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

Step	Hyp	Ref	Expression
1		0zd 10973	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> 0 e. ZZ )
2		pire 23492	. . . . . . . . . 10 |- pi e. RR
3	2	renegcli 9955	. . . . . . . . 9 |- -u pi e. RR
4		iccssre 11741	. . . . . . . . 9 |- ( ( -u pi e. RR /\ pi e. RR ) -> ( -u pi [,] pi ) C_ RR )
5	3, 2, 4	mp2an 686	. . . . . . . 8 |- ( -u pi [,] pi ) C_ RR
6		eldifi 3544	. . . . . . . 8 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A e. ( -u pi [,] pi ) )
7	5, 6	sseldi 3416	. . . . . . 7 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A e. RR )
8	7	adantr 472	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> A e. RR )
9		2re 10701	. . . . . . . 8 |- 2 e. RR
10	9, 2	remulcli 9675	. . . . . . 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 468	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> 0 < A )
13		2pos 10723	. . . . . . . 8 |- 0 < 2
14		pipos 23494	. . . . . . . 8 |- 0 < pi
15	9, 2, 13, 14	mulgt0ii 9785	. . . . . . 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 10564	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> 0 < ( A / ( 2 x. pi ) ) )
18	11, 16	elrpd 11361	. . . . . . . 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 9711	. . . . . . . . . . . 12 |- -u pi e. RR*
22	21	a1i 11	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u pi e. RR* )
23	19	rexrd 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 )
25	22, 23, 6, 24	syl3anc 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 ) )
28	26, 27	mp1i 13	. . . . . . . . . 10 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> pi < ( 2 x. pi ) )
29	7, 19, 20, 25, 28	lelttrd 9810	. . . . . . . . 9 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A < ( 2 x. pi ) )
30	29	adantr 472	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> A < ( 2 x. pi ) )
31	8, 11, 18, 30	ltdiv1dd 11418	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> ( A / ( 2 x. pi ) ) < ( ( 2 x. pi ) / ( 2 x. pi ) ) )
32	10	recni 9673	. . . . . . . 8 |- ( 2 x. pi ) e. CC
33	10, 15	gt0ne0ii 10171	. . . . . . . 8 |- ( 2 x. pi ) =/= 0
34	32, 33	dividi 10362	. . . . . . 7 |- ( ( 2 x. pi ) / ( 2 x. pi ) ) = 1
35	31, 34	syl6breq 4435	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> ( A / ( 2 x. pi ) ) < 1 )
36		0p1e1 10743	. . . . . 6 |- ( 0 + 1 ) = 1
37	35, 36	syl6breqr 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 )
39	1, 17, 37, 38	syl3anc 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 } ) )
41	7	adantr 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 )
44	41, 42, 43	nltled 9802	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> A <_ 0 )
45		eldifsni 4089	. . . . . . . 8 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A =/= 0 )
46	45	necomd 2698	. . . . . . 7 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> 0 =/= A )
47	46	adantr 472	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> 0 =/= A )
48	41, 42, 44, 47	leneltd 9806	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> A < 0 )
49		neg1z 10997	. . . . . . 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 10457	. . . . . . . 8 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( A / ( 2 x. pi ) ) e. RR )
53	52	adantr 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 )
55	7	recnd 9687	. . . . . . . . . 10 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A e. CC )
56	55	adantr 472	. . . . . . . . 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 10418	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u ( A / ( 2 x. pi ) ) = ( -u A / ( 2 x. pi ) ) )
60	7	renegcld 10067	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u A e. RR )
61	60	adantr 472	. . . . . . . . . 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 11330	. . . . . . . . . . . 12 |- 2 e. RR+
64		rpmulcl 11347	. . . . . . . . . . . 12 |- ( ( 2 e. RR+ /\ pi e. RR+ ) -> ( 2 x. pi ) e. RR+ )
65	63, 26, 64	mp2an 686	. . . . . . . . . . 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 11716	. . . . . . . . . . . . . 14 |- ( ( -u pi e. RR* /\ pi e. RR* /\ A e. ( -u pi [,] pi ) ) -> -u pi <_ A )
68	22, 23, 6, 67	syl3anc 1292	. . . . . . . . . . . . 13 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u pi <_ A )
69	19, 7, 68	lenegcon1d 10216	. . . . . . . . . . . 12 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u A <_ pi )
70	60, 19, 20, 69, 28	lelttrd 9810	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u A < ( 2 x. pi ) )
71	70	adantr 472	. . . . . . . . . 10 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u A < ( 2 x. pi ) )
72	61, 62, 66, 71	ltdiv1dd 11418	. . . . . . . . 9 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( -u A / ( 2 x. pi ) ) < ( ( 2 x. pi ) / ( 2 x. pi ) ) )
73	72, 34	syl6breq 4435	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( -u A / ( 2 x. pi ) ) < 1 )
74	59, 73	eqbrtrd 4416	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u ( A / ( 2 x. pi ) ) < 1 )
75	53, 54, 74	ltnegcon1d 10214	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u 1 < ( A / ( 2 x. pi ) ) )
76	7	adantr 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 )
78	76, 66, 77	divlt0gt0d 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
81	79, 80	addcomi 9842	. . . . . . . 8 |- ( -u 1 + 1 ) = ( 1 + -u 1 )
82		1pneg1e0 10740	. . . . . . . 8 |- ( 1 + -u 1 ) = 0
83	81, 82	eqtr2i 2494	. . . . . . 7 |- 0 = ( -u 1 + 1 )
84	78, 83	syl6breq 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 )
86	50, 75, 84, 85	syl3anc 1292	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -. ( A / ( 2 x. pi ) ) e. ZZ )
87	40, 48, 86	syl2anc 673	. . . 4 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> -. ( A / ( 2 x. pi ) ) e. ZZ )
88	39, 87	pm2.61dan 808	. . 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 12136	. . . 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 673	. . 3 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( ( A mod ( 2 x. pi ) ) = 0 <-> ( A / ( 2 x. pi ) ) e. ZZ ) )
92	88, 91	mtbird 308	. 2 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -. ( A mod ( 2 x. pi ) ) = 0 )
93	92	neqned 2650	1 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( A mod ( 2 x. pi ) ) =/= 0 )

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