Theorem fourierdlem24 40348

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 11389	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> 0 e. ZZ )
2		pire 24210	. . . . . . . . . 10 |- pi e. RR
3	2	renegcli 10342	. . . . . . . . 9 |- -u pi e. RR
4		iccssre 12255	. . . . . . . . 9 |- ( ( -u pi e. RR /\ pi e. RR ) -> ( -u pi [,] pi ) C_ RR )
5	3, 2, 4	mp2an 708	. . . . . . . 8 |- ( -u pi [,] pi ) C_ RR
6		eldifi 3732	. . . . . . . 8 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A e. ( -u pi [,] pi ) )
7	5, 6	sseldi 3601	. . . . . . 7 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A e. RR )
8	7	adantr 481	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> A e. RR )
9		2re 11090	. . . . . . . 8 |- 2 e. RR
10	9, 2	remulcli 10054	. . . . . . 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 477	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> 0 < A )
13		2pos 11112	. . . . . . . 8 |- 0 < 2
14		pipos 24212	. . . . . . . 8 |- 0 < pi
15	9, 2, 13, 14	mulgt0ii 10170	. . . . . . 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 10959	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> 0 < ( A / ( 2 x. pi ) ) )
18	11, 16	elrpd 11869	. . . . . . . 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 10097	. . . . . . . . . . . 12 |- -u pi e. RR*
22	21	a1i 11	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u pi e. RR* )
23	19	rexrd 10089	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> pi e. RR* )
24		iccleub 12229	. . . . . . . . . . 11 |- ( ( -u pi e. RR* /\ pi e. RR* /\ A e. ( -u pi [,] pi ) ) -> A <_ pi )
25	22, 23, 6, 24	syl3anc 1326	. . . . . . . . . 10 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A <_ pi )
26		pirp 24213	. . . . . . . . . . 11 |- pi e. RR+
27		2timesgt 39500	. . . . . . . . . . 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 10195	. . . . . . . . 9 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A < ( 2 x. pi ) )
30	29	adantr 481	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> A < ( 2 x. pi ) )
31	8, 11, 18, 30	ltdiv1dd 11929	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> ( A / ( 2 x. pi ) ) < ( ( 2 x. pi ) / ( 2 x. pi ) ) )
32	10	recni 10052	. . . . . . . 8 |- ( 2 x. pi ) e. CC
33	10, 15	gt0ne0ii 10564	. . . . . . . 8 |- ( 2 x. pi ) =/= 0
34	32, 33	dividi 10758	. . . . . . 7 |- ( ( 2 x. pi ) / ( 2 x. pi ) ) = 1
35	31, 34	syl6breq 4694	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> ( A / ( 2 x. pi ) ) < 1 )
36		0p1e1 11132	. . . . . 6 |- ( 0 + 1 ) = 1
37	35, 36	syl6breqr 4695	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> ( A / ( 2 x. pi ) ) < ( 0 + 1 ) )
38		btwnnz 11453	. . . . 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 1326	. . . 4 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ 0 < A ) -> -. ( A / ( 2 x. pi ) ) e. ZZ )
40		simpl 473	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> A e. ( ( -u pi [,] pi ) \ { 0 } ) )
41	7	adantr 481	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> A e. RR )
42		0red 10041	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> 0 e. RR )
43		simpr 477	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> -. 0 < A )
44	41, 42, 43	nltled 10187	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> A <_ 0 )
45		eldifsni 4320	. . . . . . . 8 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A =/= 0 )
46	45	necomd 2849	. . . . . . 7 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> 0 =/= A )
47	46	adantr 481	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> 0 =/= A )
48	41, 42, 44, 47	leneltd 10191	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> A < 0 )
49		neg1z 11413	. . . . . . 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 10853	. . . . . . . 8 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( A / ( 2 x. pi ) ) e. RR )
53	52	adantr 481	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( A / ( 2 x. pi ) ) e. RR )
54		1red 10055	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> 1 e. RR )
55	7	recnd 10068	. . . . . . . . . 10 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> A e. CC )
56	55	adantr 481	. . . . . . . . 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 10814	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u ( A / ( 2 x. pi ) ) = ( -u A / ( 2 x. pi ) ) )
60	7	renegcld 10457	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u A e. RR )
61	60	adantr 481	. . . . . . . . . 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 11837	. . . . . . . . . . . 12 |- 2 e. RR+
64		rpmulcl 11855	. . . . . . . . . . . 12 |- ( ( 2 e. RR+ /\ pi e. RR+ ) -> ( 2 x. pi ) e. RR+ )
65	63, 26, 64	mp2an 708	. . . . . . . . . . 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 12230	. . . . . . . . . . . . . 14 |- ( ( -u pi e. RR* /\ pi e. RR* /\ A e. ( -u pi [,] pi ) ) -> -u pi <_ A )
68	22, 23, 6, 67	syl3anc 1326	. . . . . . . . . . . . 13 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u pi <_ A )
69	19, 7, 68	lenegcon1d 10609	. . . . . . . . . . . 12 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u A <_ pi )
70	60, 19, 20, 69, 28	lelttrd 10195	. . . . . . . . . . 11 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -u A < ( 2 x. pi ) )
71	70	adantr 481	. . . . . . . . . 10 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u A < ( 2 x. pi ) )
72	61, 62, 66, 71	ltdiv1dd 11929	. . . . . . . . 9 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( -u A / ( 2 x. pi ) ) < ( ( 2 x. pi ) / ( 2 x. pi ) ) )
73	72, 34	syl6breq 4694	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( -u A / ( 2 x. pi ) ) < 1 )
74	59, 73	eqbrtrd 4675	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u ( A / ( 2 x. pi ) ) < 1 )
75	53, 54, 74	ltnegcon1d 10607	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -u 1 < ( A / ( 2 x. pi ) ) )
76	7	adantr 481	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> A e. RR )
77		simpr 477	. . . . . . . 8 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> A < 0 )
78	76, 66, 77	divlt0gt0d 39498	. . . . . . 7 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( A / ( 2 x. pi ) ) < 0 )
79		neg1cn 11124	. . . . . . . . 9 |- -u 1 e. CC
80		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
81	79, 80	addcomi 10227	. . . . . . . 8 |- ( -u 1 + 1 ) = ( 1 + -u 1 )
82		1pneg1e0 11129	. . . . . . . 8 |- ( 1 + -u 1 ) = 0
83	81, 82	eqtr2i 2645	. . . . . . 7 |- 0 = ( -u 1 + 1 )
84	78, 83	syl6breq 4694	. . . . . 6 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> ( A / ( 2 x. pi ) ) < ( -u 1 + 1 ) )
85		btwnnz 11453	. . . . . 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 1326	. . . . 5 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ A < 0 ) -> -. ( A / ( 2 x. pi ) ) e. ZZ )
87	40, 48, 86	syl2anc 693	. . . 4 |- ( ( A e. ( ( -u pi [,] pi ) \ { 0 } ) /\ -. 0 < A ) -> -. ( A / ( 2 x. pi ) ) e. ZZ )
88	39, 87	pm2.61dan 832	. . 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 12675	. . . 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 693	. . 3 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( ( A mod ( 2 x. pi ) ) = 0 <-> ( A / ( 2 x. pi ) ) e. ZZ ) )
92	88, 91	mtbird 315	. 2 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> -. ( A mod ( 2 x. pi ) ) = 0 )
93	92	neqned 2801	1 |- ( A e. ( ( -u pi [,] pi ) \ { 0 } ) -> ( A mod ( 2 x. pi ) ) =/= 0 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   -ucneg 10267   / cdiv 10684   2c2 11070   ZZcz 11377   RR+crp 11832   [,]cicc 12178   mod cmo 12668   picpi 14797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  fourierdlem66  40389