . 8  |-  ( ph  ->  B  e.  RR* )
52 iocleub 37696 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( X  +  ( J  x.  T ) )  e.  ( A (,] B
) )  ->  ( X  +  ( J  x.  T ) )  <_  B )
5350, 51, 20, 52syl3anc 1292 . . . . . . 7  |-  ( ph  ->  ( X  +  ( J  x.  T ) )  <_  B )
5453adantr 472 . . . . . 6  |-  ( (
ph  /\  J  =  ( I  +  1
) )  ->  ( X  +  ( J  x.  T ) )  <_  B )
552adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  J  =  ( I  +  1
) )  ->  A  e.  RR )
564, 2resubcld 10068 . . . . . . . . . . . . 13  |-  ( ph  ->  ( B  -  A
)  e.  RR )
578, 56syl5eqel 2553 . . . . . . . . . . . 12  |-  ( ph  ->  T  e.  RR )
5829, 57remulcld 9689 . . . . . . . . . . 11  |-  ( ph  ->  ( I  x.  T
)  e.  RR )
599, 58readdcld 9688 . . . . . . . . . 10  |-  ( ph  ->  ( X  +  ( I  x.  T ) )  e.  RR )
6059adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  J  =  ( I  +  1
) )  ->  ( X  +  ( I  x.  T ) )  e.  RR )
6157adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  J  =  ( I  +  1
) )  ->  T  e.  RR )
62 iocgtlb 37695 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( X  +  ( I  x.  T ) )  e.  ( A (,] B
) )  ->  A  <  ( X  +  ( I  x.  T ) ) )
6350, 51, 17, 62syl3anc 1292 . . . . . . . . . 10  |-  ( ph  ->  A  <  ( X  +  ( I  x.  T ) ) )
6463adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  J  =  ( I  +  1
) )  ->  A  <  ( X  +  ( I  x.  T ) ) )
6555, 60, 61, 64ltadd1dd 10245 . . . . . . . 8  |-  ( (
ph  /\  J  =  ( I  +  1
) )  ->  ( A  +  T )  <  ( ( X  +  ( I  x.  T
) )  +  T
) )
668eqcomi 2480 . . . . . . . . . . 11  |-  ( B  -  A )  =  T
674recnd 9687 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  CC )
682recnd 9687 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CC )
6957recnd 9687 . . . . . . . . . . . 12  |-  ( ph  ->  T  e.  CC )
7067, 68, 69subaddd 10023 . . . . . . . . . . 11  |-  ( ph  ->  ( ( B  -  A )  =  T  <-> 
( A  +  T
)  =  B ) )
7166, 70mpbii 216 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  T
)  =  B )
7271eqcomd 2477 . . . . . . . . 9  |-  ( ph  ->  B  =  ( A  +  T ) )
7372adantr 472 . . . . . . . 8  |-  ( (
ph  /\  J  =  ( I  +  1
) )  ->  B  =  ( A  +  T ) )
749recnd 9687 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  CC )
7558recnd 9687 . . . . . . . . . . 11  |-  ( ph  ->  ( I  x.  T
)  e.  CC )
7674, 75, 69addassd 9683 . . . . . . . . . 10  |-  ( ph  ->  ( ( X  +  ( I  x.  T
) )  +  T
)  =  ( X  +  ( ( I  x.  T )  +  T ) ) )
7776adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  J  =  ( I  +  1
) )  ->  (
( X  +  ( I  x.  T ) )  +  T )  =  ( X  +  ( ( I  x.  T )  +  T
) ) )
7829recnd 9687 . . . . . . . . . . . . 13  |-  ( ph  ->  I  e.  CC )
7978, 69adddirp1d 37598 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( I  + 
1 )  x.  T
)  =  ( ( I  x.  T )  +  T ) )
8079eqcomd 2477 . . . . . . . . . . 11  |-  ( ph  ->  ( ( I  x.  T )  +  T
)  =  ( ( I  +  1 )  x.  T ) )
8180oveq2d 6324 . . . . . . . . . 10  |-  ( ph  ->  ( X  +  ( ( I  x.  T
)  +  T ) )  =  ( X  +  ( ( I  +  1 )  x.  T ) ) )
8281adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  J  =  ( I  +  1
) )  ->  ( X  +  ( (
I  x.  T )  +  T ) )  =  ( X  +  ( ( I  + 
1 )  x.  T
) ) )
83 oveq1 6315 . . . . . . . . . . . 12  |-  ( J  =  ( I  + 
1 )  ->  ( J  x.  T )  =  ( ( I  +  1 )  x.  T ) )
8483eqcomd 2477 . . . . . . . . . . 11  |-  ( J  =  ( I  + 
1 )  ->  (
( I  +  1 )  x.  T )  =  ( J  x.  T ) )
8584oveq2d 6324 . . . . . . . . . 10  |-  ( J  =  ( I  + 
1 )  ->  ( X  +  ( (
I  +  1 )  x.  T ) )  =  ( X  +  ( J  x.  T
) ) )
8685adantl 473 . . . . . . . . 9  |-  ( (
ph  /\  J  =  ( I  +  1
) )  ->  ( X  +  ( (
I  +  1 )  x.  T ) )  =  ( X  +  ( J  x.  T
) ) )
8777, 82, 863eqtrrd 2510 . . . . . . . 8  |-  ( (
ph  /\  J  =  ( I  +  1
) )  ->  ( X  +  ( J  x.  T ) )  =  ( ( X  +  ( I  x.  T
) )  +  T
) )
8865, 73, 873brtr4d 4426 . . . . . . 7  |-  ( (
ph  /\  J  =  ( I  +  1
) )  ->  B  <  ( X  +  ( J  x.  T ) ) )
894adantr 472 . . . . . . . 8  |-  ( (
ph  /\  J  =  ( I  +  1
) )  ->  B  e.  RR )
9027, 57remulcld 9689 . . . . . . . . . 10  |-  ( ph  ->  ( J  x.  T
)  e.  RR )
919, 90readdcld 9688 . . . . . . . . 9  |-  ( ph  ->  ( X  +  ( J  x.  T ) )  e.  RR )
9291adantr 472 . . . . . . . 8  |-  ( (
ph  /\  J  =  ( I  +  1
) )  ->  ( X  +  ( J  x.  T ) )  e.  RR )
9389, 92ltnled 9799 . . . . . . 7  |-  ( (
ph  /\  J  =  ( I  +  1
) )  ->  ( B  <  ( X  +  ( J  x.  T
) )  <->  -.  ( X  +  ( J  x.  T ) )  <_  B ) )
9488, 93mpbid 215 . . . . . 6  |-  ( (
ph  /\  J  =  ( I  +  1
) )  ->  -.  ( X  +  ( J  x.  T )
)  <_  B )
9554, 94pm2.65da 586 . . . . 5  |-  ( ph  ->  -.  J  =  ( I  +  1 ) )
96 iocleub 37696 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( X  +  ( I  x.  T ) )  e.  ( A (,] B
) )  ->  ( X  +  ( I  x.  T ) )  <_  B )
9750, 51, 17, 96syl3anc 1292 . . . . . . 7  |-  ( ph  ->  ( X  +  ( I  x.  T ) )  <_  B )
9897adantr 472 . . . . . 6  |-  ( (
ph  /\  I  =  ( J  +  1
) )  ->  ( X  +  ( I  x.  T ) )  <_  B )
992adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  I  =  ( J  +  1
) )  ->  A  e.  RR )
10091adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  I  =  ( J  +  1
) )  ->  ( X  +  ( J  x.  T ) )  e.  RR )
10157adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  I  =  ( J  +  1
) )  ->  T  e.  RR )
102 iocgtlb 37695 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( X  +  ( J  x.  T ) )  e.  ( A (,] B
) )  ->  A  <  ( X  +  ( J  x.  T ) ) )
10350, 51, 20, 102syl3anc 1292 . . . . . . . . . 10  |-  ( ph  ->  A  <  ( X  +  ( J  x.  T ) ) )
104103adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  I  =  ( J  +  1
) )  ->  A  <  ( X  +  ( J  x.  T ) ) )
10599, 100, 101, 104ltadd1dd 10245 . . . . . . . 8  |-  ( (
ph  /\  I  =  ( J  +  1
) )  ->  ( A  +  T )  <  ( ( X  +  ( J  x.  T
) )  +  T
) )
10672adantr 472 . . . . . . . 8  |-  ( (
ph  /\  I  =  ( J  +  1
) )  ->  B  =  ( A  +  T ) )
10790recnd 9687 . . . . . . . . . . 11  |-  ( ph  ->  ( J  x.  T
)  e.  CC )
10874, 107, 69addassd 9683 . . . . . . . . . 10  |-  ( ph  ->  ( ( X  +  ( J  x.  T
) )  +  T
)  =  ( X  +  ( ( J  x.  T )  +  T ) ) )
109108adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  I  =  ( J  +  1
) )  ->  (
( X  +  ( J  x.  T ) )  +  T )  =  ( X  +  ( ( J  x.  T )  +  T
) ) )
11027recnd 9687 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  CC )
111110, 69adddirp1d 37598 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( J  + 
1 )  x.  T
)  =  ( ( J  x.  T )  +  T ) )
112111eqcomd 2477 . . . . . . . . . . 11  |-  ( ph  ->  ( ( J  x.  T )  +  T
)  =  ( ( J  +  1 )  x.  T ) )
113112oveq2d 6324 . . . . . . . . . 10  |-  ( ph  ->  ( X  +  ( ( J  x.  T
)  +  T ) )  =  ( X  +  ( ( J  +  1 )  x.  T ) ) )
114113adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  I  =  ( J  +  1
) )  ->  ( X  +  ( ( J  x.  T )  +  T ) )  =  ( X  +  ( ( J  +  1 )  x.  T ) ) )
115 oveq1 6315 . . . . . . . . . . . 12  |-  ( I  =  ( J  + 
1 )  ->  (
I  x.  T )  =  ( ( J  +  1 )  x.  T ) )
116115eqcomd 2477 . . . . . . . . . . 11  |-  ( I  =  ( J  + 
1 )  ->  (
( J  +  1 )  x.  T )  =  ( I  x.  T ) )
117116oveq2d 6324 . . . . . . . . . 10  |-  ( I  =  ( J  + 
1 )  ->  ( X  +  ( ( J  +  1 )  x.  T ) )  =  ( X  +  ( I  x.  T
) ) )
118117adantl 473 . . . . . . . . 9  |-  ( (
ph  /\  I  =  ( J  +  1
) )  ->  ( X  +  ( ( J  +  1 )  x.  T ) )  =  ( X  +  ( I  x.  T
) ) )
119109, 114, 1183eqtrrd 2510 . . . . . . . 8  |-  ( (
ph  /\  I  =  ( J  +  1
) )  ->  ( X  +  ( I  x.  T ) )  =  ( ( X  +  ( J  x.  T
) )  +  T
) )
120105, 106, 1193brtr4d 4426 . . . . . . 7  |-  ( (
ph  /\  I  =  ( J  +  1
) )  ->  B  <  ( X  +  ( I  x.  T ) ) )
1214adantr 472 . . . . . . . 8  |-  ( (
ph  /\  I  =  ( J  +  1
) )  ->  B  e.  RR )
12259adantr 472 . . . . . . . 8  |-  ( (
ph  /\  I  =  ( J  +  1
) )  ->  ( X  +  ( I  x.  T ) )  e.  RR )
123121, 122ltnled 9799 . . . . . . 7  |-  ( (
ph  /\  I  =  ( J  +  1
) )  ->  ( B  <  ( X  +  ( I  x.  T
) )  <->  -.  ( X  +  ( I  x.  T ) )  <_  B ) )
124120, 123mpbid 215 . . . . . 6  |-  ( (
ph  /\  I  =  ( J  +  1
) )  ->  -.  ( X  +  (
I  x.  T ) )  <_  B )
12598, 124pm2.65da 586 . . . . 5  |-  ( ph  ->  -.  I  =  ( J  +  1 ) )
12695, 125jca 541 . . . 4  |-  ( ph  ->  ( -.  J  =  ( I  +  1 )  /\  -.  I  =  ( J  + 
1 ) ) )
127126adantr 472 . . 3  |-  ( (
ph  /\  -.  I  =  J )  ->  ( -.  J  =  (
I  +  1 )  /\  -.  I  =  ( J  +  1 ) ) )
128 pm4.56 503 . . 3  |-  ( ( -.  J  =  ( I  +  1 )  /\  -.  I  =  ( J  +  1 ) )  <->  -.  ( J  =  ( I  +  1 )  \/  I  =  ( J  +  1 ) ) )
129127, 128sylib 201 . 2  |-  ( (
ph  /\  -.  I  =  J )  ->  -.  ( J  =  (
I  +  1 )  \/  I  =  ( J  +  1 ) ) )
13049, 129condan 811 1  |-  ( ph  ->  I  =  J )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   class class class wbr 4395  (class class class)co 6308   RRcr 9556   1c1 9558    + caddc 9560    x. cmul 9562   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880   ZZcz 10961   (,]cioc 11661   [,]cicc 11663
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-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
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-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-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  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-n0 10894  df-z 10962  df-rp 11326  df-ioc 11665  df-icc 11667
This theorem is referenced by:  fourierdlem51  38133
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