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

Step	Hyp	Ref	Expression
1		0zd 10883	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> 0 e. ZZ )
2		pire 22829	. . . . . . . . . 10 |- pi e. RR
3	2	renegcli 9885	. . . . . . . . 9 |- -u pi e. RR
4		iccssre 11617	. . . . . . . . 9 |- ( ( -u pi e. RR /\ pi e. RR ) -> ( -u pi [,] pi ) C_ RR )
5	3, 2, 4	mp2an 672	. . . . . . . 8 |- ( -u pi [,] pi ) C_ RR
6		eldifi 3611	. . . . . . . 8 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A e. ( -u pi [,] pi ) )
7	5, 6	sseldi 3487	. . . . . . 7 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A e. RR )
8	7	adantr 465	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> A e. RR )
9		2re 10612	. . . . . . . 8 |- 2 e. RR
10	9, 2	remulcli 9613	. . . . . . 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 461	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> 0 < A )
13		2pos 10634	. . . . . . . 8 |- 0 < 2
14		pipos 22831	. . . . . . . 8 |- 0 < pi
15	9, 2, 13, 14	mulgt0ii 9721	. . . . . . 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 10488	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> 0 < ( A / ( 2 x. pi ) ) )
18	11, 16	elrpd 11265	. . . . . . . 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 9649	. . . . . . . . . . . 12 |- -u pi e. RR*
22	21	a1i 11	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u pi e. RR* )
23	19	rexrd 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 )
25	22, 23, 6, 24	syl3anc 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 ) )
28	26, 27	mp1i 12	. . . . . . . . . 10 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> pi < ( 2 x. pi ) )
29	7, 19, 20, 25, 28	lelttrd 9743	. . . . . . . . 9 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A < ( 2 x. pi ) )
30	29	adantr 465	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> A < ( 2 x. pi ) )
31	8, 11, 18, 30	ltdiv1dd 11320	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> ( A / ( 2 x. pi ) ) < ( ( 2 x. pi ) / ( 2 x. pi ) ) )
32	10	recni 9611	. . . . . . . 8 |- ( 2 x. pi ) e. CC
33	10, 15	gt0ne0ii 10096	. . . . . . . 8 |- ( 2 x. pi ) =/= 0
34	32, 33	dividi 10284	. . . . . . 7 |- ( ( 2 x. pi ) / ( 2 x. pi ) ) = 1
35	31, 34	syl6breq 4476	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> ( A / ( 2 x. pi ) ) < 1 )
36		0p1e1 10654	. . . . . 6 |- ( 0 + 1 ) = 1
37	35, 36	syl6breqr 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 )
39	1, 17, 37, 38	syl3anc 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 } ) )
41	7	adantr 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 )
44	41, 42, 43	nltled 31431	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> A <_ 0 )
45		eldifsni 4141	. . . . . . . 8 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A =/= 0 )
46	45	necomd 2714	. . . . . . 7 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> 0 =/= A )
47	46	adantr 465	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> 0 =/= A )
48	41, 42, 44, 47	leneltd 31448	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> A < 0 )
49		neg1z 10907	. . . . . . 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 10379	. . . . . . . 8 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( A / ( 2 x. pi ) ) e. RR )
53	52	adantr 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 )
55	7	recnd 9625	. . . . . . . . . 10 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A e. CC )
56	55	adantr 465	. . . . . . . . 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 10340	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u ( A / ( 2 x. pi ) ) = ( -u A / ( 2 x. pi ) ) )
60	7	renegcld 9993	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u A e. RR )
61	60	adantr 465	. . . . . . . . . 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 11236	. . . . . . . . . . . 12 |- 2 e. RR+
64		rpmulcl 11252	. . . . . . . . . . . 12 |- ( ( 2 e. RR+ /\ pi e. RR+ ) -> ( 2 x. pi ) e. RR+ )
65	63, 26, 64	mp2an 672	. . . . . . . . . . 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 11592	. . . . . . . . . . . . . 14 |- ( ( -u pi e. RR* /\ pi e. RR* /\ A e. ( -u pi [,] pi ) ) -> -u pi <_ A )
68	22, 23, 6, 67	syl3anc 1229	. . . . . . . . . . . . 13 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u pi <_ A )
69	19, 7, 68	lenegcon1d 10141	. . . . . . . . . . . 12 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u A <_ pi )
70	60, 19, 20, 69, 28	lelttrd 9743	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u A < ( 2 x. pi ) )
71	70	adantr 465	. . . . . . . . . 10 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u A < ( 2 x. pi ) )
72	61, 62, 66, 71	ltdiv1dd 11320	. . . . . . . . 9 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( -u A / ( 2 x. pi ) ) < ( ( 2 x. pi ) / ( 2 x. pi ) ) )
73	72, 34	syl6breq 4476	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( -u A / ( 2 x. pi ) ) < 1 )
74	59, 73	eqbrtrd 4457	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u ( A / ( 2 x. pi ) ) < 1 )
75	53, 54, 74	ltnegcon1d 10139	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u 1 < ( A / ( 2 x. pi ) ) )
76	7	adantr 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 )
78	76, 66, 77	divlt0gt0d 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
81	79, 80	addcomi 9774	. . . . . . . 8 |- ( -u 1 + 1 ) = ( 1 + -u 1 )
82		1pneg1e0 10651	. . . . . . . 8 |- ( 1 + -u 1 ) = 0
83	81, 82	eqtr2i 2473	. . . . . . 7 |- 0 = ( -u 1 + 1 )
84	78, 83	syl6breq 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 )
86	50, 75, 84, 85	syl3anc 1229	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -. ( A / ( 2 x. pi ) ) e. ZZ )
87	40, 48, 86	syl2anc 661	. . . 4 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> -. ( A / ( 2 x. pi ) ) e. ZZ )
88	39, 87	pm2.61dan 791	. . 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 11985	. . . 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 661	. . 3 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( ( A mod ( 2 x. pi ) ) = 0 <-> ( A / ( 2 x. pi ) ) e. ZZ ) )
92	88, 91	mtbird 301	. 2 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -. ( A mod ( 2 x. pi ) ) = 0 )
93	92	neqned 2646	1 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( A mod ( 2 x. pi ) ) =/= 0 )

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