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Theorem fourierdlem24 37566
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 10938 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  0  e.  ZZ )
2 pire 23275 . . . . . . . . . 10  |-  pi  e.  RR
32renegcli 9924 . . . . . . . . 9  |-  -u pi  e.  RR
4 iccssre 11705 . . . . . . . . 9  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR )  ->  ( -u pi [,] pi )  C_  RR )
53, 2, 4mp2an 676 . . . . . . . 8  |-  ( -u pi [,] pi )  C_  RR
6 eldifi 3584 . . . . . . . 8  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  e.  (
-u pi [,] pi ) )
75, 6sseldi 3459 . . . . . . 7  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  e.  RR )
87adantr 466 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  A  e.  RR )
9 2re 10668 . . . . . . . 8  |-  2  e.  RR
109, 2remulcli 9646 . . . . . . 7  |-  ( 2  x.  pi )  e.  RR
1110a1i 11 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  (
2  x.  pi )  e.  RR )
12 simpr 462 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  0  <  A )
13 2pos 10690 . . . . . . . 8  |-  0  <  2
14 pipos 23277 . . . . . . . 8  |-  0  <  pi
159, 2, 13, 14mulgt0ii 9757 . . . . . . 7  |-  0  <  ( 2  x.  pi )
1615a1i 11 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  0  <  ( 2  x.  pi ) )
178, 11, 12, 16divgt0d 10531 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  0  <  ( A  /  (
2  x.  pi ) ) )
1811, 16elrpd 11327 . . . . . . . 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 9682 . . . . . . . . . . . 12  |-  -u pi  e.  RR*
2221a1i 11 . . . . . . . . . . 11  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u pi  e.  RR* )
2319rexrd 9679 . . . . . . . . . . 11  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  pi  e.  RR* )
24 iccleub 11679 . . . . . . . . . . 11  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  A  e.  ( -u pi [,] pi ) )  ->  A  <_  pi )
2522, 23, 6, 24syl3anc 1264 . . . . . . . . . 10  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  <_  pi )
26 pirp 23278 . . . . . . . . . . 11  |-  pi  e.  RR+
27 2timesgt 37114 . . . . . . . . . . 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 9782 . . . . . . . . 9  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  <  (
2  x.  pi ) )
3029adantr 466 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  A  <  ( 2  x.  pi ) )
318, 11, 18, 30ltdiv1dd 11384 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  ( A  /  ( 2  x.  pi ) )  < 
( ( 2  x.  pi )  /  (
2  x.  pi ) ) )
3210recni 9644 . . . . . . . 8  |-  ( 2  x.  pi )  e.  CC
3310, 15gt0ne0ii 10139 . . . . . . . 8  |-  ( 2  x.  pi )  =/=  0
3432, 33dividi 10329 . . . . . . 7  |-  ( ( 2  x.  pi )  /  ( 2  x.  pi ) )  =  1
3531, 34syl6breq 4456 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  ( A  /  ( 2  x.  pi ) )  <  1 )
36 0p1e1 10710 . . . . . 6  |-  ( 0  +  1 )  =  1
3735, 36syl6breqr 4457 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  ( A  /  ( 2  x.  pi ) )  < 
( 0  +  1 ) )
38 btwnnz 11001 . . . . 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 1264 . . . 4  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  -.  ( A  /  (
2  x.  pi ) )  e.  ZZ )
40 simpl 458 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  A  e.  ( ( -u pi [,] pi )  \  {
0 } ) )
417adantr 466 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  A  e.  RR )
42 0red 9633 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  0  e.  RR )
43 simpr 462 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  -.  0  <  A )
4441, 42, 43nltled 9774 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  A  <_  0 )
45 eldifsni 4120 . . . . . . . 8  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  =/=  0
)
4645necomd 2693 . . . . . . 7  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  0  =/=  A
)
4746adantr 466 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  0  =/=  A )
4841, 42, 44, 47leneltd 9778 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  A  <  0 )
49 neg1z 10962 . . . . . . 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 10424 . . . . . . . 8  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  ( A  / 
( 2  x.  pi ) )  e.  RR )
5352adantr 466 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( A  /  ( 2  x.  pi ) )  e.  RR )
54 1red 9647 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  1  e.  RR )
557recnd 9658 . . . . . . . . . 10  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  e.  CC )
5655adantr 466 . . . . . . . . 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 10385 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u ( A  /  ( 2  x.  pi ) )  =  ( -u A  / 
( 2  x.  pi ) ) )
607renegcld 10035 . . . . . . . . . . 11  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u A  e.  RR )
6160adantr 466 . . . . . . . . . 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 11296 . . . . . . . . . . . 12  |-  2  e.  RR+
64 rpmulcl 11313 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR+  /\  pi  e.  RR+ )  ->  (
2  x.  pi )  e.  RR+ )
6563, 26, 64mp2an 676 . . . . . . . . . . 11  |-  ( 2  x.  pi )  e.  RR+
6665a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  (
2  x.  pi )  e.  RR+ )
67 iccgelb 11680 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  A  e.  ( -u pi [,] pi ) )  ->  -u pi  <_  A )
6822, 23, 6, 67syl3anc 1264 . . . . . . . . . . . . 13  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u pi  <_  A
)
6919, 7, 68lenegcon1d 10184 . . . . . . . . . . . 12  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u A  <_  pi )
7060, 19, 20, 69, 28lelttrd 9782 . . . . . . . . . . 11  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u A  <  (
2  x.  pi ) )
7170adantr 466 . . . . . . . . . 10  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u A  <  ( 2  x.  pi ) )
7261, 62, 66, 71ltdiv1dd 11384 . . . . . . . . 9  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( -u A  /  ( 2  x.  pi ) )  <  ( ( 2  x.  pi )  / 
( 2  x.  pi ) ) )
7372, 34syl6breq 4456 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( -u A  /  ( 2  x.  pi ) )  <  1 )
7459, 73eqbrtrd 4437 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u ( A  /  ( 2  x.  pi ) )  <  1 )
7553, 54, 74ltnegcon1d 10182 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u 1  <  ( A  /  (
2  x.  pi ) ) )
767adantr 466 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  A  e.  RR )
77 simpr 462 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  A  <  0 )
7876, 66, 77divlt0gt0d 37109 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( A  /  ( 2  x.  pi ) )  <  0 )
79 neg1cn 10702 . . . . . . . . 9  |-  -u 1  e.  CC
80 ax-1cn 9586 . . . . . . . . 9  |-  1  e.  CC
8179, 80addcomi 9813 . . . . . . . 8  |-  ( -u
1  +  1 )  =  ( 1  + 
-u 1 )
82 1pneg1e0 10707 . . . . . . . 8  |-  ( 1  +  -u 1 )  =  0
8381, 82eqtr2i 2450 . . . . . . 7  |-  0  =  ( -u 1  +  1 )
8478, 83syl6breq 4456 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( A  /  ( 2  x.  pi ) )  < 
( -u 1  +  1 ) )
85 btwnnz 11001 . . . . . 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 1264 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -.  ( A  /  (
2  x.  pi ) )  e.  ZZ )
8740, 48, 86syl2anc 665 . . . 4  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  -.  ( A  /  (
2  x.  pi ) )  e.  ZZ )
8839, 87pm2.61dan 798 . . 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 12089 . . . 4  |-  ( ( A  e.  RR  /\  ( 2  x.  pi )  e.  RR+ )  -> 
( ( A  mod  ( 2  x.  pi ) )  =  0  <-> 
( A  /  (
2  x.  pi ) )  e.  ZZ ) )
917, 89, 90syl2anc 665 . . 3  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  ( ( A  mod  ( 2  x.  pi ) )  =  0  <->  ( A  / 
( 2  x.  pi ) )  e.  ZZ ) )
9288, 91mtbird 302 . 2  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -.  ( A  mod  ( 2  x.  pi ) )  =  0 )
9392neqned 2625 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 187    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616    \ cdif 3430    C_ wss 3433   {csn 3993   class class class wbr 4417  (class class class)co 6296   CCcc 9526   RRcr 9527   0cc0 9528   1c1 9529    + caddc 9531    x. cmul 9533   RR*cxr 9663    < clt 9664    <_ cle 9665   -ucneg 9850    / cdiv 10258   2c2 10648   ZZcz 10926   RR+crp 11291   [,]cicc 11627    mod cmo 12082   picpi 14086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606  ax-addf 9607  ax-mulf 9608
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6917  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8016  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-q 11254  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-ioo 11628  df-ioc 11629  df-ico 11630  df-icc 11631  df-fz 11772  df-fzo 11903  df-fl 12014  df-mod 12083  df-seq 12200  df-exp 12259  df-fac 12446  df-bc 12474  df-hash 12502  df-shft 13098  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-limsup 13493  df-clim 13519  df-rlim 13520  df-sum 13720  df-ef 14088  df-sin 14090  df-cos 14091  df-pi 14093  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-ress 15080  df-plusg 15155  df-mulr 15156  df-starv 15157  df-sca 15158  df-vsca 15159  df-ip 15160  df-tset 15161  df-ple 15162  df-ds 15164  df-unif 15165  df-hom 15166  df-cco 15167  df-rest 15273  df-topn 15274  df-0g 15292  df-gsum 15293  df-topgen 15294  df-pt 15295  df-prds 15298  df-xrs 15352  df-qtop 15357  df-imas 15358  df-xps 15360  df-mre 15436  df-mrc 15437  df-acs 15439  df-mgm 16432  df-sgrp 16471  df-mnd 16481  df-submnd 16527  df-mulg 16620  df-cntz 16915  df-cmn 17360  df-psmet 18890  df-xmet 18891  df-met 18892  df-bl 18893  df-mopn 18894  df-fbas 18895  df-fg 18896  df-cnfld 18899  df-top 19845  df-bases 19846  df-topon 19847  df-topsp 19848  df-cld 19958  df-ntr 19959  df-cls 19960  df-nei 20038  df-lp 20076  df-perf 20077  df-cn 20167  df-cnp 20168  df-haus 20255  df-tx 20501  df-hmeo 20694  df-fil 20785  df-fm 20877  df-flim 20878  df-flf 20879  df-xms 21259  df-ms 21260  df-tms 21261  df-cncf 21799  df-limc 22695  df-dv 22696
This theorem is referenced by:  fourierdlem66  37608
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