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Theorem fourierdlem24 37987
Description: A sufficient condition for module being nonzero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
fourierdlem24  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  ( A  mod  ( 2  x.  pi ) )  =/=  0
)

Proof of Theorem fourierdlem24
StepHypRef Expression
1 0zd 10946 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  0  e.  ZZ )
2 pire 23406 . . . . . . . . . 10  |-  pi  e.  RR
32renegcli 9932 . . . . . . . . 9  |-  -u pi  e.  RR
4 iccssre 11713 . . . . . . . . 9  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR )  ->  ( -u pi [,] pi )  C_  RR )
53, 2, 4mp2an 677 . . . . . . . 8  |-  ( -u pi [,] pi )  C_  RR
6 eldifi 3554 . . . . . . . 8  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  e.  (
-u pi [,] pi ) )
75, 6sseldi 3429 . . . . . . 7  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  e.  RR )
87adantr 467 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  A  e.  RR )
9 2re 10676 . . . . . . . 8  |-  2  e.  RR
109, 2remulcli 9654 . . . . . . 7  |-  ( 2  x.  pi )  e.  RR
1110a1i 11 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  (
2  x.  pi )  e.  RR )
12 simpr 463 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  0  <  A )
13 2pos 10698 . . . . . . . 8  |-  0  <  2
14 pipos 23408 . . . . . . . 8  |-  0  <  pi
159, 2, 13, 14mulgt0ii 9765 . . . . . . 7  |-  0  <  ( 2  x.  pi )
1615a1i 11 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  0  <  ( 2  x.  pi ) )
178, 11, 12, 16divgt0d 10539 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  0  <  ( A  /  (
2  x.  pi ) ) )
1811, 16elrpd 11335 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  (
2  x.  pi )  e.  RR+ )
192a1i 11 . . . . . . . . . 10  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  pi  e.  RR )
2010a1i 11 . . . . . . . . . 10  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  ( 2  x.  pi )  e.  RR )
213rexri 9690 . . . . . . . . . . . 12  |-  -u pi  e.  RR*
2221a1i 11 . . . . . . . . . . 11  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u pi  e.  RR* )
2319rexrd 9687 . . . . . . . . . . 11  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  pi  e.  RR* )
24 iccleub 11687 . . . . . . . . . . 11  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  A  e.  ( -u pi [,] pi ) )  ->  A  <_  pi )
2522, 23, 6, 24syl3anc 1267 . . . . . . . . . 10  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  <_  pi )
26 pirp 23409 . . . . . . . . . . 11  |-  pi  e.  RR+
27 2timesgt 37495 . . . . . . . . . . 11  |-  ( pi  e.  RR+  ->  pi  <  ( 2  x.  pi ) )
2826, 27mp1i 13 . . . . . . . . . 10  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  pi  <  (
2  x.  pi ) )
297, 19, 20, 25, 28lelttrd 9790 . . . . . . . . 9  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  <  (
2  x.  pi ) )
3029adantr 467 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  A  <  ( 2  x.  pi ) )
318, 11, 18, 30ltdiv1dd 11392 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  ( A  /  ( 2  x.  pi ) )  < 
( ( 2  x.  pi )  /  (
2  x.  pi ) ) )
3210recni 9652 . . . . . . . 8  |-  ( 2  x.  pi )  e.  CC
3310, 15gt0ne0ii 10147 . . . . . . . 8  |-  ( 2  x.  pi )  =/=  0
3432, 33dividi 10337 . . . . . . 7  |-  ( ( 2  x.  pi )  /  ( 2  x.  pi ) )  =  1
3531, 34syl6breq 4441 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  ( A  /  ( 2  x.  pi ) )  <  1 )
36 0p1e1 10718 . . . . . 6  |-  ( 0  +  1 )  =  1
3735, 36syl6breqr 4442 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  ( A  /  ( 2  x.  pi ) )  < 
( 0  +  1 ) )
38 btwnnz 11009 . . . . 5  |-  ( ( 0  e.  ZZ  /\  0  <  ( A  / 
( 2  x.  pi ) )  /\  ( A  /  ( 2  x.  pi ) )  < 
( 0  +  1 ) )  ->  -.  ( A  /  (
2  x.  pi ) )  e.  ZZ )
391, 17, 37, 38syl3anc 1267 . . . 4  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  -.  ( A  /  (
2  x.  pi ) )  e.  ZZ )
40 simpl 459 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  A  e.  ( ( -u pi [,] pi )  \  {
0 } ) )
417adantr 467 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  A  e.  RR )
42 0red 9641 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  0  e.  RR )
43 simpr 463 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  -.  0  <  A )
4441, 42, 43nltled 9782 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  A  <_  0 )
45 eldifsni 4097 . . . . . . . 8  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  =/=  0
)
4645necomd 2678 . . . . . . 7  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  0  =/=  A
)
4746adantr 467 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  0  =/=  A )
4841, 42, 44, 47leneltd 9786 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  A  <  0 )
49 neg1z 10970 . . . . . . 7  |-  -u 1  e.  ZZ
5049a1i 11 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u 1  e.  ZZ )
5133a1i 11 . . . . . . . . 9  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  ( 2  x.  pi )  =/=  0
)
527, 20, 51redivcld 10432 . . . . . . . 8  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  ( A  / 
( 2  x.  pi ) )  e.  RR )
5352adantr 467 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( A  /  ( 2  x.  pi ) )  e.  RR )
54 1red 9655 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  1  e.  RR )
557recnd 9666 . . . . . . . . . 10  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  e.  CC )
5655adantr 467 . . . . . . . . 9  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  A  e.  CC )
5732a1i 11 . . . . . . . . 9  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  (
2  x.  pi )  e.  CC )
5833a1i 11 . . . . . . . . 9  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  (
2  x.  pi )  =/=  0 )
5956, 57, 58divnegd 10393 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u ( A  /  ( 2  x.  pi ) )  =  ( -u A  / 
( 2  x.  pi ) ) )
607renegcld 10043 . . . . . . . . . . 11  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u A  e.  RR )
6160adantr 467 . . . . . . . . . 10  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u A  e.  RR )
6210a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  (
2  x.  pi )  e.  RR )
63 2rp 11304 . . . . . . . . . . . 12  |-  2  e.  RR+
64 rpmulcl 11321 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR+  /\  pi  e.  RR+ )  ->  (
2  x.  pi )  e.  RR+ )
6563, 26, 64mp2an 677 . . . . . . . . . . 11  |-  ( 2  x.  pi )  e.  RR+
6665a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  (
2  x.  pi )  e.  RR+ )
67 iccgelb 11688 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  A  e.  ( -u pi [,] pi ) )  ->  -u pi  <_  A )
6822, 23, 6, 67syl3anc 1267 . . . . . . . . . . . . 13  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u pi  <_  A
)
6919, 7, 68lenegcon1d 10192 . . . . . . . . . . . 12  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u A  <_  pi )
7060, 19, 20, 69, 28lelttrd 9790 . . . . . . . . . . 11  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u A  <  (
2  x.  pi ) )
7170adantr 467 . . . . . . . . . 10  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u A  <  ( 2  x.  pi ) )
7261, 62, 66, 71ltdiv1dd 11392 . . . . . . . . 9  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( -u A  /  ( 2  x.  pi ) )  <  ( ( 2  x.  pi )  / 
( 2  x.  pi ) ) )
7372, 34syl6breq 4441 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( -u A  /  ( 2  x.  pi ) )  <  1 )
7459, 73eqbrtrd 4422 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u ( A  /  ( 2  x.  pi ) )  <  1 )
7553, 54, 74ltnegcon1d 10190 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u 1  <  ( A  /  (
2  x.  pi ) ) )
767adantr 467 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  A  e.  RR )
77 simpr 463 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  A  <  0 )
7876, 66, 77divlt0gt0d 37490 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( A  /  ( 2  x.  pi ) )  <  0 )
79 neg1cn 10710 . . . . . . . . 9  |-  -u 1  e.  CC
80 ax-1cn 9594 . . . . . . . . 9  |-  1  e.  CC
8179, 80addcomi 9821 . . . . . . . 8  |-  ( -u
1  +  1 )  =  ( 1  + 
-u 1 )
82 1pneg1e0 10715 . . . . . . . 8  |-  ( 1  +  -u 1 )  =  0
8381, 82eqtr2i 2473 . . . . . . 7  |-  0  =  ( -u 1  +  1 )
8478, 83syl6breq 4441 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( A  /  ( 2  x.  pi ) )  < 
( -u 1  +  1 ) )
85 btwnnz 11009 . . . . . 6  |-  ( (
-u 1  e.  ZZ  /\  -u 1  <  ( A  /  ( 2  x.  pi ) )  /\  ( A  /  (
2  x.  pi ) )  <  ( -u
1  +  1 ) )  ->  -.  ( A  /  ( 2  x.  pi ) )  e.  ZZ )
8650, 75, 84, 85syl3anc 1267 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -.  ( A  /  (
2  x.  pi ) )  e.  ZZ )
8740, 48, 86syl2anc 666 . . . 4  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  -.  ( A  /  (
2  x.  pi ) )  e.  ZZ )
8839, 87pm2.61dan 799 . . 3  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -.  ( A  /  ( 2  x.  pi ) )  e.  ZZ )
8965a1i 11 . . . 4  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  ( 2  x.  pi )  e.  RR+ )
90 mod0 12100 . . . 4  |-  ( ( A  e.  RR  /\  ( 2  x.  pi )  e.  RR+ )  -> 
( ( A  mod  ( 2  x.  pi ) )  =  0  <-> 
( A  /  (
2  x.  pi ) )  e.  ZZ ) )
917, 89, 90syl2anc 666 . . 3  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  ( ( A  mod  ( 2  x.  pi ) )  =  0  <->  ( A  / 
( 2  x.  pi ) )  e.  ZZ ) )
9288, 91mtbird 303 . 2  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -.  ( A  mod  ( 2  x.  pi ) )  =  0 )
9392neqned 2630 1  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  ( A  mod  ( 2  x.  pi ) )  =/=  0
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621    \ cdif 3400    C_ wss 3403   {csn 3967   class class class wbr 4401  (class class class)co 6288   CCcc 9534   RRcr 9535   0cc0 9536   1c1 9537    + caddc 9539    x. cmul 9541   RR*cxr 9671    < clt 9672    <_ cle 9673   -ucneg 9858    / cdiv 10266   2c2 10656   ZZcz 10934   RR+crp 11299   [,]cicc 11635    mod cmo 12093   picpi 14112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ioc 11637  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13123  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-sin 14116  df-cos 14117  df-pi 14119  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-ntr 20028  df-cls 20029  df-nei 20107  df-lp 20145  df-perf 20146  df-cn 20236  df-cnp 20237  df-haus 20324  df-tx 20570  df-hmeo 20763  df-fil 20854  df-fm 20946  df-flim 20947  df-flf 20948  df-xms 21328  df-ms 21329  df-tms 21330  df-cncf 21903  df-limc 22814  df-dv 22815
This theorem is referenced by:  fourierdlem66  38030
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