Theorem fourierdlem24 37987

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 10946	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> 0 e. ZZ )
2		pire 23406	. . . . . . . . . 10 |- pi e. RR
3	2	renegcli 9932	. . . . . . . . 9 |- -u pi e. RR
4		iccssre 11713	. . . . . . . . 9 |- ( ( -u pi e. RR /\ pi e. RR ) -> ( -u pi [,] pi ) C_ RR )
5	3, 2, 4	mp2an 677	. . . . . . . 8 |- ( -u pi [,] pi ) C_ RR
6		eldifi 3554	. . . . . . . 8 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A e. ( -u pi [,] pi ) )
7	5, 6	sseldi 3429	. . . . . . 7 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A e. RR )
8	7	adantr 467	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> A e. RR )
9		2re 10676	. . . . . . . 8 |- 2 e. RR
10	9, 2	remulcli 9654	. . . . . . 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 463	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> 0 < A )
13		2pos 10698	. . . . . . . 8 |- 0 < 2
14		pipos 23408	. . . . . . . 8 |- 0 < pi
15	9, 2, 13, 14	mulgt0ii 9765	. . . . . . 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 10539	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> 0 < ( A / ( 2 x. pi ) ) )
18	11, 16	elrpd 11335	. . . . . . . 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 9690	. . . . . . . . . . . 12 |- -u pi e. RR*
22	21	a1i 11	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u pi e. RR* )
23	19	rexrd 9687	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> pi e. RR* )
24		iccleub 11687	. . . . . . . . . . 11 |- ( ( -u pi e. RR* /\ pi e. RR* /\ A e. ( -u pi [,] pi ) ) -> A <_ pi )
25	22, 23, 6, 24	syl3anc 1267	. . . . . . . . . 10 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A <_ pi )
26		pirp 23409	. . . . . . . . . . 11 |- pi e. RR+
27		2timesgt 37495	. . . . . . . . . . 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 9790	. . . . . . . . 9 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A < ( 2 x. pi ) )
30	29	adantr 467	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> A < ( 2 x. pi ) )
31	8, 11, 18, 30	ltdiv1dd 11392	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> ( A / ( 2 x. pi ) ) < ( ( 2 x. pi ) / ( 2 x. pi ) ) )
32	10	recni 9652	. . . . . . . 8 |- ( 2 x. pi ) e. CC
33	10, 15	gt0ne0ii 10147	. . . . . . . 8 |- ( 2 x. pi ) =/= 0
34	32, 33	dividi 10337	. . . . . . 7 |- ( ( 2 x. pi ) / ( 2 x. pi ) ) = 1
35	31, 34	syl6breq 4441	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> ( A / ( 2 x. pi ) ) < 1 )
36		0p1e1 10718	. . . . . 6 |- ( 0 + 1 ) = 1
37	35, 36	syl6breqr 4442	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> ( A / ( 2 x. pi ) ) < ( 0 + 1 ) )
38		btwnnz 11009	. . . . 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 1267	. . . 4 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> -. ( A / ( 2 x. pi ) ) e. ZZ )
40		simpl 459	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> A e. ( ( -u pi [,] pi ) \ { 0 } ) )
41	7	adantr 467	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> A e. RR )
42		0red 9641	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> 0 e. RR )
43		simpr 463	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> -. 0 < A )
44	41, 42, 43	nltled 9782	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> A <_ 0 )
45		eldifsni 4097	. . . . . . . 8 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A =/= 0 )
46	45	necomd 2678	. . . . . . 7 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> 0 =/= A )
47	46	adantr 467	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> 0 =/= A )
48	41, 42, 44, 47	leneltd 9786	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> A < 0 )
49		neg1z 10970	. . . . . . 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 10432	. . . . . . . 8 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( A / ( 2 x. pi ) ) e. RR )
53	52	adantr 467	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( A / ( 2 x. pi ) ) e. RR )
54		1red 9655	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> 1 e. RR )
55	7	recnd 9666	. . . . . . . . . 10 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A e. CC )
56	55	adantr 467	. . . . . . . . 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 10393	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u ( A / ( 2 x. pi ) ) = ( -u A / ( 2 x. pi ) ) )
60	7	renegcld 10043	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u A e. RR )
61	60	adantr 467	. . . . . . . . . 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 11304	. . . . . . . . . . . 12 |- 2 e. RR+
64		rpmulcl 11321	. . . . . . . . . . . 12 |- ( ( 2 e. RR+ /\ pi e. RR+ ) -> ( 2 x. pi ) e. RR+ )
65	63, 26, 64	mp2an 677	. . . . . . . . . . 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 11688	. . . . . . . . . . . . . 14 |- ( ( -u pi e. RR* /\ pi e. RR* /\ A e. ( -u pi [,] pi ) ) -> -u pi <_ A )
68	22, 23, 6, 67	syl3anc 1267	. . . . . . . . . . . . 13 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u pi <_ A )
69	19, 7, 68	lenegcon1d 10192	. . . . . . . . . . . 12 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u A <_ pi )
70	60, 19, 20, 69, 28	lelttrd 9790	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u A < ( 2 x. pi ) )
71	70	adantr 467	. . . . . . . . . 10 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u A < ( 2 x. pi ) )
72	61, 62, 66, 71	ltdiv1dd 11392	. . . . . . . . 9 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( -u A / ( 2 x. pi ) ) < ( ( 2 x. pi ) / ( 2 x. pi ) ) )
73	72, 34	syl6breq 4441	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( -u A / ( 2 x. pi ) ) < 1 )
74	59, 73	eqbrtrd 4422	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u ( A / ( 2 x. pi ) ) < 1 )
75	53, 54, 74	ltnegcon1d 10190	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u 1 < ( A / ( 2 x. pi ) ) )
76	7	adantr 467	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> A e. RR )
77		simpr 463	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> A < 0 )
78	76, 66, 77	divlt0gt0d 37490	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( A / ( 2 x. pi ) ) < 0 )
79		neg1cn 10710	. . . . . . . . 9 |- -u 1 e. CC
80		ax-1cn 9594	. . . . . . . . 9 |- 1 e. CC
81	79, 80	addcomi 9821	. . . . . . . 8 |- ( -u 1 + 1 ) = ( 1 + -u 1 )
82		1pneg1e0 10715	. . . . . . . 8 |- ( 1 + -u 1 ) = 0
83	81, 82	eqtr2i 2473	. . . . . . 7 |- 0 = ( -u 1 + 1 )
84	78, 83	syl6breq 4441	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( A / ( 2 x. pi ) ) < ( -u 1 + 1 ) )
85		btwnnz 11009	. . . . . 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 1267	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -. ( A / ( 2 x. pi ) ) e. ZZ )
87	40, 48, 86	syl2anc 666	. . . 4 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> -. ( A / ( 2 x. pi ) ) e. ZZ )
88	39, 87	pm2.61dan 799	. . 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 12100	. . . 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 666	. . 3 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( ( A mod ( 2 x. pi ) ) = 0 <-> ( A / ( 2 x. pi ) ) e. ZZ ) )
92	88, 91	mtbird 303	. 2 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -. ( A mod ( 2 x. pi ) ) = 0 )
93	92	neqned 2630	1 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( A mod ( 2 x. pi ) ) =/= 0 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1443   e. wcel 1886   =/= wne 2621   \ cdif 3400   C_ wss 3403   {csn 3967   class class class wbr 4401  (class class class)co 6288   CCcc 9534   RRcr 9535   0cc0 9536   1c1 9537   + caddc 9539   x. cmul 9541   RR*cxr 9671   < clt 9672   <_ cle 9673   -ucneg 9858   / cdiv 10266   2c2 10656   ZZcz 10934   RR+crp 11299   [,]cicc 11635   mod cmo 12093   picpi 14112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ioc 11637  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13123  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-sin 14116  df-cos 14117  df-pi 14119  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-ntr 20028  df-cls 20029  df-nei 20107  df-lp 20145  df-perf 20146  df-cn 20236  df-cnp 20237  df-haus 20324  df-tx 20570  df-hmeo 20763  df-fil 20854  df-fm 20946  df-flim 20947  df-flf 20948  df-xms 21328  df-ms 21329  df-tms 21330  df-cncf 21903  df-limc 22814  df-dv 22815

This theorem is referenced by:  fourierdlem66  38030