Theorem fperiodmul 39518

Description: A function with period T is also periodic with period multiple of T. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fperiodmul.f	|- ( ph -> F : RR --> CC )
fperiodmul.t	|- ( ph -> T e. RR )
fperiodmul.n	|- ( ph -> N e. ZZ )
fperiodmul.x	|- ( ph -> X e. RR )
fperiodmul.per	|- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )

Assertion

Ref	Expression
fperiodmul	|- ( ph -> ( F ` ( X + ( N x. T ) ) ) = ( F ` X ) )

Distinct variable groups:   x, F   x, N   x, T   x, X   ph, x

Proof of Theorem fperiodmul

Step	Hyp	Ref	Expression
1		fperiodmul.f	. . . 4 |- ( ph -> F : RR --> CC )
2	1	adantr 481	. . 3 |- ( ( ph /\ N e. NN0 ) -> F : RR --> CC )
3		fperiodmul.t	. . . 4 |- ( ph -> T e. RR )
4	3	adantr 481	. . 3 |- ( ( ph /\ N e. NN0 ) -> T e. RR )
5		simpr 477	. . 3 |- ( ( ph /\ N e. NN0 ) -> N e. NN0 )
6		fperiodmul.x	. . . 4 |- ( ph -> X e. RR )
7	6	adantr 481	. . 3 |- ( ( ph /\ N e. NN0 ) -> X e. RR )
8		fperiodmul.per	. . . 4 |- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
9	8	adantlr 751	. . 3 |- ( ( ( ph /\ N e. NN0 ) /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
10	2, 4, 5, 7, 9	fperiodmullem 39517	. 2 |- ( ( ph /\ N e. NN0 ) -> ( F ` ( X + ( N x. T ) ) ) = ( F ` X ) )
11	6	recnd 10068	. . . . . . 7 |- ( ph -> X e. CC )
12		fperiodmul.n	. . . . . . . . 9 |- ( ph -> N e. ZZ )
13	12	zcnd 11483	. . . . . . . 8 |- ( ph -> N e. CC )
14	3	recnd 10068	. . . . . . . 8 |- ( ph -> T e. CC )
15	13, 14	mulcld 10060	. . . . . . 7 |- ( ph -> ( N x. T ) e. CC )
16	11, 15	subnegd 10399	. . . . . 6 |- ( ph -> ( X - -u ( N x. T ) ) = ( X + ( N x. T ) ) )
17	13, 14	mulneg1d 10483	. . . . . . . 8 |- ( ph -> ( -u N x. T ) = -u ( N x. T ) )
18	17	eqcomd 2628	. . . . . . 7 |- ( ph -> -u ( N x. T ) = ( -u N x. T ) )
19	18	oveq2d 6666	. . . . . 6 |- ( ph -> ( X - -u ( N x. T ) ) = ( X - ( -u N x. T ) ) )
20	16, 19	eqtr3d 2658	. . . . 5 |- ( ph -> ( X + ( N x. T ) ) = ( X - ( -u N x. T ) ) )
21	20	fveq2d 6195	. . . 4 |- ( ph -> ( F ` ( X + ( N x. T ) ) ) = ( F ` ( X - ( -u N x. T ) ) ) )
22	21	adantr 481	. . 3 |- ( ( ph /\ -. N e. NN0 ) -> ( F ` ( X + ( N x. T ) ) ) = ( F ` ( X - ( -u N x. T ) ) ) )
23	1	adantr 481	. . . 4 |- ( ( ph /\ -. N e. NN0 ) -> F : RR --> CC )
24	3	adantr 481	. . . 4 |- ( ( ph /\ -. N e. NN0 ) -> T e. RR )
25		znnn0nn 11489	. . . . . 6 |- ( ( N e. ZZ /\ -. N e. NN0 ) -> -u N e. NN )
26	12, 25	sylan 488	. . . . 5 |- ( ( ph /\ -. N e. NN0 ) -> -u N e. NN )
27	26	nnnn0d 11351	. . . 4 |- ( ( ph /\ -. N e. NN0 ) -> -u N e. NN0 )
28	6	adantr 481	. . . . 5 |- ( ( ph /\ -. N e. NN0 ) -> X e. RR )
29	12	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. N e. NN0 ) -> N e. ZZ )
30	29	zred 11482	. . . . . . 7 |- ( ( ph /\ -. N e. NN0 ) -> N e. RR )
31	30	renegcld 10457	. . . . . 6 |- ( ( ph /\ -. N e. NN0 ) -> -u N e. RR )
32	31, 24	remulcld 10070	. . . . 5 |- ( ( ph /\ -. N e. NN0 ) -> ( -u N x. T ) e. RR )
33	28, 32	resubcld 10458	. . . 4 |- ( ( ph /\ -. N e. NN0 ) -> ( X - ( -u N x. T ) ) e. RR )
34	8	adantlr 751	. . . 4 |- ( ( ( ph /\ -. N e. NN0 ) /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
35	23, 24, 27, 33, 34	fperiodmullem 39517	. . 3 |- ( ( ph /\ -. N e. NN0 ) -> ( F ` ( ( X - ( -u N x. T ) ) + ( -u N x. T ) ) ) = ( F ` ( X - ( -u N x. T ) ) ) )
36	28	recnd 10068	. . . . 5 |- ( ( ph /\ -. N e. NN0 ) -> X e. CC )
37	30	recnd 10068	. . . . . . 7 |- ( ( ph /\ -. N e. NN0 ) -> N e. CC )
38	37	negcld 10379	. . . . . 6 |- ( ( ph /\ -. N e. NN0 ) -> -u N e. CC )
39	24	recnd 10068	. . . . . 6 |- ( ( ph /\ -. N e. NN0 ) -> T e. CC )
40	38, 39	mulcld 10060	. . . . 5 |- ( ( ph /\ -. N e. NN0 ) -> ( -u N x. T ) e. CC )
41	36, 40	npcand 10396	. . . 4 |- ( ( ph /\ -. N e. NN0 ) -> ( ( X - ( -u N x. T ) ) + ( -u N x. T ) ) = X )
42	41	fveq2d 6195	. . 3 |- ( ( ph /\ -. N e. NN0 ) -> ( F ` ( ( X - ( -u N x. T ) ) + ( -u N x. T ) ) ) = ( F ` X ) )
43	22, 35, 42	3eqtr2d 2662	. 2 |- ( ( ph /\ -. N e. NN0 ) -> ( F ` ( X + ( N x. T ) ) ) = ( F ` X ) )
44	10, 43	pm2.61dan 832	1 |- ( ph -> ( F ` ( X + ( N x. T ) ) ) = ( F ` X ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   NNcn 11020   NN0cn0 11292   ZZcz 11377

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-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-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-nn 11021  df-n0 11293  df-z 11378

This theorem is referenced by:  fourierdlem89  40412  fourierdlem90  40413  fourierdlem91  40414  fourierdlem94  40417  fourierdlem97  40420  fourierdlem113  40436