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 483 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝐹:ℝ⟶ℂ) |
3 | fperiodmul.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ ℝ) | |
4 | 3 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑇 ∈ ℝ) |
5 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
6 | fperiodmul.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
7 | 6 | adantr 483 | . . 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 41576 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹‘𝑋)) |
11 | 6 | recnd 10672 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
12 | fperiodmul.n | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
13 | 12 | zcnd 12091 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
14 | 3 | recnd 10672 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
15 | 13, 14 | mulcld 10664 | . . . . . . 7 ⊢ (𝜑 → (𝑁 · 𝑇) ∈ ℂ) |
16 | 11, 15 | subnegd 11007 | . . . . . 6 ⊢ (𝜑 → (𝑋 − -(𝑁 · 𝑇)) = (𝑋 + (𝑁 · 𝑇))) |
17 | 13, 14 | mulneg1d 11096 | . . . . . . . 8 ⊢ (𝜑 → (-𝑁 · 𝑇) = -(𝑁 · 𝑇)) |
18 | 17 | eqcomd 2830 | . . . . . . 7 ⊢ (𝜑 → -(𝑁 · 𝑇) = (-𝑁 · 𝑇)) |
19 | 18 | oveq2d 7175 | . . . . . 6 ⊢ (𝜑 → (𝑋 − -(𝑁 · 𝑇)) = (𝑋 − (-𝑁 · 𝑇))) |
20 | 16, 19 | eqtr3d 2861 | . . . . 5 ⊢ (𝜑 → (𝑋 + (𝑁 · 𝑇)) = (𝑋 − (-𝑁 · 𝑇))) |
21 | 20 | fveq2d 6677 | . . . 4 ⊢ (𝜑 → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹‘(𝑋 − (-𝑁 · 𝑇)))) |
22 | 21 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹‘(𝑋 − (-𝑁 · 𝑇)))) |
23 | 1 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝐹:ℝ⟶ℂ) |
24 | 3 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑇 ∈ ℝ) |
25 | znnn0nn 12097 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℕ) | |
26 | 12, 25 | sylan 582 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℕ) |
27 | 26 | nnnn0d 11958 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℕ0) |
28 | 6 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑋 ∈ ℝ) |
29 | 12 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℤ) |
30 | 29 | zred 12090 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℝ) |
31 | 30 | renegcld 11070 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℝ) |
32 | 31, 24 | remulcld 10674 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (-𝑁 · 𝑇) ∈ ℝ) |
33 | 28, 32 | resubcld 11071 | . . . 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 41576 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (𝐹‘((𝑋 − (-𝑁 · 𝑇)) + (-𝑁 · 𝑇))) = (𝐹‘(𝑋 − (-𝑁 · 𝑇)))) |
36 | 28 | recnd 10672 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑋 ∈ ℂ) |
37 | 30 | recnd 10672 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℂ) |
38 | 37 | negcld 10987 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℂ) |
39 | 24 | recnd 10672 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑇 ∈ ℂ) |
40 | 38, 39 | mulcld 10664 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (-𝑁 · 𝑇) ∈ ℂ) |
41 | 36, 40 | npcand 11004 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → ((𝑋 − (-𝑁 · 𝑇)) + (-𝑁 · 𝑇)) = 𝑋) |
42 | 41 | fveq2d 6677 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (𝐹‘((𝑋 − (-𝑁 · 𝑇)) + (-𝑁 · 𝑇))) = (𝐹‘𝑋)) |
43 | 22, 35, 42 | 3eqtr2d 2865 | . 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 398 = wceq 1536 ∈ wcel 2113 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ℝcr 10539 + caddc 10543 · cmul 10545 − cmin 10873 -cneg 10874 ℕcn 11641 ℕ0cn0 11900 ℤcz 11984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-z 11985 |
This theorem is referenced by: fourierdlem89 42487 fourierdlem90 42488 fourierdlem91 42489 fourierdlem94 42492 fourierdlem97 42495 fourierdlem113 42511 |
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