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 41577 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹‘𝑋)) |
11 | 6 | recnd 10671 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
12 | fperiodmul.n | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
13 | 12 | zcnd 12091 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
14 | 3 | recnd 10671 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
15 | 13, 14 | mulcld 10663 | . . . . . . 7 ⊢ (𝜑 → (𝑁 · 𝑇) ∈ ℂ) |
16 | 11, 15 | subnegd 11006 | . . . . . 6 ⊢ (𝜑 → (𝑋 − -(𝑁 · 𝑇)) = (𝑋 + (𝑁 · 𝑇))) |
17 | 13, 14 | mulneg1d 11095 | . . . . . . . 8 ⊢ (𝜑 → (-𝑁 · 𝑇) = -(𝑁 · 𝑇)) |
18 | 17 | eqcomd 2829 | . . . . . . 7 ⊢ (𝜑 → -(𝑁 · 𝑇) = (-𝑁 · 𝑇)) |
19 | 18 | oveq2d 7174 | . . . . . 6 ⊢ (𝜑 → (𝑋 − -(𝑁 · 𝑇)) = (𝑋 − (-𝑁 · 𝑇))) |
20 | 16, 19 | eqtr3d 2860 | . . . . 5 ⊢ (𝜑 → (𝑋 + (𝑁 · 𝑇)) = (𝑋 − (-𝑁 · 𝑇))) |
21 | 20 | fveq2d 6676 | . . . 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 11069 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℝ) |
32 | 31, 24 | remulcld 10673 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (-𝑁 · 𝑇) ∈ ℝ) |
33 | 28, 32 | resubcld 11070 | . . . 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 41577 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (𝐹‘((𝑋 − (-𝑁 · 𝑇)) + (-𝑁 · 𝑇))) = (𝐹‘(𝑋 − (-𝑁 · 𝑇)))) |
36 | 28 | recnd 10671 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑋 ∈ ℂ) |
37 | 30 | recnd 10671 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℂ) |
38 | 37 | negcld 10986 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → -𝑁 ∈ ℂ) |
39 | 24 | recnd 10671 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → 𝑇 ∈ ℂ) |
40 | 38, 39 | mulcld 10663 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (-𝑁 · 𝑇) ∈ ℂ) |
41 | 36, 40 | npcand 11003 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → ((𝑋 − (-𝑁 · 𝑇)) + (-𝑁 · 𝑇)) = 𝑋) |
42 | 41 | fveq2d 6676 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 ∈ ℕ0) → (𝐹‘((𝑋 − (-𝑁 · 𝑇)) + (-𝑁 · 𝑇))) = (𝐹‘𝑋)) |
43 | 22, 35, 42 | 3eqtr2d 2864 | . 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 1537 ∈ wcel 2114 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 + caddc 10542 · cmul 10544 − cmin 10872 -cneg 10873 ℕcn 11640 ℕ0cn0 11900 ℤcz 11984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 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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