Theorem fourierdlem7 40331

Description: The difference between a point and it's periodic image in the interval, is decreasing. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem7.a	|- ( ph -> A e. RR )
fourierdlem7.b	|- ( ph -> B e. RR )
fourierdlem7.altb	|- ( ph -> A < B )
fourierdlem7.t	|- T = ( B - A )
fourierdlem7.e	|- E = ( x e. RR |-> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) )
fourierdlem7.x	|- ( ph -> X e. RR )
fourierdlem7.y	|- ( ph -> Y e. RR )
fourierdlem7.xlty	|- ( ph -> X <_ Y )

Assertion

Ref	Expression
fourierdlem7	|- ( ph -> ( ( E ` Y ) - Y ) <_ ( ( E ` X ) - X ) )

Distinct variable groups:   x, B   x, T   x, X   x, Y   ph, x

Allowed substitution hints:   A( x)   E( x)

Proof of Theorem fourierdlem7

Step	Hyp	Ref	Expression
1		fourierdlem7.b	. . . . . 6 |- ( ph -> B e. RR )
2		fourierdlem7.y	. . . . . 6 |- ( ph -> Y e. RR )
3	1, 2	resubcld 10458	. . . . 5 |- ( ph -> ( B - Y ) e. RR )
4		fourierdlem7.t	. . . . . 6 |- T = ( B - A )
5		fourierdlem7.a	. . . . . . 7 |- ( ph -> A e. RR )
6	1, 5	resubcld 10458	. . . . . 6 |- ( ph -> ( B - A ) e. RR )
7	4, 6	syl5eqel 2705	. . . . 5 |- ( ph -> T e. RR )
8		fourierdlem7.altb	. . . . . . . 8 |- ( ph -> A < B )
9	5, 1	posdifd 10614	. . . . . . . 8 |- ( ph -> ( A < B <-> 0 < ( B - A ) ) )
10	8, 9	mpbid 222	. . . . . . 7 |- ( ph -> 0 < ( B - A ) )
11	10, 4	syl6breqr 4695	. . . . . 6 |- ( ph -> 0 < T )
12	11	gt0ne0d 10592	. . . . 5 |- ( ph -> T =/= 0 )
13	3, 7, 12	redivcld 10853	. . . 4 |- ( ph -> ( ( B - Y ) / T ) e. RR )
14		fourierdlem7.x	. . . . . 6 |- ( ph -> X e. RR )
15	1, 14	resubcld 10458	. . . . 5 |- ( ph -> ( B - X ) e. RR )
16	15, 7, 12	redivcld 10853	. . . 4 |- ( ph -> ( ( B - X ) / T ) e. RR )
17	7, 11	elrpd 11869	. . . . 5 |- ( ph -> T e. RR+ )
18		fourierdlem7.xlty	. . . . . 6 |- ( ph -> X <_ Y )
19	14, 2, 1, 18	lesub2dd 10644	. . . . 5 |- ( ph -> ( B - Y ) <_ ( B - X ) )
20	3, 15, 17, 19	lediv1dd 11930	. . . 4 |- ( ph -> ( ( B - Y ) / T ) <_ ( ( B - X ) / T ) )
21		flwordi 12613	. . . 4 |- ( ( ( ( B - Y ) / T ) e. RR /\ ( ( B - X ) / T ) e. RR /\ ( ( B - Y ) / T ) <_ ( ( B - X ) / T ) ) -> ( |_ ` ( ( B - Y ) / T ) ) <_ ( |_ ` ( ( B - X ) / T ) ) )
22	13, 16, 20, 21	syl3anc 1326	. . 3 |- ( ph -> ( |_ ` ( ( B - Y ) / T ) ) <_ ( |_ ` ( ( B - X ) / T ) ) )
23	13	flcld 12599	. . . . 5 |- ( ph -> ( |_ ` ( ( B - Y ) / T ) ) e. ZZ )
24	23	zred 11482	. . . 4 |- ( ph -> ( |_ ` ( ( B - Y ) / T ) ) e. RR )
25	16	flcld 12599	. . . . 5 |- ( ph -> ( |_ ` ( ( B - X ) / T ) ) e. ZZ )
26	25	zred 11482	. . . 4 |- ( ph -> ( |_ ` ( ( B - X ) / T ) ) e. RR )
27	24, 26, 17	lemul1d 11915	. . 3 |- ( ph -> ( ( |_ ` ( ( B - Y ) / T ) ) <_ ( |_ ` ( ( B - X ) / T ) ) <-> ( ( |_ ` ( ( B - Y ) / T ) ) x. T ) <_ ( ( |_ ` ( ( B - X ) / T ) ) x. T ) ) )
28	22, 27	mpbid 222	. 2 |- ( ph -> ( ( |_ ` ( ( B - Y ) / T ) ) x. T ) <_ ( ( |_ ` ( ( B - X ) / T ) ) x. T ) )
29		fourierdlem7.e	. . . . . 6 |- E = ( x e. RR |-> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) )
30	29	a1i 11	. . . . 5 |- ( ph -> E = ( x e. RR |-> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) ) )
31		id 22	. . . . . . 7 |- ( x = Y -> x = Y )
32		oveq2 6658	. . . . . . . . . 10 |- ( x = Y -> ( B - x ) = ( B - Y ) )
33	32	oveq1d 6665	. . . . . . . . 9 |- ( x = Y -> ( ( B - x ) / T ) = ( ( B - Y ) / T ) )
34	33	fveq2d 6195	. . . . . . . 8 |- ( x = Y -> ( |_ ` ( ( B - x ) / T ) ) = ( |_ ` ( ( B - Y ) / T ) ) )
35	34	oveq1d 6665	. . . . . . 7 |- ( x = Y -> ( ( |_ ` ( ( B - x ) / T ) ) x. T ) = ( ( |_ ` ( ( B - Y ) / T ) ) x. T ) )
36	31, 35	oveq12d 6668	. . . . . 6 |- ( x = Y -> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) = ( Y + ( ( |_ ` ( ( B - Y ) / T ) ) x. T ) ) )
37	36	adantl 482	. . . . 5 |- ( ( ph /\ x = Y ) -> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) = ( Y + ( ( |_ ` ( ( B - Y ) / T ) ) x. T ) ) )
38	24, 7	remulcld 10070	. . . . . 6 |- ( ph -> ( ( |_ ` ( ( B - Y ) / T ) ) x. T ) e. RR )
39	2, 38	readdcld 10069	. . . . 5 |- ( ph -> ( Y + ( ( |_ ` ( ( B - Y ) / T ) ) x. T ) ) e. RR )
40	30, 37, 2, 39	fvmptd 6288	. . . 4 |- ( ph -> ( E ` Y ) = ( Y + ( ( |_ ` ( ( B - Y ) / T ) ) x. T ) ) )
41	40	oveq1d 6665	. . 3 |- ( ph -> ( ( E ` Y ) - Y ) = ( ( Y + ( ( |_ ` ( ( B - Y ) / T ) ) x. T ) ) - Y ) )
42	2	recnd 10068	. . . 4 |- ( ph -> Y e. CC )
43	38	recnd 10068	. . . 4 |- ( ph -> ( ( |_ ` ( ( B - Y ) / T ) ) x. T ) e. CC )
44	42, 43	pncan2d 10394	. . 3 |- ( ph -> ( ( Y + ( ( |_ ` ( ( B - Y ) / T ) ) x. T ) ) - Y ) = ( ( |_ ` ( ( B - Y ) / T ) ) x. T ) )
45	41, 44	eqtrd 2656	. 2 |- ( ph -> ( ( E ` Y ) - Y ) = ( ( |_ ` ( ( B - Y ) / T ) ) x. T ) )
46		id 22	. . . . . . 7 |- ( x = X -> x = X )
47		oveq2 6658	. . . . . . . . . 10 |- ( x = X -> ( B - x ) = ( B - X ) )
48	47	oveq1d 6665	. . . . . . . . 9 |- ( x = X -> ( ( B - x ) / T ) = ( ( B - X ) / T ) )
49	48	fveq2d 6195	. . . . . . . 8 |- ( x = X -> ( |_ ` ( ( B - x ) / T ) ) = ( |_ ` ( ( B - X ) / T ) ) )
50	49	oveq1d 6665	. . . . . . 7 |- ( x = X -> ( ( |_ ` ( ( B - x ) / T ) ) x. T ) = ( ( |_ ` ( ( B - X ) / T ) ) x. T ) )
51	46, 50	oveq12d 6668	. . . . . 6 |- ( x = X -> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) = ( X + ( ( |_ ` ( ( B - X ) / T ) ) x. T ) ) )
52	51	adantl 482	. . . . 5 |- ( ( ph /\ x = X ) -> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) = ( X + ( ( |_ ` ( ( B - X ) / T ) ) x. T ) ) )
53	26, 7	remulcld 10070	. . . . . 6 |- ( ph -> ( ( |_ ` ( ( B - X ) / T ) ) x. T ) e. RR )
54	14, 53	readdcld 10069	. . . . 5 |- ( ph -> ( X + ( ( |_ ` ( ( B - X ) / T ) ) x. T ) ) e. RR )
55	30, 52, 14, 54	fvmptd 6288	. . . 4 |- ( ph -> ( E ` X ) = ( X + ( ( |_ ` ( ( B - X ) / T ) ) x. T ) ) )
56	55	oveq1d 6665	. . 3 |- ( ph -> ( ( E ` X ) - X ) = ( ( X + ( ( |_ ` ( ( B - X ) / T ) ) x. T ) ) - X ) )
57	14	recnd 10068	. . . 4 |- ( ph -> X e. CC )
58	53	recnd 10068	. . . 4 |- ( ph -> ( ( |_ ` ( ( B - X ) / T ) ) x. T ) e. CC )
59	57, 58	pncan2d 10394	. . 3 |- ( ph -> ( ( X + ( ( |_ ` ( ( B - X ) / T ) ) x. T ) ) - X ) = ( ( |_ ` ( ( B - X ) / T ) ) x. T ) )
60	56, 59	eqtrd 2656	. 2 |- ( ph -> ( ( E ` X ) - X ) = ( ( |_ ` ( ( B - X ) / T ) ) x. T ) )
61	28, 45, 60	3brtr4d 4685	1 |- ( ph -> ( ( E ` Y ) - Y ) <_ ( ( E ` X ) - X ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   |_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-fl 12593

This theorem is referenced by:  fourierdlem63  40386