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Mirrors > Home > ILE Home > Th. List > iexpcyc | GIF version |
Description: Taking i to the 𝐾-th power is the same as using the 𝐾 mod 4 -th power instead, by i4 10398. (Contributed by Mario Carneiro, 7-Jul-2014.) |
Ref | Expression |
---|---|
iexpcyc | ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zq 9421 | . . . 4 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℚ) | |
2 | 4z 9087 | . . . . . 6 ⊢ 4 ∈ ℤ | |
3 | zq 9421 | . . . . . 6 ⊢ (4 ∈ ℤ → 4 ∈ ℚ) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ 4 ∈ ℚ |
5 | 4pos 8820 | . . . . 5 ⊢ 0 < 4 | |
6 | modqval 10100 | . . . . 5 ⊢ ((𝐾 ∈ ℚ ∧ 4 ∈ ℚ ∧ 0 < 4) → (𝐾 mod 4) = (𝐾 − (4 · (⌊‘(𝐾 / 4))))) | |
7 | 4, 5, 6 | mp3an23 1307 | . . . 4 ⊢ (𝐾 ∈ ℚ → (𝐾 mod 4) = (𝐾 − (4 · (⌊‘(𝐾 / 4))))) |
8 | 1, 7 | syl 14 | . . 3 ⊢ (𝐾 ∈ ℤ → (𝐾 mod 4) = (𝐾 − (4 · (⌊‘(𝐾 / 4))))) |
9 | 8 | oveq2d 5790 | . 2 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4)))))) |
10 | 4nn 8886 | . . . . . . 7 ⊢ 4 ∈ ℕ | |
11 | znq 9419 | . . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 4 ∈ ℕ) → (𝐾 / 4) ∈ ℚ) | |
12 | 10, 11 | mpan2 421 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → (𝐾 / 4) ∈ ℚ) |
13 | 12 | flqcld 10053 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (⌊‘(𝐾 / 4)) ∈ ℤ) |
14 | zmulcl 9110 | . . . . 5 ⊢ ((4 ∈ ℤ ∧ (⌊‘(𝐾 / 4)) ∈ ℤ) → (4 · (⌊‘(𝐾 / 4))) ∈ ℤ) | |
15 | 2, 13, 14 | sylancr 410 | . . . 4 ⊢ (𝐾 ∈ ℤ → (4 · (⌊‘(𝐾 / 4))) ∈ ℤ) |
16 | ax-icn 7718 | . . . . 5 ⊢ i ∈ ℂ | |
17 | iap0 8946 | . . . . 5 ⊢ i # 0 | |
18 | expsubap 10344 | . . . . 5 ⊢ (((i ∈ ℂ ∧ i # 0) ∧ (𝐾 ∈ ℤ ∧ (4 · (⌊‘(𝐾 / 4))) ∈ ℤ)) → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4)))))) | |
19 | 16, 17, 18 | mpanl12 432 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ (4 · (⌊‘(𝐾 / 4))) ∈ ℤ) → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4)))))) |
20 | 15, 19 | mpdan 417 | . . 3 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4)))))) |
21 | expmulzap 10342 | . . . . . . . 8 ⊢ (((i ∈ ℂ ∧ i # 0) ∧ (4 ∈ ℤ ∧ (⌊‘(𝐾 / 4)) ∈ ℤ)) → (i↑(4 · (⌊‘(𝐾 / 4)))) = ((i↑4)↑(⌊‘(𝐾 / 4)))) | |
22 | 16, 17, 21 | mpanl12 432 | . . . . . . 7 ⊢ ((4 ∈ ℤ ∧ (⌊‘(𝐾 / 4)) ∈ ℤ) → (i↑(4 · (⌊‘(𝐾 / 4)))) = ((i↑4)↑(⌊‘(𝐾 / 4)))) |
23 | 2, 13, 22 | sylancr 410 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → (i↑(4 · (⌊‘(𝐾 / 4)))) = ((i↑4)↑(⌊‘(𝐾 / 4)))) |
24 | i4 10398 | . . . . . . . 8 ⊢ (i↑4) = 1 | |
25 | 24 | oveq1i 5784 | . . . . . . 7 ⊢ ((i↑4)↑(⌊‘(𝐾 / 4))) = (1↑(⌊‘(𝐾 / 4))) |
26 | 1exp 10325 | . . . . . . . 8 ⊢ ((⌊‘(𝐾 / 4)) ∈ ℤ → (1↑(⌊‘(𝐾 / 4))) = 1) | |
27 | 13, 26 | syl 14 | . . . . . . 7 ⊢ (𝐾 ∈ ℤ → (1↑(⌊‘(𝐾 / 4))) = 1) |
28 | 25, 27 | syl5eq 2184 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → ((i↑4)↑(⌊‘(𝐾 / 4))) = 1) |
29 | 23, 28 | eqtrd 2172 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (i↑(4 · (⌊‘(𝐾 / 4)))) = 1) |
30 | 29 | oveq2d 5790 | . . . 4 ⊢ (𝐾 ∈ ℤ → ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / 1)) |
31 | expclzap 10321 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ i # 0 ∧ 𝐾 ∈ ℤ) → (i↑𝐾) ∈ ℂ) | |
32 | 16, 17, 31 | mp3an12 1305 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (i↑𝐾) ∈ ℂ) |
33 | 32 | div1d 8543 | . . . 4 ⊢ (𝐾 ∈ ℤ → ((i↑𝐾) / 1) = (i↑𝐾)) |
34 | 30, 33 | eqtrd 2172 | . . 3 ⊢ (𝐾 ∈ ℤ → ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4))))) = (i↑𝐾)) |
35 | 20, 34 | eqtrd 2172 | . 2 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = (i↑𝐾)) |
36 | 9, 35 | eqtrd 2172 | 1 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 ℂcc 7621 0cc0 7623 1c1 7624 ici 7625 · cmul 7628 < clt 7803 − cmin 7936 # cap 8346 / cdiv 8435 ℕcn 8723 4c4 8776 ℤcz 9057 ℚcq 9414 ⌊cfl 10044 mod cmo 10098 ↑cexp 10295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 ax-pre-mulext 7741 ax-arch 7742 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-reap 8340 df-ap 8347 df-div 8436 df-inn 8724 df-2 8782 df-3 8783 df-4 8784 df-n0 8981 df-z 9058 df-uz 9330 df-q 9415 df-rp 9445 df-fl 10046 df-mod 10099 df-seqfrec 10222 df-exp 10296 |
This theorem is referenced by: (None) |
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