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Theorem fourierdlem24 38105
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 10973 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  0  e.  ZZ )
2 pire 23492 . . . . . . . . . 10  |-  pi  e.  RR
32renegcli 9955 . . . . . . . . 9  |-  -u pi  e.  RR
4 iccssre 11741 . . . . . . . . 9  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR )  ->  ( -u pi [,] pi )  C_  RR )
53, 2, 4mp2an 686 . . . . . . . 8  |-  ( -u pi [,] pi )  C_  RR
6 eldifi 3544 . . . . . . . 8  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  e.  (
-u pi [,] pi ) )
75, 6sseldi 3416 . . . . . . 7  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  e.  RR )
87adantr 472 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  A  e.  RR )
9 2re 10701 . . . . . . . 8  |-  2  e.  RR
109, 2remulcli 9675 . . . . . . 7  |-  ( 2  x.  pi )  e.  RR
1110a1i 11 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  (
2  x.  pi )  e.  RR )
12 simpr 468 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  0  <  A )
13 2pos 10723 . . . . . . . 8  |-  0  <  2
14 pipos 23494 . . . . . . . 8  |-  0  <  pi
159, 2, 13, 14mulgt0ii 9785 . . . . . . 7  |-  0  <  ( 2  x.  pi )
1615a1i 11 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  0  <  ( 2  x.  pi ) )
178, 11, 12, 16divgt0d 10564 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  0  <  ( A  /  (
2  x.  pi ) ) )
1811, 16elrpd 11361 . . . . . . . 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 9711 . . . . . . . . . . . 12  |-  -u pi  e.  RR*
2221a1i 11 . . . . . . . . . . 11  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u pi  e.  RR* )
2319rexrd 9708 . . . . . . . . . . 11  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  pi  e.  RR* )
24 iccleub 11715 . . . . . . . . . . 11  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  A  e.  ( -u pi [,] pi ) )  ->  A  <_  pi )
2522, 23, 6, 24syl3anc 1292 . . . . . . . . . 10  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  <_  pi )
26 pirp 23495 . . . . . . . . . . 11  |-  pi  e.  RR+
27 2timesgt 37590 . . . . . . . . . . 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 9810 . . . . . . . . 9  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  <  (
2  x.  pi ) )
3029adantr 472 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  A  <  ( 2  x.  pi ) )
318, 11, 18, 30ltdiv1dd 11418 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  ( A  /  ( 2  x.  pi ) )  < 
( ( 2  x.  pi )  /  (
2  x.  pi ) ) )
3210recni 9673 . . . . . . . 8  |-  ( 2  x.  pi )  e.  CC
3310, 15gt0ne0ii 10171 . . . . . . . 8  |-  ( 2  x.  pi )  =/=  0
3432, 33dividi 10362 . . . . . . 7  |-  ( ( 2  x.  pi )  /  ( 2  x.  pi ) )  =  1
3531, 34syl6breq 4435 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  ( A  /  ( 2  x.  pi ) )  <  1 )
36 0p1e1 10743 . . . . . 6  |-  ( 0  +  1 )  =  1
3735, 36syl6breqr 4436 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  ( A  /  ( 2  x.  pi ) )  < 
( 0  +  1 ) )
38 btwnnz 11035 . . . . 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 1292 . . . 4  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  0  < 
A )  ->  -.  ( A  /  (
2  x.  pi ) )  e.  ZZ )
40 simpl 464 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  A  e.  ( ( -u pi [,] pi )  \  {
0 } ) )
417adantr 472 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  A  e.  RR )
42 0red 9662 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  0  e.  RR )
43 simpr 468 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  -.  0  <  A )
4441, 42, 43nltled 9802 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  A  <_  0 )
45 eldifsni 4089 . . . . . . . 8  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  =/=  0
)
4645necomd 2698 . . . . . . 7  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  0  =/=  A
)
4746adantr 472 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  0  =/=  A )
4841, 42, 44, 47leneltd 9806 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  A  <  0 )
49 neg1z 10997 . . . . . . 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 10457 . . . . . . . 8  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  ( A  / 
( 2  x.  pi ) )  e.  RR )
5352adantr 472 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( A  /  ( 2  x.  pi ) )  e.  RR )
54 1red 9676 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  1  e.  RR )
557recnd 9687 . . . . . . . . . 10  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  A  e.  CC )
5655adantr 472 . . . . . . . . 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 10418 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u ( A  /  ( 2  x.  pi ) )  =  ( -u A  / 
( 2  x.  pi ) ) )
607renegcld 10067 . . . . . . . . . . 11  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u A  e.  RR )
6160adantr 472 . . . . . . . . . 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 11330 . . . . . . . . . . . 12  |-  2  e.  RR+
64 rpmulcl 11347 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR+  /\  pi  e.  RR+ )  ->  (
2  x.  pi )  e.  RR+ )
6563, 26, 64mp2an 686 . . . . . . . . . . 11  |-  ( 2  x.  pi )  e.  RR+
6665a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  (
2  x.  pi )  e.  RR+ )
67 iccgelb 11716 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  A  e.  ( -u pi [,] pi ) )  ->  -u pi  <_  A )
6822, 23, 6, 67syl3anc 1292 . . . . . . . . . . . . 13  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u pi  <_  A
)
6919, 7, 68lenegcon1d 10216 . . . . . . . . . . . 12  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u A  <_  pi )
7060, 19, 20, 69, 28lelttrd 9810 . . . . . . . . . . 11  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -u A  <  (
2  x.  pi ) )
7170adantr 472 . . . . . . . . . 10  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u A  <  ( 2  x.  pi ) )
7261, 62, 66, 71ltdiv1dd 11418 . . . . . . . . 9  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( -u A  /  ( 2  x.  pi ) )  <  ( ( 2  x.  pi )  / 
( 2  x.  pi ) ) )
7372, 34syl6breq 4435 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( -u A  /  ( 2  x.  pi ) )  <  1 )
7459, 73eqbrtrd 4416 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u ( A  /  ( 2  x.  pi ) )  <  1 )
7553, 54, 74ltnegcon1d 10214 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -u 1  <  ( A  /  (
2  x.  pi ) ) )
767adantr 472 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  A  e.  RR )
77 simpr 468 . . . . . . . 8  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  A  <  0 )
7876, 66, 77divlt0gt0d 37586 . . . . . . 7  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( A  /  ( 2  x.  pi ) )  <  0 )
79 neg1cn 10735 . . . . . . . . 9  |-  -u 1  e.  CC
80 ax-1cn 9615 . . . . . . . . 9  |-  1  e.  CC
8179, 80addcomi 9842 . . . . . . . 8  |-  ( -u
1  +  1 )  =  ( 1  + 
-u 1 )
82 1pneg1e0 10740 . . . . . . . 8  |-  ( 1  +  -u 1 )  =  0
8381, 82eqtr2i 2494 . . . . . . 7  |-  0  =  ( -u 1  +  1 )
8478, 83syl6breq 4435 . . . . . 6  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  ( A  /  ( 2  x.  pi ) )  < 
( -u 1  +  1 ) )
85 btwnnz 11035 . . . . . 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 1292 . . . . 5  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  A  <  0 )  ->  -.  ( A  /  (
2  x.  pi ) )  e.  ZZ )
8740, 48, 86syl2anc 673 . . . 4  |-  ( ( A  e.  ( (
-u pi [,] pi )  \  { 0 } )  /\  -.  0  <  A )  ->  -.  ( A  /  (
2  x.  pi ) )  e.  ZZ )
8839, 87pm2.61dan 808 . . 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 12136 . . . 4  |-  ( ( A  e.  RR  /\  ( 2  x.  pi )  e.  RR+ )  -> 
( ( A  mod  ( 2  x.  pi ) )  =  0  <-> 
( A  /  (
2  x.  pi ) )  e.  ZZ ) )
917, 89, 90syl2anc 673 . . 3  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  ( ( A  mod  ( 2  x.  pi ) )  =  0  <->  ( A  / 
( 2  x.  pi ) )  e.  ZZ ) )
9288, 91mtbird 308 . 2  |-  ( A  e.  ( ( -u pi [,] pi )  \  { 0 } )  ->  -.  ( A  mod  ( 2  x.  pi ) )  =  0 )
9392neqned 2650 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 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641    \ cdif 3387    C_ wss 3390   {csn 3959   class class class wbr 4395  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   RR*cxr 9692    < clt 9693    <_ cle 9694   -ucneg 9881    / cdiv 10291   2c2 10681   ZZcz 10961   RR+crp 11325   [,]cicc 11663    mod cmo 12129   picpi 14196
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-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  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  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  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-int 4227  df-iun 4271  df-iin 4272  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-se 4799  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-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  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-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901
This theorem is referenced by:  fourierdlem66  38148
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