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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fperiodmul | Structured version Visualization version GIF version |
Description: A function with period T is also periodic with period multiple of T. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fperiodmul.f | ⊢ (𝜑 → 𝐹:ℝ⟶ℂ) |
fperiodmul.t | ⊢ (𝜑 → 𝑇 ∈ ℝ) |
fperiodmul.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
fperiodmul.x | ⊢ (𝜑 → 𝑋 ∈ ℝ) |
fperiodmul.per | ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
Ref | Expression |
---|---|
fperiodmul | ⊢ (𝜑 → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fperiodmul.f | . . . 4 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝐹:ℝ⟶ℂ) |
3 | fperiodmul.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ ℝ) | |
4 | 3 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑇 ∈ ℝ) |
5 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
6 | fperiodmul.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
7 | 6 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑋 ∈ ℝ) |
8 | fperiodmul.per | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) | |
9 | 8 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ0) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
10 | 2, 4, 5, 7, 9 | fperiodmullem 43442 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹‘𝑋)) |
11 | 6 | recnd 11141 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
12 | fperiodmul.n | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
13 | 12 | zcnd 12566 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
14 | 3 | recnd 11141 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
15 | 13, 14 | mulcld 11133 | . . . . . . 7 ⊢ (𝜑 → (𝑁 · 𝑇) ∈ ℂ) |
16 | 11, 15 | subnegd 11477 | . . . . . 6 ⊢ (𝜑 → (𝑋 − -(𝑁 · 𝑇)) = (𝑋 + (𝑁 · 𝑇))) |
17 | 13, 14 | mulneg1d 11566 | . . . . . . . 8 ⊢ (𝜑 → (-𝑁 · 𝑇) = -(𝑁 · 𝑇)) |
18 | 17 | eqcomd 2743 | . . . . . . 7 ⊢ (𝜑 → -(𝑁 · 𝑇) = (-𝑁 · 𝑇)) |
19 | 18 | oveq2d 7367 | . . . . . 6 ⊢ (𝜑 → (𝑋 − -(𝑁 · 𝑇)) = (𝑋 − (-𝑁 · 𝑇))) |
20 | 16, 19 | eqtr3d 2779 | . . . . 5 ⊢ (𝜑 → (𝑋 + (𝑁 · 𝑇)) = (𝑋 − (-𝑁 · 𝑇))) |
21 | 20 | fveq2d 6843 | . . . 4 ⊢ (𝜑 → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹‘(𝑋 − (-𝑁 · 𝑇)))) |
22 | 21 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹‘(𝑋 − (-𝑁 · 𝑇)))) |
23 | 1 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝐹:ℝ⟶ℂ) |
24 | 3 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑇 ∈ ℝ) |
25 | znnn0nn 12572 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℕ) | |
26 | 12, 25 | sylan 580 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℕ) |
27 | 26 | nnnn0d 12431 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℕ0) |
28 | 6 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑋 ∈ ℝ) |
29 | 12 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℤ) |
30 | 29 | zred 12565 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℝ) |
31 | 30 | renegcld 11540 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℝ) |
32 | 31, 24 | remulcld 11143 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (-𝑁 · 𝑇) ∈ ℝ) |
33 | 28, 32 | resubcld 11541 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (𝑋 − (-𝑁 · 𝑇)) ∈ ℝ) |
34 | 8 | adantlr 713 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
35 | 23, 24, 27, 33, 34 | fperiodmullem 43442 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (𝐹‘((𝑋 − (-𝑁 · 𝑇)) + (-𝑁 · 𝑇))) = (𝐹‘(𝑋 − (-𝑁 · 𝑇)))) |
36 | 28 | recnd 11141 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑋 ∈ ℂ) |
37 | 30 | recnd 11141 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℂ) |
38 | 37 | negcld 11457 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℂ) |
39 | 24 | recnd 11141 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑇 ∈ ℂ) |
40 | 38, 39 | mulcld 11133 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (-𝑁 · 𝑇) ∈ ℂ) |
41 | 36, 40 | npcand 11474 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → ((𝑋 − (-𝑁 · 𝑇)) + (-𝑁 · 𝑇)) = 𝑋) |
42 | 41 | fveq2d 6843 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (𝐹‘((𝑋 − (-𝑁 · 𝑇)) + (-𝑁 · 𝑇))) = (𝐹‘𝑋)) |
43 | 22, 35, 42 | 3eqtr2d 2783 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹‘𝑋)) |
44 | 10, 43 | pm2.61dan 811 | 1 ⊢ (𝜑 → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 ℂcc 11007 ℝcr 11008 + caddc 11012 · cmul 11014 − cmin 11343 -cneg 11344 ℕcn 12111 ℕ0cn0 12371 ℤcz 12457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-n0 12372 df-z 12458 |
This theorem is referenced by: fourierdlem89 44337 fourierdlem90 44338 fourierdlem91 44339 fourierdlem94 44342 fourierdlem97 44345 fourierdlem113 44361 |
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