Theorem fperiodmullem 39517

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

Hypotheses

Ref	Expression
fperiodmullem.f	|- ( ph -> F : RR --> CC )
fperiodmullem.t	|- ( ph -> T e. RR )
fperiodmullem.n	|- ( ph -> N e. NN0 )
fperiodmullem.x	|- ( ph -> X e. RR )
fperiodmullem.per	|- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )

Assertion

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

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

Allowed substitution hint:   N( x)

Proof of Theorem fperiodmullem

Dummy variables m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fperiodmullem.n	. 2 |- ( ph -> N e. NN0 )
2		oveq1 6657	. . . . . . 7 |- ( n = 0 -> ( n x. T ) = ( 0 x. T ) )
3	2	oveq2d 6666	. . . . . 6 |- ( n = 0 -> ( X + ( n x. T ) ) = ( X + ( 0 x. T ) ) )
4	3	fveq2d 6195	. . . . 5 |- ( n = 0 -> ( F ` ( X + ( n x. T ) ) ) = ( F ` ( X + ( 0 x. T ) ) ) )
5	4	eqeq1d 2624	. . . 4 |- ( n = 0 -> ( ( F ` ( X + ( n x. T ) ) ) = ( F ` X ) <-> ( F ` ( X + ( 0 x. T ) ) ) = ( F ` X ) ) )
6	5	imbi2d 330	. . 3 |- ( n = 0 -> ( ( ph -> ( F ` ( X + ( n x. T ) ) ) = ( F ` X ) ) <-> ( ph -> ( F ` ( X + ( 0 x. T ) ) ) = ( F ` X ) ) ) )
7		oveq1 6657	. . . . . . 7 |- ( n = m -> ( n x. T ) = ( m x. T ) )
8	7	oveq2d 6666	. . . . . 6 |- ( n = m -> ( X + ( n x. T ) ) = ( X + ( m x. T ) ) )
9	8	fveq2d 6195	. . . . 5 |- ( n = m -> ( F ` ( X + ( n x. T ) ) ) = ( F ` ( X + ( m x. T ) ) ) )
10	9	eqeq1d 2624	. . . 4 |- ( n = m -> ( ( F ` ( X + ( n x. T ) ) ) = ( F ` X ) <-> ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) ) )
11	10	imbi2d 330	. . 3 |- ( n = m -> ( ( ph -> ( F ` ( X + ( n x. T ) ) ) = ( F ` X ) ) <-> ( ph -> ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) ) ) )
12		oveq1 6657	. . . . . . 7 |- ( n = ( m + 1 ) -> ( n x. T ) = ( ( m + 1 ) x. T ) )
13	12	oveq2d 6666	. . . . . 6 |- ( n = ( m + 1 ) -> ( X + ( n x. T ) ) = ( X + ( ( m + 1 ) x. T ) ) )
14	13	fveq2d 6195	. . . . 5 |- ( n = ( m + 1 ) -> ( F ` ( X + ( n x. T ) ) ) = ( F ` ( X + ( ( m + 1 ) x. T ) ) ) )
15	14	eqeq1d 2624	. . . 4 |- ( n = ( m + 1 ) -> ( ( F ` ( X + ( n x. T ) ) ) = ( F ` X ) <-> ( F ` ( X + ( ( m + 1 ) x. T ) ) ) = ( F ` X ) ) )
16	15	imbi2d 330	. . 3 |- ( n = ( m + 1 ) -> ( ( ph -> ( F ` ( X + ( n x. T ) ) ) = ( F ` X ) ) <-> ( ph -> ( F ` ( X + ( ( m + 1 ) x. T ) ) ) = ( F ` X ) ) ) )
17		oveq1 6657	. . . . . . 7 |- ( n = N -> ( n x. T ) = ( N x. T ) )
18	17	oveq2d 6666	. . . . . 6 |- ( n = N -> ( X + ( n x. T ) ) = ( X + ( N x. T ) ) )
19	18	fveq2d 6195	. . . . 5 |- ( n = N -> ( F ` ( X + ( n x. T ) ) ) = ( F ` ( X + ( N x. T ) ) ) )
20	19	eqeq1d 2624	. . . 4 |- ( n = N -> ( ( F ` ( X + ( n x. T ) ) ) = ( F ` X ) <-> ( F ` ( X + ( N x. T ) ) ) = ( F ` X ) ) )
21	20	imbi2d 330	. . 3 |- ( n = N -> ( ( ph -> ( F ` ( X + ( n x. T ) ) ) = ( F ` X ) ) <-> ( ph -> ( F ` ( X + ( N x. T ) ) ) = ( F ` X ) ) ) )
22		fperiodmullem.t	. . . . . . . 8 |- ( ph -> T e. RR )
23	22	recnd 10068	. . . . . . 7 |- ( ph -> T e. CC )
24	23	mul02d 10234	. . . . . 6 |- ( ph -> ( 0 x. T ) = 0 )
25	24	oveq2d 6666	. . . . 5 |- ( ph -> ( X + ( 0 x. T ) ) = ( X + 0 ) )
26		fperiodmullem.x	. . . . . . 7 |- ( ph -> X e. RR )
27	26	recnd 10068	. . . . . 6 |- ( ph -> X e. CC )
28	27	addid1d 10236	. . . . 5 |- ( ph -> ( X + 0 ) = X )
29	25, 28	eqtrd 2656	. . . 4 |- ( ph -> ( X + ( 0 x. T ) ) = X )
30	29	fveq2d 6195	. . 3 |- ( ph -> ( F ` ( X + ( 0 x. T ) ) ) = ( F ` X ) )
31		simp3 1063	. . . . 5 |- ( ( m e. NN0 /\ ( ph -> ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) ) /\ ph ) -> ph )
32		simp1 1061	. . . . 5 |- ( ( m e. NN0 /\ ( ph -> ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) ) /\ ph ) -> m e. NN0 )
33		simpr 477	. . . . . . 7 |- ( ( ( ph -> ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) ) /\ ph ) -> ph )
34		simpl 473	. . . . . . 7 |- ( ( ( ph -> ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) ) /\ ph ) -> ( ph -> ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) ) )
35	33, 34	mpd 15	. . . . . 6 |- ( ( ( ph -> ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) ) /\ ph ) -> ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) )
36	35	3adant1 1079	. . . . 5 |- ( ( m e. NN0 /\ ( ph -> ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) ) /\ ph ) -> ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) )
37		nn0cn 11302	. . . . . . . . . . . 12 |- ( m e. NN0 -> m e. CC )
38	37	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN0 ) -> m e. CC )
39		1cnd 10056	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN0 ) -> 1 e. CC )
40	23	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN0 ) -> T e. CC )
41	38, 39, 40	adddird 10065	. . . . . . . . . 10 |- ( ( ph /\ m e. NN0 ) -> ( ( m + 1 ) x. T ) = ( ( m x. T ) + ( 1 x. T ) ) )
42	41	oveq2d 6666	. . . . . . . . 9 |- ( ( ph /\ m e. NN0 ) -> ( X + ( ( m + 1 ) x. T ) ) = ( X + ( ( m x. T ) + ( 1 x. T ) ) ) )
43	27	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ m e. NN0 ) -> X e. CC )
44	38, 40	mulcld 10060	. . . . . . . . . 10 |- ( ( ph /\ m e. NN0 ) -> ( m x. T ) e. CC )
45	39, 40	mulcld 10060	. . . . . . . . . 10 |- ( ( ph /\ m e. NN0 ) -> ( 1 x. T ) e. CC )
46	43, 44, 45	addassd 10062	. . . . . . . . 9 |- ( ( ph /\ m e. NN0 ) -> ( ( X + ( m x. T ) ) + ( 1 x. T ) ) = ( X + ( ( m x. T ) + ( 1 x. T ) ) ) )
47	40	mulid2d 10058	. . . . . . . . . 10 |- ( ( ph /\ m e. NN0 ) -> ( 1 x. T ) = T )
48	47	oveq2d 6666	. . . . . . . . 9 |- ( ( ph /\ m e. NN0 ) -> ( ( X + ( m x. T ) ) + ( 1 x. T ) ) = ( ( X + ( m x. T ) ) + T ) )
49	42, 46, 48	3eqtr2d 2662	. . . . . . . 8 |- ( ( ph /\ m e. NN0 ) -> ( X + ( ( m + 1 ) x. T ) ) = ( ( X + ( m x. T ) ) + T ) )
50	49	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ m e. NN0 ) -> ( F ` ( X + ( ( m + 1 ) x. T ) ) ) = ( F ` ( ( X + ( m x. T ) ) + T ) ) )
51	50	3adant3 1081	. . . . . 6 |- ( ( ph /\ m e. NN0 /\ ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) ) -> ( F ` ( X + ( ( m + 1 ) x. T ) ) ) = ( F ` ( ( X + ( m x. T ) ) + T ) ) )
52	26	adantr 481	. . . . . . . . 9 |- ( ( ph /\ m e. NN0 ) -> X e. RR )
53		nn0re 11301	. . . . . . . . . . 11 |- ( m e. NN0 -> m e. RR )
54	53	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ m e. NN0 ) -> m e. RR )
55	22	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ m e. NN0 ) -> T e. RR )
56	54, 55	remulcld 10070	. . . . . . . . 9 |- ( ( ph /\ m e. NN0 ) -> ( m x. T ) e. RR )
57	52, 56	readdcld 10069	. . . . . . . 8 |- ( ( ph /\ m e. NN0 ) -> ( X + ( m x. T ) ) e. RR )
58	57	ex 450	. . . . . . . . 9 |- ( ph -> ( m e. NN0 -> ( X + ( m x. T ) ) e. RR ) )
59	58	imdistani 726	. . . . . . . 8 |- ( ( ph /\ m e. NN0 ) -> ( ph /\ ( X + ( m x. T ) ) e. RR ) )
60		eleq1 2689	. . . . . . . . . . 11 |- ( x = ( X + ( m x. T ) ) -> ( x e. RR <-> ( X + ( m x. T ) ) e. RR ) )
61	60	anbi2d 740	. . . . . . . . . 10 |- ( x = ( X + ( m x. T ) ) -> ( ( ph /\ x e. RR ) <-> ( ph /\ ( X + ( m x. T ) ) e. RR ) ) )
62		oveq1 6657	. . . . . . . . . . . 12 |- ( x = ( X + ( m x. T ) ) -> ( x + T ) = ( ( X + ( m x. T ) ) + T ) )
63	62	fveq2d 6195	. . . . . . . . . . 11 |- ( x = ( X + ( m x. T ) ) -> ( F ` ( x + T ) ) = ( F ` ( ( X + ( m x. T ) ) + T ) ) )
64		fveq2 6191	. . . . . . . . . . 11 |- ( x = ( X + ( m x. T ) ) -> ( F ` x ) = ( F ` ( X + ( m x. T ) ) ) )
65	63, 64	eqeq12d 2637	. . . . . . . . . 10 |- ( x = ( X + ( m x. T ) ) -> ( ( F ` ( x + T ) ) = ( F ` x ) <-> ( F ` ( ( X + ( m x. T ) ) + T ) ) = ( F ` ( X + ( m x. T ) ) ) ) )
66	61, 65	imbi12d 334	. . . . . . . . 9 |- ( x = ( X + ( m x. T ) ) -> ( ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) ) <-> ( ( ph /\ ( X + ( m x. T ) ) e. RR ) -> ( F ` ( ( X + ( m x. T ) ) + T ) ) = ( F ` ( X + ( m x. T ) ) ) ) ) )
67		fperiodmullem.per	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
68	66, 67	vtoclg 3266	. . . . . . . 8 |- ( ( X + ( m x. T ) ) e. RR -> ( ( ph /\ ( X + ( m x. T ) ) e. RR ) -> ( F ` ( ( X + ( m x. T ) ) + T ) ) = ( F ` ( X + ( m x. T ) ) ) ) )
69	57, 59, 68	sylc 65	. . . . . . 7 |- ( ( ph /\ m e. NN0 ) -> ( F ` ( ( X + ( m x. T ) ) + T ) ) = ( F ` ( X + ( m x. T ) ) ) )
70	69	3adant3 1081	. . . . . 6 |- ( ( ph /\ m e. NN0 /\ ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) ) -> ( F ` ( ( X + ( m x. T ) ) + T ) ) = ( F ` ( X + ( m x. T ) ) ) )
71		simp3 1063	. . . . . 6 |- ( ( ph /\ m e. NN0 /\ ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) ) -> ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) )
72	51, 70, 71	3eqtrd 2660	. . . . 5 |- ( ( ph /\ m e. NN0 /\ ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) ) -> ( F ` ( X + ( ( m + 1 ) x. T ) ) ) = ( F ` X ) )
73	31, 32, 36, 72	syl3anc 1326	. . . 4 |- ( ( m e. NN0 /\ ( ph -> ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) ) /\ ph ) -> ( F ` ( X + ( ( m + 1 ) x. T ) ) ) = ( F ` X ) )
74	73	3exp 1264	. . 3 |- ( m e. NN0 -> ( ( ph -> ( F ` ( X + ( m x. T ) ) ) = ( F ` X ) ) -> ( ph -> ( F ` ( X + ( ( m + 1 ) x. T ) ) ) = ( F ` X ) ) ) )
75	6, 11, 16, 21, 30, 74	nn0ind 11472	. 2 |- ( N e. NN0 -> ( ph -> ( F ` ( X + ( N x. T ) ) ) = ( F ` X ) ) )
76	1, 75	mpcom 38	1 |- ( ph -> ( F ` ( X + ( N x. T ) ) ) = ( F ` X ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NN0cn0 11292

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:  fperiodmul  39518