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

Step	Hyp	Ref	Expression
1		0zd 10938	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> 0 e. ZZ )
2		pire 23275	. . . . . . . . . 10 |- pi e. RR
3	2	renegcli 9924	. . . . . . . . 9 |- -u pi e. RR
4		iccssre 11705	. . . . . . . . 9 |- ( ( -u pi e. RR /\ pi e. RR ) -> ( -u pi [,] pi ) C_ RR )
5	3, 2, 4	mp2an 676	. . . . . . . 8 |- ( -u pi [,] pi ) C_ RR
6		eldifi 3584	. . . . . . . 8 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A e. ( -u pi [,] pi ) )
7	5, 6	sseldi 3459	. . . . . . 7 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A e. RR )
8	7	adantr 466	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> A e. RR )
9		2re 10668	. . . . . . . 8 |- 2 e. RR
10	9, 2	remulcli 9646	. . . . . . 7 |- ( 2 x. pi ) e. RR
11	10	a1i 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
15	9, 2, 13, 14	mulgt0ii 9757	. . . . . . 7 |- 0 < ( 2 x. pi )
16	15	a1i 11	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> 0 < ( 2 x. pi ) )
17	8, 11, 12, 16	divgt0d 10531	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> 0 < ( A / ( 2 x. pi ) ) )
18	11, 16	elrpd 11327	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> ( 2 x. pi ) e. RR+ )
19	2	a1i 11	. . . . . . . . . 10 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> pi e. RR )
20	10	a1i 11	. . . . . . . . . 10 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( 2 x. pi ) e. RR )
21	3	rexri 9682	. . . . . . . . . . . 12 |- -u pi e. RR*
22	21	a1i 11	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u pi e. RR* )
23	19	rexrd 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 )
25	22, 23, 6, 24	syl3anc 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 ) )
28	26, 27	mp1i 13	. . . . . . . . . 10 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> pi < ( 2 x. pi ) )
29	7, 19, 20, 25, 28	lelttrd 9782	. . . . . . . . 9 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A < ( 2 x. pi ) )
30	29	adantr 466	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> A < ( 2 x. pi ) )
31	8, 11, 18, 30	ltdiv1dd 11384	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> ( A / ( 2 x. pi ) ) < ( ( 2 x. pi ) / ( 2 x. pi ) ) )
32	10	recni 9644	. . . . . . . 8 |- ( 2 x. pi ) e. CC
33	10, 15	gt0ne0ii 10139	. . . . . . . 8 |- ( 2 x. pi ) =/= 0
34	32, 33	dividi 10329	. . . . . . 7 |- ( ( 2 x. pi ) / ( 2 x. pi ) ) = 1
35	31, 34	syl6breq 4456	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> ( A / ( 2 x. pi ) ) < 1 )
36		0p1e1 10710	. . . . . 6 |- ( 0 + 1 ) = 1
37	35, 36	syl6breqr 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 )
39	1, 17, 37, 38	syl3anc 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 } ) )
41	7	adantr 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 )
44	41, 42, 43	nltled 9774	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> A <_ 0 )
45		eldifsni 4120	. . . . . . . 8 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A =/= 0 )
46	45	necomd 2693	. . . . . . 7 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> 0 =/= A )
47	46	adantr 466	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> 0 =/= A )
48	41, 42, 44, 47	leneltd 9778	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> A < 0 )
49		neg1z 10962	. . . . . . 7 |- -u 1 e. ZZ
50	49	a1i 11	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u 1 e. ZZ )
51	33	a1i 11	. . . . . . . . 9 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( 2 x. pi ) =/= 0 )
52	7, 20, 51	redivcld 10424	. . . . . . . 8 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( A / ( 2 x. pi ) ) e. RR )
53	52	adantr 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 )
55	7	recnd 9658	. . . . . . . . . 10 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A e. CC )
56	55	adantr 466	. . . . . . . . 9 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> A e. CC )
57	32	a1i 11	. . . . . . . . 9 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( 2 x. pi ) e. CC )
58	33	a1i 11	. . . . . . . . 9 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( 2 x. pi ) =/= 0 )
59	56, 57, 58	divnegd 10385	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u ( A / ( 2 x. pi ) ) = ( -u A / ( 2 x. pi ) ) )
60	7	renegcld 10035	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u A e. RR )
61	60	adantr 466	. . . . . . . . . 10 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u A e. RR )
62	10	a1i 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+ )
65	63, 26, 64	mp2an 676	. . . . . . . . . . 11 |- ( 2 x. pi ) e. RR+
66	65	a1i 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 )
68	22, 23, 6, 67	syl3anc 1264	. . . . . . . . . . . . 13 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u pi <_ A )
69	19, 7, 68	lenegcon1d 10184	. . . . . . . . . . . 12 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u A <_ pi )
70	60, 19, 20, 69, 28	lelttrd 9782	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u A < ( 2 x. pi ) )
71	70	adantr 466	. . . . . . . . . 10 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u A < ( 2 x. pi ) )
72	61, 62, 66, 71	ltdiv1dd 11384	. . . . . . . . 9 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( -u A / ( 2 x. pi ) ) < ( ( 2 x. pi ) / ( 2 x. pi ) ) )
73	72, 34	syl6breq 4456	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( -u A / ( 2 x. pi ) ) < 1 )
74	59, 73	eqbrtrd 4437	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u ( A / ( 2 x. pi ) ) < 1 )
75	53, 54, 74	ltnegcon1d 10182	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u 1 < ( A / ( 2 x. pi ) ) )
76	7	adantr 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 )
78	76, 66, 77	divlt0gt0d 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
81	79, 80	addcomi 9813	. . . . . . . 8 |- ( -u 1 + 1 ) = ( 1 + -u 1 )
82		1pneg1e0 10707	. . . . . . . 8 |- ( 1 + -u 1 ) = 0
83	81, 82	eqtr2i 2450	. . . . . . 7 |- 0 = ( -u 1 + 1 )
84	78, 83	syl6breq 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 )
86	50, 75, 84, 85	syl3anc 1264	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -. ( A / ( 2 x. pi ) ) e. ZZ )
87	40, 48, 86	syl2anc 665	. . . 4 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> -. ( A / ( 2 x. pi ) ) e. ZZ )
88	39, 87	pm2.61dan 798	. . 3 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -. ( A / ( 2 x. pi ) ) e. ZZ )
89	65	a1i 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 ) )
91	7, 89, 90	syl2anc 665	. . 3 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( ( A mod ( 2 x. pi ) ) = 0 <-> ( A / ( 2 x. pi ) ) e. ZZ ) )
92	88, 91	mtbird 302	. 2 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -. ( A mod ( 2 x. pi ) ) = 0 )
93	92	neqned 2625	1 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( A mod ( 2 x. pi ) ) =/= 0 )

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