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Theorem fourierdlem44 37956
Description: A condition for having  ( sin `  ( A  /  2
) ) non zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
fourierdlem44  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  =/=  0 )  -> 
( sin `  ( A  /  2 ) )  =/=  0 )

Proof of Theorem fourierdlem44
StepHypRef Expression
1 0xr 9695 . . . . . 6  |-  0  e.  RR*
21a1i 11 . . . . 5  |-  ( ( A  e.  ( -u pi [,] pi )  /\  0  <  A )  -> 
0  e.  RR* )
3 2re 10687 . . . . . . . 8  |-  2  e.  RR
4 pire 23412 . . . . . . . 8  |-  pi  e.  RR
53, 4remulcli 9665 . . . . . . 7  |-  ( 2  x.  pi )  e.  RR
65rexri 9701 . . . . . 6  |-  ( 2  x.  pi )  e. 
RR*
76a1i 11 . . . . 5  |-  ( ( A  e.  ( -u pi [,] pi )  /\  0  <  A )  -> 
( 2  x.  pi )  e.  RR* )
84renegcli 9943 . . . . . . . 8  |-  -u pi  e.  RR
98a1i 11 . . . . . . 7  |-  ( A  e.  ( -u pi [,] pi )  ->  -u pi  e.  RR )
104a1i 11 . . . . . . 7  |-  ( A  e.  ( -u pi [,] pi )  ->  pi  e.  RR )
11 id 22 . . . . . . 7  |-  ( A  e.  ( -u pi [,] pi )  ->  A  e.  ( -u pi [,] pi ) )
12 eliccre 37553 . . . . . . 7  |-  ( (
-u pi  e.  RR  /\  pi  e.  RR  /\  A  e.  ( -u pi [,] pi ) )  ->  A  e.  RR )
139, 10, 11, 12syl3anc 1264 . . . . . 6  |-  ( A  e.  ( -u pi [,] pi )  ->  A  e.  RR )
1413adantr 466 . . . . 5  |-  ( ( A  e.  ( -u pi [,] pi )  /\  0  <  A )  ->  A  e.  RR )
15 simpr 462 . . . . 5  |-  ( ( A  e.  ( -u pi [,] pi )  /\  0  <  A )  -> 
0  <  A )
165a1i 11 . . . . . . 7  |-  ( A  e.  ( -u pi [,] pi )  ->  (
2  x.  pi )  e.  RR )
179rexrd 9698 . . . . . . . 8  |-  ( A  e.  ( -u pi [,] pi )  ->  -u pi  e.  RR* )
1810rexrd 9698 . . . . . . . 8  |-  ( A  e.  ( -u pi [,] pi )  ->  pi  e.  RR* )
19 iccleub 11698 . . . . . . . 8  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  A  e.  ( -u pi [,] pi ) )  ->  A  <_  pi )
2017, 18, 11, 19syl3anc 1264 . . . . . . 7  |-  ( A  e.  ( -u pi [,] pi )  ->  A  <_  pi )
21 pirp 23415 . . . . . . . . 9  |-  pi  e.  RR+
22 2timesgt 37456 . . . . . . . . 9  |-  ( pi  e.  RR+  ->  pi  <  ( 2  x.  pi ) )
2321, 22ax-mp 5 . . . . . . . 8  |-  pi  <  ( 2  x.  pi )
2423a1i 11 . . . . . . 7  |-  ( A  e.  ( -u pi [,] pi )  ->  pi  <  ( 2  x.  pi ) )
2513, 10, 16, 20, 24lelttrd 9801 . . . . . 6  |-  ( A  e.  ( -u pi [,] pi )  ->  A  <  ( 2  x.  pi ) )
2625adantr 466 . . . . 5  |-  ( ( A  e.  ( -u pi [,] pi )  /\  0  <  A )  ->  A  <  ( 2  x.  pi ) )
272, 7, 14, 15, 26eliood 37545 . . . 4  |-  ( ( A  e.  ( -u pi [,] pi )  /\  0  <  A )  ->  A  e.  ( 0 (,) ( 2  x.  pi ) ) )
2827adantlr 719 . . 3  |-  ( ( ( A  e.  (
-u pi [,] pi )  /\  A  =/=  0
)  /\  0  <  A )  ->  A  e.  ( 0 (,) (
2  x.  pi ) ) )
29 sinaover2ne0 37684 . . 3  |-  ( A  e.  ( 0 (,) ( 2  x.  pi ) )  ->  ( sin `  ( A  / 
2 ) )  =/=  0 )
3028, 29syl 17 . 2  |-  ( ( ( A  e.  (
-u pi [,] pi )  /\  A  =/=  0
)  /\  0  <  A )  ->  ( sin `  ( A  /  2
) )  =/=  0
)
31 simpll 758 . . 3  |-  ( ( ( A  e.  (
-u pi [,] pi )  /\  A  =/=  0
)  /\  -.  0  <  A )  ->  A  e.  ( -u pi [,] pi ) )
3231, 13syl 17 . . . 4  |-  ( ( ( A  e.  (
-u pi [,] pi )  /\  A  =/=  0
)  /\  -.  0  <  A )  ->  A  e.  RR )
33 0red 9652 . . . 4  |-  ( ( ( A  e.  (
-u pi [,] pi )  /\  A  =/=  0
)  /\  -.  0  <  A )  ->  0  e.  RR )
34 simplr 760 . . . 4  |-  ( ( ( A  e.  (
-u pi [,] pi )  /\  A  =/=  0
)  /\  -.  0  <  A )  ->  A  =/=  0 )
35 simpr 462 . . . 4  |-  ( ( ( A  e.  (
-u pi [,] pi )  /\  A  =/=  0
)  /\  -.  0  <  A )  ->  -.  0  <  A )
3632, 33, 34, 35lttri5d 37472 . . 3  |-  ( ( ( A  e.  (
-u pi [,] pi )  /\  A  =/=  0
)  /\  -.  0  <  A )  ->  A  <  0 )
3713recnd 9677 . . . . . . . . . . 11  |-  ( A  e.  ( -u pi [,] pi )  ->  A  e.  CC )
3837halfcld 10865 . . . . . . . . . 10  |-  ( A  e.  ( -u pi [,] pi )  ->  ( A  /  2 )  e.  CC )
39 sinneg 14200 . . . . . . . . . 10  |-  ( ( A  /  2 )  e.  CC  ->  ( sin `  -u ( A  / 
2 ) )  = 
-u ( sin `  ( A  /  2 ) ) )
4038, 39syl 17 . . . . . . . . 9  |-  ( A  e.  ( -u pi [,] pi )  ->  ( sin `  -u ( A  / 
2 ) )  = 
-u ( sin `  ( A  /  2 ) ) )
41 2cnd 10690 . . . . . . . . . . 11  |-  ( A  e.  ( -u pi [,] pi )  ->  2  e.  CC )
42 2ne0 10710 . . . . . . . . . . . 12  |-  2  =/=  0
4342a1i 11 . . . . . . . . . . 11  |-  ( A  e.  ( -u pi [,] pi )  ->  2  =/=  0 )
4437, 41, 43divnegd 10404 . . . . . . . . . 10  |-  ( A  e.  ( -u pi [,] pi )  ->  -u ( A  /  2 )  =  ( -u A  / 
2 ) )
4544fveq2d 5886 . . . . . . . . 9  |-  ( A  e.  ( -u pi [,] pi )  ->  ( sin `  -u ( A  / 
2 ) )  =  ( sin `  ( -u A  /  2 ) ) )
4640, 45eqtr3d 2465 . . . . . . . 8  |-  ( A  e.  ( -u pi [,] pi )  ->  -u ( sin `  ( A  / 
2 ) )  =  ( sin `  ( -u A  /  2 ) ) )
4746adantr 466 . . . . . . 7  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  ->  -u ( sin `  ( A  /  2 ) )  =  ( sin `  ( -u A  /  2 ) ) )
481a1i 11 . . . . . . . . 9  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  -> 
0  e.  RR* )
496a1i 11 . . . . . . . . 9  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  -> 
( 2  x.  pi )  e.  RR* )
5013renegcld 10054 . . . . . . . . . 10  |-  ( A  e.  ( -u pi [,] pi )  ->  -u A  e.  RR )
5150adantr 466 . . . . . . . . 9  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  ->  -u A  e.  RR )
52 simpr 462 . . . . . . . . . 10  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  ->  A  <  0 )
5313adantr 466 . . . . . . . . . . 11  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  ->  A  e.  RR )
5453lt0neg1d 10191 . . . . . . . . . 10  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  -> 
( A  <  0  <->  0  <  -u A ) )
5552, 54mpbid 213 . . . . . . . . 9  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  -> 
0  <  -u A )
565renegcli 9943 . . . . . . . . . . . . 13  |-  -u (
2  x.  pi )  e.  RR
5756a1i 11 . . . . . . . . . . . 12  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  ->  -u ( 2  x.  pi )  e.  RR )
588a1i 11 . . . . . . . . . . . 12  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  ->  -u pi  e.  RR )
594, 5ltnegi 10166 . . . . . . . . . . . . . 14  |-  ( pi 
<  ( 2  x.  pi )  <->  -u ( 2  x.  pi )  <  -u pi )
6023, 59mpbi 211 . . . . . . . . . . . . 13  |-  -u (
2  x.  pi )  <  -u pi
6160a1i 11 . . . . . . . . . . . 12  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  ->  -u ( 2  x.  pi )  <  -u pi )
62 iccgelb 11699 . . . . . . . . . . . . . 14  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR*  /\  A  e.  ( -u pi [,] pi ) )  ->  -u pi  <_  A )
6317, 18, 11, 62syl3anc 1264 . . . . . . . . . . . . 13  |-  ( A  e.  ( -u pi [,] pi )  ->  -u pi  <_  A )
6463adantr 466 . . . . . . . . . . . 12  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  ->  -u pi  <_  A )
6557, 58, 53, 61, 64ltletrd 9803 . . . . . . . . . . 11  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  ->  -u ( 2  x.  pi )  <  A )
6657, 53ltnegd 10199 . . . . . . . . . . 11  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  -> 
( -u ( 2  x.  pi )  <  A  <->  -u A  <  -u -u (
2  x.  pi ) ) )
6765, 66mpbid 213 . . . . . . . . . 10  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  ->  -u A  <  -u -u (
2  x.  pi ) )
6816recnd 9677 . . . . . . . . . . . 12  |-  ( A  e.  ( -u pi [,] pi )  ->  (
2  x.  pi )  e.  CC )
6968negnegd 9985 . . . . . . . . . . 11  |-  ( A  e.  ( -u pi [,] pi )  ->  -u -u (
2  x.  pi )  =  ( 2  x.  pi ) )
7069adantr 466 . . . . . . . . . 10  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  ->  -u -u ( 2  x.  pi )  =  ( 2  x.  pi ) )
7167, 70breqtrd 4448 . . . . . . . . 9  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  ->  -u A  <  ( 2  x.  pi ) )
7248, 49, 51, 55, 71eliood 37545 . . . . . . . 8  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  ->  -u A  e.  ( 0 (,) ( 2  x.  pi ) ) )
73 sinaover2ne0 37684 . . . . . . . 8  |-  ( -u A  e.  ( 0 (,) ( 2  x.  pi ) )  -> 
( sin `  ( -u A  /  2 ) )  =/=  0 )
7472, 73syl 17 . . . . . . 7  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  -> 
( sin `  ( -u A  /  2 ) )  =/=  0 )
7547, 74eqnetrd 2713 . . . . . 6  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  ->  -u ( sin `  ( A  /  2 ) )  =/=  0 )
7675neneqd 2621 . . . . 5  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  ->  -.  -u ( sin `  ( A  /  2 ) )  =  0 )
7738sincld 14184 . . . . . . 7  |-  ( A  e.  ( -u pi [,] pi )  ->  ( sin `  ( A  / 
2 ) )  e.  CC )
7877adantr 466 . . . . . 6  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  -> 
( sin `  ( A  /  2 ) )  e.  CC )
7978negeq0d 9986 . . . . 5  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  -> 
( ( sin `  ( A  /  2 ) )  =  0  <->  -u ( sin `  ( A  /  2
) )  =  0 ) )
8076, 79mtbird 302 . . . 4  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  ->  -.  ( sin `  ( A  /  2 ) )  =  0 )
8180neqned 2623 . . 3  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  <  0 )  -> 
( sin `  ( A  /  2 ) )  =/=  0 )
8231, 36, 81syl2anc 665 . 2  |-  ( ( ( A  e.  (
-u pi [,] pi )  /\  A  =/=  0
)  /\  -.  0  <  A )  ->  ( sin `  ( A  / 
2 ) )  =/=  0 )
8330, 82pm2.61dan 798 1  |-  ( ( A  e.  ( -u pi [,] pi )  /\  A  =/=  0 )  -> 
( sin `  ( A  /  2 ) )  =/=  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   class class class wbr 4423   ` cfv 5601  (class class class)co 6306   CCcc 9545   RRcr 9546   0cc0 9547    x. cmul 9552   RR*cxr 9682    < clt 9683    <_ cle 9684   -ucneg 9869    / cdiv 10277   2c2 10667   RR+crp 11310   (,)cioo 11643   [,]cicc 11646   sincsin 14116   picpi 14119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-inf2 8156  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624  ax-pre-sup 9625  ax-addf 9626  ax-mulf 9627
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 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-of 6546  df-om 6708  df-1st 6808  df-2nd 6809  df-supp 6927  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-2o 7195  df-oadd 7198  df-er 7375  df-map 7486  df-pm 7487  df-ixp 7535  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-fsupp 7894  df-fi 7935  df-sup 7966  df-inf 7967  df-oi 8035  df-card 8382  df-cda 8606  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-div 10278  df-nn 10618  df-2 10676  df-3 10677  df-4 10678  df-5 10679  df-6 10680  df-7 10681  df-8 10682  df-9 10683  df-10 10684  df-n0 10878  df-z 10946  df-dec 11060  df-uz 11168  df-q 11273  df-rp 11311  df-xneg 11417  df-xadd 11418  df-xmul 11419  df-ioo 11647  df-ioc 11648  df-ico 11649  df-icc 11650  df-fz 11793  df-fzo 11924  df-fl 12035  df-mod 12104  df-seq 12221  df-exp 12280  df-fac 12467  df-bc 12495  df-hash 12523  df-shft 13131  df-cj 13163  df-re 13164  df-im 13165  df-sqrt 13299  df-abs 13300  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121  df-sin 14123  df-cos 14124  df-pi 14126  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19920  df-bases 19921  df-topon 19922  df-topsp 19923  df-cld 20033  df-ntr 20034  df-cls 20035  df-nei 20113  df-lp 20151  df-perf 20152  df-cn 20242  df-cnp 20243  df-haus 20330  df-tx 20576  df-hmeo 20769  df-fil 20860  df-fm 20952  df-flim 20953  df-flf 20954  df-xms 21334  df-ms 21335  df-tms 21336  df-cncf 21909  df-limc 22820  df-dv 22821
This theorem is referenced by:  fourierdlem56  37967  fourierdlem57  37968  fourierdlem58  37969  fourierdlem62  37973  fourierdlem66  37977  fourierdlem68  37979  fourierdlem72  37983  fourierdlem76  37987  fourierdlem78  37989  fourierdlem80  37991  fourierdlem103  38014  fourierdlem104  38015
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