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 484 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝐹:ℝ⟶ℂ) |
3 | fperiodmul.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ ℝ) | |
4 | 3 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑇 ∈ ℝ) |
5 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
6 | fperiodmul.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
7 | 6 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑋 ∈ ℝ) |
8 | fperiodmul.per | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) | |
9 | 8 | adantlr 715 | . . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ0) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
10 | 2, 4, 5, 7, 9 | fperiodmullem 42515 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹‘𝑋)) |
11 | 6 | recnd 10861 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
12 | fperiodmul.n | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
13 | 12 | zcnd 12283 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
14 | 3 | recnd 10861 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
15 | 13, 14 | mulcld 10853 | . . . . . . 7 ⊢ (𝜑 → (𝑁 · 𝑇) ∈ ℂ) |
16 | 11, 15 | subnegd 11196 | . . . . . 6 ⊢ (𝜑 → (𝑋 − -(𝑁 · 𝑇)) = (𝑋 + (𝑁 · 𝑇))) |
17 | 13, 14 | mulneg1d 11285 | . . . . . . . 8 ⊢ (𝜑 → (-𝑁 · 𝑇) = -(𝑁 · 𝑇)) |
18 | 17 | eqcomd 2743 | . . . . . . 7 ⊢ (𝜑 → -(𝑁 · 𝑇) = (-𝑁 · 𝑇)) |
19 | 18 | oveq2d 7229 | . . . . . 6 ⊢ (𝜑 → (𝑋 − -(𝑁 · 𝑇)) = (𝑋 − (-𝑁 · 𝑇))) |
20 | 16, 19 | eqtr3d 2779 | . . . . 5 ⊢ (𝜑 → (𝑋 + (𝑁 · 𝑇)) = (𝑋 − (-𝑁 · 𝑇))) |
21 | 20 | fveq2d 6721 | . . . 4 ⊢ (𝜑 → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹‘(𝑋 − (-𝑁 · 𝑇)))) |
22 | 21 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹‘(𝑋 − (-𝑁 · 𝑇)))) |
23 | 1 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝐹:ℝ⟶ℂ) |
24 | 3 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑇 ∈ ℝ) |
25 | znnn0nn 12289 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℕ) | |
26 | 12, 25 | sylan 583 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℕ) |
27 | 26 | nnnn0d 12150 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℕ0) |
28 | 6 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑋 ∈ ℝ) |
29 | 12 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℤ) |
30 | 29 | zred 12282 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℝ) |
31 | 30 | renegcld 11259 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℝ) |
32 | 31, 24 | remulcld 10863 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (-𝑁 · 𝑇) ∈ ℝ) |
33 | 28, 32 | resubcld 11260 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (𝑋 − (-𝑁 · 𝑇)) ∈ ℝ) |
34 | 8 | adantlr 715 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
35 | 23, 24, 27, 33, 34 | fperiodmullem 42515 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (𝐹‘((𝑋 − (-𝑁 · 𝑇)) + (-𝑁 · 𝑇))) = (𝐹‘(𝑋 − (-𝑁 · 𝑇)))) |
36 | 28 | recnd 10861 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑋 ∈ ℂ) |
37 | 30 | recnd 10861 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℂ) |
38 | 37 | negcld 11176 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℂ) |
39 | 24 | recnd 10861 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑇 ∈ ℂ) |
40 | 38, 39 | mulcld 10853 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (-𝑁 · 𝑇) ∈ ℂ) |
41 | 36, 40 | npcand 11193 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → ((𝑋 − (-𝑁 · 𝑇)) + (-𝑁 · 𝑇)) = 𝑋) |
42 | 41 | fveq2d 6721 | . . 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 813 | 1 ⊢ (𝜑 → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℝcr 10728 + caddc 10732 · cmul 10734 − cmin 11062 -cneg 11063 ℕcn 11830 ℕ0cn0 12090 ℤcz 12176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 |
This theorem is referenced by: fourierdlem89 43411 fourierdlem90 43412 fourierdlem91 43413 fourierdlem94 43416 fourierdlem97 43419 fourierdlem113 43435 |
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