Theorem fourierdlem4 40328

Description: E is a function that maps any point to a periodic corresponding point in ( A , B ]. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem4.a	|- ( ph -> A e. RR )
fourierdlem4.b	|- ( ph -> B e. RR )
fourierdlem4.altb	|- ( ph -> A < B )
fourierdlem4.t	|- T = ( B - A )
fourierdlem4.e	|- E = ( x e. RR |-> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) )

Assertion

Ref	Expression
fourierdlem4	|- ( ph -> E : RR --> ( A (,] B ) )

Distinct variable groups:   x, A   x, B   x, T   ph, x

Allowed substitution hint:   E( x)

Proof of Theorem fourierdlem4

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 |- ( ( ph /\ x e. RR ) -> x e. RR )
2		fourierdlem4.b	. . . . . . . . . 10 |- ( ph -> B e. RR )
3	2	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> B e. RR )
4	3, 1	resubcld 10458	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> ( B - x ) e. RR )
5		fourierdlem4.t	. . . . . . . . . 10 |- T = ( B - A )
6		fourierdlem4.a	. . . . . . . . . . 11 |- ( ph -> A e. RR )
7	2, 6	resubcld 10458	. . . . . . . . . 10 |- ( ph -> ( B - A ) e. RR )
8	5, 7	syl5eqel 2705	. . . . . . . . 9 |- ( ph -> T e. RR )
9	8	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> T e. RR )
10	5	a1i 11	. . . . . . . . . 10 |- ( ph -> T = ( B - A ) )
11	2	recnd 10068	. . . . . . . . . . 11 |- ( ph -> B e. CC )
12	6	recnd 10068	. . . . . . . . . . 11 |- ( ph -> A e. CC )
13		fourierdlem4.altb	. . . . . . . . . . . 12 |- ( ph -> A < B )
14	6, 13	gtned 10172	. . . . . . . . . . 11 |- ( ph -> B =/= A )
15	11, 12, 14	subne0d 10401	. . . . . . . . . 10 |- ( ph -> ( B - A ) =/= 0 )
16	10, 15	eqnetrd 2861	. . . . . . . . 9 |- ( ph -> T =/= 0 )
17	16	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> T =/= 0 )
18	4, 9, 17	redivcld 10853	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( ( B - x ) / T ) e. RR )
19	18	flcld 12599	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( |_ ` ( ( B - x ) / T ) ) e. ZZ )
20	19	zred 11482	. . . . 5 |- ( ( ph /\ x e. RR ) -> ( |_ ` ( ( B - x ) / T ) ) e. RR )
21	20, 9	remulcld 10070	. . . 4 |- ( ( ph /\ x e. RR ) -> ( ( |_ ` ( ( B - x ) / T ) ) x. T ) e. RR )
22	1, 21	readdcld 10069	. . 3 |- ( ( ph /\ x e. RR ) -> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) e. RR )
23	6	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> A e. RR )
24	23, 1	resubcld 10458	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( A - x ) e. RR )
25	24, 9, 17	redivcld 10853	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( ( A - x ) / T ) e. RR )
26	25, 9	remulcld 10070	. . . . 5 |- ( ( ph /\ x e. RR ) -> ( ( ( A - x ) / T ) x. T ) e. RR )
27	11	addid1d 10236	. . . . . . . . . . . . . . 15 |- ( ph -> ( B + 0 ) = B )
28	27	eqcomd 2628	. . . . . . . . . . . . . 14 |- ( ph -> B = ( B + 0 ) )
29	11, 12	subcld 10392	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( B - A ) e. CC )
30	29	subidd 10380	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( B - A ) - ( B - A ) ) = 0 )
31	30	eqcomd 2628	. . . . . . . . . . . . . . 15 |- ( ph -> 0 = ( ( B - A ) - ( B - A ) ) )
32	31	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ph -> ( B + 0 ) = ( B + ( ( B - A ) - ( B - A ) ) ) )
33	11, 29, 29	addsub12d 10415	. . . . . . . . . . . . . . 15 |- ( ph -> ( B + ( ( B - A ) - ( B - A ) ) ) = ( ( B - A ) + ( B - ( B - A ) ) ) )
34	11, 12	nncand 10397	. . . . . . . . . . . . . . . 16 |- ( ph -> ( B - ( B - A ) ) = A )
35	34	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( B - A ) + ( B - ( B - A ) ) ) = ( ( B - A ) + A ) )
36	29, 12	addcomd 10238	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( B - A ) + A ) = ( A + ( B - A ) ) )
37	10	eqcomd 2628	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( B - A ) = T )
38	37	oveq2d 6666	. . . . . . . . . . . . . . . 16 |- ( ph -> ( A + ( B - A ) ) = ( A + T ) )
39	36, 38	eqtrd 2656	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( B - A ) + A ) = ( A + T ) )
40	33, 35, 39	3eqtrd 2660	. . . . . . . . . . . . . 14 |- ( ph -> ( B + ( ( B - A ) - ( B - A ) ) ) = ( A + T ) )
41	28, 32, 40	3eqtrd 2660	. . . . . . . . . . . . 13 |- ( ph -> B = ( A + T ) )
42	41	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR ) -> B = ( A + T ) )
43	42	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ph /\ x e. RR ) -> ( B - x ) = ( ( A + T ) - x ) )
44	12	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR ) -> A e. CC )
45	9	recnd 10068	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR ) -> T e. CC )
46	1	recnd 10068	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR ) -> x e. CC )
47	44, 45, 46	addsubd 10413	. . . . . . . . . . 11 |- ( ( ph /\ x e. RR ) -> ( ( A + T ) - x ) = ( ( A - x ) + T ) )
48	43, 47	eqtrd 2656	. . . . . . . . . 10 |- ( ( ph /\ x e. RR ) -> ( B - x ) = ( ( A - x ) + T ) )
49	48	oveq1d 6665	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> ( ( B - x ) / T ) = ( ( ( A - x ) + T ) / T ) )
50	44, 46	subcld 10392	. . . . . . . . . 10 |- ( ( ph /\ x e. RR ) -> ( A - x ) e. CC )
51	50, 45, 45, 17	divdird 10839	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> ( ( ( A - x ) + T ) / T ) = ( ( ( A - x ) / T ) + ( T / T ) ) )
52	5, 29	syl5eqel 2705	. . . . . . . . . . . 12 |- ( ph -> T e. CC )
53	52, 16	dividd 10799	. . . . . . . . . . 11 |- ( ph -> ( T / T ) = 1 )
54	53	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. RR ) -> ( T / T ) = 1 )
55	54	oveq2d 6666	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> ( ( ( A - x ) / T ) + ( T / T ) ) = ( ( ( A - x ) / T ) + 1 ) )
56	49, 51, 55	3eqtrd 2660	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> ( ( B - x ) / T ) = ( ( ( A - x ) / T ) + 1 ) )
57	56	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( |_ ` ( ( B - x ) / T ) ) = ( |_ ` ( ( ( A - x ) / T ) + 1 ) ) )
58	57	oveq1d 6665	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( ( |_ ` ( ( B - x ) / T ) ) x. T ) = ( ( |_ ` ( ( ( A - x ) / T ) + 1 ) ) x. T ) )
59	58, 21	eqeltrrd 2702	. . . . 5 |- ( ( ph /\ x e. RR ) -> ( ( |_ ` ( ( ( A - x ) / T ) + 1 ) ) x. T ) e. RR )
60		peano2re 10209	. . . . . . . 8 |- ( ( ( A - x ) / T ) e. RR -> ( ( ( A - x ) / T ) + 1 ) e. RR )
61	25, 60	syl 17	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( ( ( A - x ) / T ) + 1 ) e. RR )
62		reflcl 12597	. . . . . . 7 |- ( ( ( ( A - x ) / T ) + 1 ) e. RR -> ( |_ ` ( ( ( A - x ) / T ) + 1 ) ) e. RR )
63	61, 62	syl 17	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( |_ ` ( ( ( A - x ) / T ) + 1 ) ) e. RR )
64	6, 2	posdifd 10614	. . . . . . . . . 10 |- ( ph -> ( A < B <-> 0 < ( B - A ) ) )
65	13, 64	mpbid 222	. . . . . . . . 9 |- ( ph -> 0 < ( B - A ) )
66	65, 10	breqtrrd 4681	. . . . . . . 8 |- ( ph -> 0 < T )
67	8, 66	elrpd 11869	. . . . . . 7 |- ( ph -> T e. RR+ )
68	67	adantr 481	. . . . . 6 |- ( ( ph /\ x e. RR ) -> T e. RR+ )
69		flltp1 12601	. . . . . . . 8 |- ( ( ( A - x ) / T ) e. RR -> ( ( A - x ) / T ) < ( ( |_ ` ( ( A - x ) / T ) ) + 1 ) )
70	25, 69	syl 17	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( ( A - x ) / T ) < ( ( |_ ` ( ( A - x ) / T ) ) + 1 ) )
71		1zzd 11408	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> 1 e. ZZ )
72		fladdz 12626	. . . . . . . 8 |- ( ( ( ( A - x ) / T ) e. RR /\ 1 e. ZZ ) -> ( |_ ` ( ( ( A - x ) / T ) + 1 ) ) = ( ( |_ ` ( ( A - x ) / T ) ) + 1 ) )
73	25, 71, 72	syl2anc 693	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( |_ ` ( ( ( A - x ) / T ) + 1 ) ) = ( ( |_ ` ( ( A - x ) / T ) ) + 1 ) )
74	70, 73	breqtrrd 4681	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( ( A - x ) / T ) < ( |_ ` ( ( ( A - x ) / T ) + 1 ) ) )
75	25, 63, 68, 74	ltmul1dd 11927	. . . . 5 |- ( ( ph /\ x e. RR ) -> ( ( ( A - x ) / T ) x. T ) < ( ( |_ ` ( ( ( A - x ) / T ) + 1 ) ) x. T ) )
76	26, 59, 1, 75	ltadd2dd 10196	. . . 4 |- ( ( ph /\ x e. RR ) -> ( x + ( ( ( A - x ) / T ) x. T ) ) < ( x + ( ( |_ ` ( ( ( A - x ) / T ) + 1 ) ) x. T ) ) )
77	50, 45, 17	divcan1d 10802	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( ( ( A - x ) / T ) x. T ) = ( A - x ) )
78	77	oveq2d 6666	. . . . 5 |- ( ( ph /\ x e. RR ) -> ( x + ( ( ( A - x ) / T ) x. T ) ) = ( x + ( A - x ) ) )
79	46, 44	pncan3d 10395	. . . . 5 |- ( ( ph /\ x e. RR ) -> ( x + ( A - x ) ) = A )
80	78, 79	eqtrd 2656	. . . 4 |- ( ( ph /\ x e. RR ) -> ( x + ( ( ( A - x ) / T ) x. T ) ) = A )
81	58	oveq2d 6666	. . . . 5 |- ( ( ph /\ x e. RR ) -> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) = ( x + ( ( |_ ` ( ( ( A - x ) / T ) + 1 ) ) x. T ) ) )
82	81	eqcomd 2628	. . . 4 |- ( ( ph /\ x e. RR ) -> ( x + ( ( |_ ` ( ( ( A - x ) / T ) + 1 ) ) x. T ) ) = ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) )
83	76, 80, 82	3brtr3d 4684	. . 3 |- ( ( ph /\ x e. RR ) -> A < ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) )
84	18, 9	remulcld 10070	. . . . 5 |- ( ( ph /\ x e. RR ) -> ( ( ( B - x ) / T ) x. T ) e. RR )
85		flle 12600	. . . . . . 7 |- ( ( ( B - x ) / T ) e. RR -> ( |_ ` ( ( B - x ) / T ) ) <_ ( ( B - x ) / T ) )
86	18, 85	syl 17	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( |_ ` ( ( B - x ) / T ) ) <_ ( ( B - x ) / T ) )
87	20, 18, 68	lemul1d 11915	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( ( |_ ` ( ( B - x ) / T ) ) <_ ( ( B - x ) / T ) <-> ( ( |_ ` ( ( B - x ) / T ) ) x. T ) <_ ( ( ( B - x ) / T ) x. T ) ) )
88	86, 87	mpbid 222	. . . . 5 |- ( ( ph /\ x e. RR ) -> ( ( |_ ` ( ( B - x ) / T ) ) x. T ) <_ ( ( ( B - x ) / T ) x. T ) )
89	21, 84, 1, 88	leadd2dd 10642	. . . 4 |- ( ( ph /\ x e. RR ) -> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) <_ ( x + ( ( ( B - x ) / T ) x. T ) ) )
90	4	recnd 10068	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( B - x ) e. CC )
91	90, 45, 17	divcan1d 10802	. . . . . 6 |- ( ( ph /\ x e. RR ) -> ( ( ( B - x ) / T ) x. T ) = ( B - x ) )
92	91	oveq2d 6666	. . . . 5 |- ( ( ph /\ x e. RR ) -> ( x + ( ( ( B - x ) / T ) x. T ) ) = ( x + ( B - x ) ) )
93	11	adantr 481	. . . . . 6 |- ( ( ph /\ x e. RR ) -> B e. CC )
94	46, 93	pncan3d 10395	. . . . 5 |- ( ( ph /\ x e. RR ) -> ( x + ( B - x ) ) = B )
95	92, 94	eqtrd 2656	. . . 4 |- ( ( ph /\ x e. RR ) -> ( x + ( ( ( B - x ) / T ) x. T ) ) = B )
96	89, 95	breqtrd 4679	. . 3 |- ( ( ph /\ x e. RR ) -> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) <_ B )
97	23	rexrd 10089	. . . 4 |- ( ( ph /\ x e. RR ) -> A e. RR* )
98		elioc2 12236	. . . 4 |- ( ( A e. RR* /\ B e. RR ) -> ( ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) e. ( A (,] B ) <-> ( ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) e. RR /\ A < ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) /\ ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) <_ B ) ) )
99	97, 3, 98	syl2anc 693	. . 3 |- ( ( ph /\ x e. RR ) -> ( ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) e. ( A (,] B ) <-> ( ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) e. RR /\ A < ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) /\ ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) <_ B ) ) )
100	22, 83, 96, 99	mpbir3and 1245	. 2 |- ( ( ph /\ x e. RR ) -> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) e. ( A (,] B ) )
101		fourierdlem4.e	. 2 |- E = ( x e. RR |-> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) )
102	100, 101	fmptd 6385	1 |- ( ph -> E : RR --> ( A (,] B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (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   - cmin 10266   / cdiv 10684   ZZcz 11377   RR+crp 11832   (,]cioc 12176   |_cfl 12591

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  ax-pre-sup 10014

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-sup 8348  df-inf 8349  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-uz 11688  df-rp 11833  df-ioc 12180  df-fl 12593

This theorem is referenced by:  fourierdlem19  40343  fourierdlem37  40361  fourierdlem41  40365  fourierdlem48  40371  fourierdlem49  40372  fourierdlem51  40374  fourierdlem63  40386  fourierdlem65  40388  fourierdlem71  40394  fourierdlem79  40402  fourierdlem89  40412  fourierdlem90  40413  fourierdlem91  40414  fourierdlem102  40425  fourierdlem114  40437