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Theorem fourierdlem24 31867
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 10883 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  0  e.  ZZ )
2 pire 22829 . . . . . . . . . 10  |-  pi  e.  RR
32renegcli 9885 . . . . . . . . 9  |-  -u pi  e.  RR
4 iccssre 11617 . . . . . . . . 9  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR )  ->  ( -u pi [,] pi )  C_  RR )
53, 2, 4mp2an 672 . . . . . . . 8  |-  ( -u pi [,] pi )  C_  RR
6 eldifi 3611 . . . . . . . 8  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  e.  (
-u pi [,] pi ) )
75, 6sseldi 3487 . . . . . . 7  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  e.  RR )
87adantr 465 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  A  e.  RR )
9 2re 10612 . . . . . . . 8  |-  2  e.  RR
109, 2remulcli 9613 . . . . . . 7  |-  ( 2  x.  pi )  e.  RR
1110a1i 11 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  (
2  x.  pi )  e.  RR )
12 simpr 461 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  0  <  A )
13 2pos 10634 . . . . . . . 8  |-  0  <  2
14 pipos 22831 . . . . . . . 8  |-  0  <  pi
159, 2, 13, 14mulgt0ii 9721 . . . . . . 7  |-  0  <  ( 2  x.  pi )
1615a1i 11 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  0  <  ( 2  x.  pi ) )
178, 11, 12, 16divgt0d 10488 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  0  <  ( A  /  (
2  x.  pi ) ) )
1811, 16elrpd 11265 . . . . . . . 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 9649 . . . . . . . . . . . 12  |-  -u pi  e.  RR*
2221a1i 11 . . . . . . . . . . 11  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u pi  e.  RR* )
2319rexrd 9646 . . . . . . . . . . 11  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  pi  e.  RR* )
24 iccleub 11591 . . . . . . . . . . 11  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  A  e.  ( -u pi [,] pi ) )  ->  A  <_  pi )
2522, 23, 6, 24syl3anc 1229 . . . . . . . . . 10  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  <_  pi )
26 pirp 22832 . . . . . . . . . . 11  |-  pi  e.  RR+
27 2timesgt 31429 . . . . . . . . . . 11  |-  ( pi  e.  RR+  ->  pi  <  ( 2  x.  pi ) )
2826, 27mp1i 12 . . . . . . . . . 10  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  pi  <  (
2  x.  pi ) )
297, 19, 20, 25, 28lelttrd 9743 . . . . . . . . 9  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  <  (
2  x.  pi ) )
3029adantr 465 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  A  <  ( 2  x.  pi ) )
318, 11, 18, 30ltdiv1dd 11320 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  ( A  /  ( 2  x.  pi ) )  < 
( ( 2  x.  pi )  /  (
2  x.  pi ) ) )
3210recni 9611 . . . . . . . 8  |-  ( 2  x.  pi )  e.  CC
3310, 15gt0ne0ii 10096 . . . . . . . 8  |-  ( 2  x.  pi )  =/=  0
3432, 33dividi 10284 . . . . . . 7  |-  ( ( 2  x.  pi )  /  ( 2  x.  pi ) )  =  1
3531, 34syl6breq 4476 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  ( A  /  ( 2  x.  pi ) )  <  1 )
36 0p1e1 10654 . . . . . 6  |-  ( 0  +  1 )  =  1
3735, 36syl6breqr 4477 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  ( A  /  ( 2  x.  pi ) )  < 
( 0  +  1 ) )
38 btwnnz 10946 . . . . 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 1229 . . . 4  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  -.  ( A  /  (
2  x.  pi ) )  e.  ZZ )
40 simpl 457 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  A  e.  ( ( -u pi [,] pi )  \  {
0 } ) )
417adantr 465 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  A  e.  RR )
42 0red 9600 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  0  e.  RR )
43 simpr 461 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  -.  0  <  A )
4441, 42, 43nltled 31431 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  A  <_  0 )
45 eldifsni 4141 . . . . . . . 8  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  =/=  0
)
4645necomd 2714 . . . . . . 7  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  0  =/=  A
)
4746adantr 465 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  0  =/=  A )
4841, 42, 44, 47leneltd 31448 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  A  <  0 )
49 neg1z 10907 . . . . . . 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 10379 . . . . . . . 8  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  ( A  / 
( 2  x.  pi ) )  e.  RR )
5352adantr 465 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( A  /  ( 2  x.  pi ) )  e.  RR )
54 1red 9614 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  1  e.  RR )
557recnd 9625 . . . . . . . . . 10  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  e.  CC )
5655adantr 465 . . . . . . . . 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 10340 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u ( A  /  ( 2  x.  pi ) )  =  ( -u A  / 
( 2  x.  pi ) ) )
607renegcld 9993 . . . . . . . . . . 11  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u A  e.  RR )
6160adantr 465 . . . . . . . . . 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 11236 . . . . . . . . . . . 12  |-  2  e.  RR+
64 rpmulcl 11252 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR+  /\  pi  e.  RR+ )  ->  (
2  x.  pi )  e.  RR+ )
6563, 26, 64mp2an 672 . . . . . . . . . . 11  |-  ( 2  x.  pi )  e.  RR+
6665a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  (
2  x.  pi )  e.  RR+ )
67 iccgelb 11592 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  A  e.  ( -u pi [,] pi ) )  ->  -u pi  <_  A )
6822, 23, 6, 67syl3anc 1229 . . . . . . . . . . . . 13  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u pi  <_  A
)
6919, 7, 68lenegcon1d 10141 . . . . . . . . . . . 12  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u A  <_  pi )
7060, 19, 20, 69, 28lelttrd 9743 . . . . . . . . . . 11  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u A  <  (
2  x.  pi ) )
7170adantr 465 . . . . . . . . . 10  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u A  <  ( 2  x.  pi ) )
7261, 62, 66, 71ltdiv1dd 11320 . . . . . . . . 9  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( -u A  /  ( 2  x.  pi ) )  <  ( ( 2  x.  pi )  / 
( 2  x.  pi ) ) )
7372, 34syl6breq 4476 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( -u A  /  ( 2  x.  pi ) )  <  1 )
7459, 73eqbrtrd 4457 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u ( A  /  ( 2  x.  pi ) )  <  1 )
7553, 54, 74ltnegcon1d 10139 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u 1  <  ( A  /  (
2  x.  pi ) ) )
767adantr 465 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  A  e.  RR )
77 simpr 461 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  A  <  0 )
7876, 66, 77divlt0gt0d 31423 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( A  /  ( 2  x.  pi ) )  <  0 )
79 neg1cn 10646 . . . . . . . . 9  |-  -u 1  e.  CC
80 ax-1cn 9553 . . . . . . . . 9  |-  1  e.  CC
8179, 80addcomi 9774 . . . . . . . 8  |-  ( -u
1  +  1 )  =  ( 1  + 
-u 1 )
82 1pneg1e0 10651 . . . . . . . 8  |-  ( 1  +  -u 1 )  =  0
8381, 82eqtr2i 2473 . . . . . . 7  |-  0  =  ( -u 1  +  1 )
8478, 83syl6breq 4476 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( A  /  ( 2  x.  pi ) )  < 
( -u 1  +  1 ) )
85 btwnnz 10946 . . . . . 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 1229 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -.  ( A  /  (
2  x.  pi ) )  e.  ZZ )
8740, 48, 86syl2anc 661 . . . 4  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  -.  ( A  /  (
2  x.  pi ) )  e.  ZZ )
8839, 87pm2.61dan 791 . . 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 11985 . . . 4  |-  ( ( A  e.  RR  /\  ( 2  x.  pi )  e.  RR+ )  -> 
( ( A  mod  ( 2  x.  pi ) )  =  0  <-> 
( A  /  (
2  x.  pi ) )  e.  ZZ ) )
917, 89, 90syl2anc 661 . . 3  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  ( ( A  mod  ( 2  x.  pi ) )  =  0  <->  ( A  / 
( 2  x.  pi ) )  e.  ZZ ) )
9288, 91mtbird 301 . 2  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -.  ( A  mod  ( 2  x.  pi ) )  =  0 )
9392neqned 2646 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 184    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638    \ cdif 3458    C_ wss 3461   {csn 4014   class class class wbr 4437  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500   RR*cxr 9630    < clt 9631    <_ cle 9632   -ucneg 9811    / cdiv 10213   2c2 10592   ZZcz 10871   RR+crp 11231   [,]cicc 11543    mod cmo 11978   picpi 13784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-q 11194  df-rp 11232  df-xneg 11329  df-xadd 11330  df-xmul 11331  df-ioo 11544  df-ioc 11545  df-ico 11546  df-icc 11547  df-fz 11684  df-fzo 11807  df-fl 11911  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12882  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-limsup 13276  df-clim 13293  df-rlim 13294  df-sum 13491  df-ef 13785  df-sin 13787  df-cos 13788  df-pi 13790  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-starv 14694  df-sca 14695  df-vsca 14696  df-ip 14697  df-tset 14698  df-ple 14699  df-ds 14701  df-unif 14702  df-hom 14703  df-cco 14704  df-rest 14802  df-topn 14803  df-0g 14821  df-gsum 14822  df-topgen 14823  df-pt 14824  df-prds 14827  df-xrs 14881  df-qtop 14886  df-imas 14887  df-xps 14889  df-mre 14965  df-mrc 14966  df-acs 14968  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-submnd 15946  df-mulg 16039  df-cntz 16334  df-cmn 16779  df-psmet 18390  df-xmet 18391  df-met 18392  df-bl 18393  df-mopn 18394  df-fbas 18395  df-fg 18396  df-cnfld 18400  df-top 19377  df-bases 19379  df-topon 19380  df-topsp 19381  df-cld 19498  df-ntr 19499  df-cls 19500  df-nei 19577  df-lp 19615  df-perf 19616  df-cn 19706  df-cnp 19707  df-haus 19794  df-tx 20041  df-hmeo 20234  df-fil 20325  df-fm 20417  df-flim 20418  df-flf 20419  df-xms 20801  df-ms 20802  df-tms 20803  df-cncf 21360  df-limc 22248  df-dv 22249
This theorem is referenced by:  fourierdlem66  31909
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