Theorem fourierdlem6 40330

Description: X is in the periodic partition, when the considered interval is centered at X. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem6.a	|- ( ph -> A e. RR )
fourierdlem6.b	|- ( ph -> B e. RR )
fourierdlem6.altb	|- ( ph -> A < B )
fourierdlem6.t	|- T = ( B - A )
fourierdlem6.5	|- ( ph -> X e. RR )
fourierdlem6.i	|- ( ph -> I e. ZZ )
fourierdlem6.j	|- ( ph -> J e. ZZ )
fourierdlem6.iltj	|- ( ph -> I < J )
fourierdlem6.iel	|- ( ph -> ( X + ( I x. T ) ) e. ( A [,] B ) )
fourierdlem6.jel	|- ( ph -> ( X + ( J x. T ) ) e. ( A [,] B ) )

Assertion

Ref	Expression
fourierdlem6	|- ( ph -> J = ( I + 1 ) )

Proof of Theorem fourierdlem6

Step	Hyp	Ref	Expression
1		fourierdlem6.j	. . . . . . . 8 |- ( ph -> J e. ZZ )
2	1	zred 11482	. . . . . . 7 |- ( ph -> J e. RR )
3		fourierdlem6.i	. . . . . . . 8 |- ( ph -> I e. ZZ )
4	3	zred 11482	. . . . . . 7 |- ( ph -> I e. RR )
5	2, 4	resubcld 10458	. . . . . 6 |- ( ph -> ( J - I ) e. RR )
6		fourierdlem6.t	. . . . . . 7 |- T = ( B - A )
7		fourierdlem6.b	. . . . . . . 8 |- ( ph -> B e. RR )
8		fourierdlem6.a	. . . . . . . 8 |- ( ph -> A e. RR )
9	7, 8	resubcld 10458	. . . . . . 7 |- ( ph -> ( B - A ) e. RR )
10	6, 9	syl5eqel 2705	. . . . . 6 |- ( ph -> T e. RR )
11	5, 10	remulcld 10070	. . . . 5 |- ( ph -> ( ( J - I ) x. T ) e. RR )
12		fourierdlem6.altb	. . . . . . . 8 |- ( ph -> A < B )
13	8, 7	posdifd 10614	. . . . . . . 8 |- ( ph -> ( A < B <-> 0 < ( B - A ) ) )
14	12, 13	mpbid 222	. . . . . . 7 |- ( ph -> 0 < ( B - A ) )
15	14, 6	syl6breqr 4695	. . . . . 6 |- ( ph -> 0 < T )
16	10, 15	elrpd 11869	. . . . 5 |- ( ph -> T e. RR+ )
17		fourierdlem6.jel	. . . . . . 7 |- ( ph -> ( X + ( J x. T ) ) e. ( A [,] B ) )
18		fourierdlem6.iel	. . . . . . 7 |- ( ph -> ( X + ( I x. T ) ) e. ( A [,] B ) )
19	8, 7, 17, 18	iccsuble 39745	. . . . . 6 |- ( ph -> ( ( X + ( J x. T ) ) - ( X + ( I x. T ) ) ) <_ ( B - A ) )
20	2	recnd 10068	. . . . . . . 8 |- ( ph -> J e. CC )
21	4	recnd 10068	. . . . . . . 8 |- ( ph -> I e. CC )
22	10	recnd 10068	. . . . . . . 8 |- ( ph -> T e. CC )
23	20, 21, 22	subdird 10487	. . . . . . 7 |- ( ph -> ( ( J - I ) x. T ) = ( ( J x. T ) - ( I x. T ) ) )
24		fourierdlem6.5	. . . . . . . . 9 |- ( ph -> X e. RR )
25	24	recnd 10068	. . . . . . . 8 |- ( ph -> X e. CC )
26	2, 10	remulcld 10070	. . . . . . . . 9 |- ( ph -> ( J x. T ) e. RR )
27	26	recnd 10068	. . . . . . . 8 |- ( ph -> ( J x. T ) e. CC )
28	4, 10	remulcld 10070	. . . . . . . . 9 |- ( ph -> ( I x. T ) e. RR )
29	28	recnd 10068	. . . . . . . 8 |- ( ph -> ( I x. T ) e. CC )
30	25, 27, 29	pnpcand 10429	. . . . . . 7 |- ( ph -> ( ( X + ( J x. T ) ) - ( X + ( I x. T ) ) ) = ( ( J x. T ) - ( I x. T ) ) )
31	23, 30	eqtr4d 2659	. . . . . 6 |- ( ph -> ( ( J - I ) x. T ) = ( ( X + ( J x. T ) ) - ( X + ( I x. T ) ) ) )
32	6	a1i 11	. . . . . 6 |- ( ph -> T = ( B - A ) )
33	19, 31, 32	3brtr4d 4685	. . . . 5 |- ( ph -> ( ( J - I ) x. T ) <_ T )
34	11, 10, 16, 33	lediv1dd 11930	. . . 4 |- ( ph -> ( ( ( J - I ) x. T ) / T ) <_ ( T / T ) )
35	5	recnd 10068	. . . . 5 |- ( ph -> ( J - I ) e. CC )
36	15	gt0ne0d 10592	. . . . 5 |- ( ph -> T =/= 0 )
37	35, 22, 36	divcan4d 10807	. . . 4 |- ( ph -> ( ( ( J - I ) x. T ) / T ) = ( J - I ) )
38	22, 36	dividd 10799	. . . 4 |- ( ph -> ( T / T ) = 1 )
39	34, 37, 38	3brtr3d 4684	. . 3 |- ( ph -> ( J - I ) <_ 1 )
40		1red 10055	. . . 4 |- ( ph -> 1 e. RR )
41	2, 4, 40	lesubadd2d 10626	. . 3 |- ( ph -> ( ( J - I ) <_ 1 <-> J <_ ( I + 1 ) ) )
42	39, 41	mpbid 222	. 2 |- ( ph -> J <_ ( I + 1 ) )
43		fourierdlem6.iltj	. . 3 |- ( ph -> I < J )
44		zltp1le 11427	. . . 4 |- ( ( I e. ZZ /\ J e. ZZ ) -> ( I < J <-> ( I + 1 ) <_ J ) )
45	3, 1, 44	syl2anc 693	. . 3 |- ( ph -> ( I < J <-> ( I + 1 ) <_ J ) )
46	43, 45	mpbid 222	. 2 |- ( ph -> ( I + 1 ) <_ J )
47	4, 40	readdcld 10069	. . 3 |- ( ph -> ( I + 1 ) e. RR )
48	2, 47	letri3d 10179	. 2 |- ( ph -> ( J = ( I + 1 ) <-> ( J <_ ( I + 1 ) /\ ( I + 1 ) <_ J ) ) )
49	42, 46, 48	mpbir2and 957	1 |- ( ph -> J = ( I + 1 ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   ZZcz 11377   [,]cicc 12178

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-iun 4522  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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-n0 11293  df-z 11378  df-rp 11833  df-icc 12182

This theorem is referenced by:  fourierdlem35  40359  fourierdlem51  40374