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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 10547. (Contributed by Mario Carneiro, 7-Jul-2014.) |
Ref | Expression |
---|---|
iexpcyc | ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zq 9555 | . . . 4 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℚ) | |
2 | 4z 9212 | . . . . . 6 ⊢ 4 ∈ ℤ | |
3 | zq 9555 | . . . . . 6 ⊢ (4 ∈ ℤ → 4 ∈ ℚ) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ 4 ∈ ℚ |
5 | 4pos 8945 | . . . . 5 ⊢ 0 < 4 | |
6 | modqval 10249 | . . . . 5 ⊢ ((𝐾 ∈ ℚ ∧ 4 ∈ ℚ ∧ 0 < 4) → (𝐾 mod 4) = (𝐾 − (4 · (⌊‘(𝐾 / 4))))) | |
7 | 4, 5, 6 | mp3an23 1318 | . . . 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 5852 | . 2 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4)))))) |
10 | 4nn 9011 | . . . . . . 7 ⊢ 4 ∈ ℕ | |
11 | znq 9553 | . . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 4 ∈ ℕ) → (𝐾 / 4) ∈ ℚ) | |
12 | 10, 11 | mpan2 422 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → (𝐾 / 4) ∈ ℚ) |
13 | 12 | flqcld 10202 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (⌊‘(𝐾 / 4)) ∈ ℤ) |
14 | zmulcl 9235 | . . . . 5 ⊢ ((4 ∈ ℤ ∧ (⌊‘(𝐾 / 4)) ∈ ℤ) → (4 · (⌊‘(𝐾 / 4))) ∈ ℤ) | |
15 | 2, 13, 14 | sylancr 411 | . . . 4 ⊢ (𝐾 ∈ ℤ → (4 · (⌊‘(𝐾 / 4))) ∈ ℤ) |
16 | ax-icn 7839 | . . . . 5 ⊢ i ∈ ℂ | |
17 | iap0 9071 | . . . . 5 ⊢ i # 0 | |
18 | expsubap 10493 | . . . . 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 433 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ (4 · (⌊‘(𝐾 / 4))) ∈ ℤ) → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4)))))) |
20 | 15, 19 | mpdan 418 | . . 3 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4)))))) |
21 | expmulzap 10491 | . . . . . . . 8 ⊢ (((i ∈ ℂ ∧ i # 0) ∧ (4 ∈ ℤ ∧ (⌊‘(𝐾 / 4)) ∈ ℤ)) → (i↑(4 · (⌊‘(𝐾 / 4)))) = ((i↑4)↑(⌊‘(𝐾 / 4)))) | |
22 | 16, 17, 21 | mpanl12 433 | . . . . . . 7 ⊢ ((4 ∈ ℤ ∧ (⌊‘(𝐾 / 4)) ∈ ℤ) → (i↑(4 · (⌊‘(𝐾 / 4)))) = ((i↑4)↑(⌊‘(𝐾 / 4)))) |
23 | 2, 13, 22 | sylancr 411 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → (i↑(4 · (⌊‘(𝐾 / 4)))) = ((i↑4)↑(⌊‘(𝐾 / 4)))) |
24 | i4 10547 | . . . . . . . 8 ⊢ (i↑4) = 1 | |
25 | 24 | oveq1i 5846 | . . . . . . 7 ⊢ ((i↑4)↑(⌊‘(𝐾 / 4))) = (1↑(⌊‘(𝐾 / 4))) |
26 | 1exp 10474 | . . . . . . . 8 ⊢ ((⌊‘(𝐾 / 4)) ∈ ℤ → (1↑(⌊‘(𝐾 / 4))) = 1) | |
27 | 13, 26 | syl 14 | . . . . . . 7 ⊢ (𝐾 ∈ ℤ → (1↑(⌊‘(𝐾 / 4))) = 1) |
28 | 25, 27 | syl5eq 2209 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → ((i↑4)↑(⌊‘(𝐾 / 4))) = 1) |
29 | 23, 28 | eqtrd 2197 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (i↑(4 · (⌊‘(𝐾 / 4)))) = 1) |
30 | 29 | oveq2d 5852 | . . . 4 ⊢ (𝐾 ∈ ℤ → ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / 1)) |
31 | expclzap 10470 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ i # 0 ∧ 𝐾 ∈ ℤ) → (i↑𝐾) ∈ ℂ) | |
32 | 16, 17, 31 | mp3an12 1316 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (i↑𝐾) ∈ ℂ) |
33 | 32 | div1d 8667 | . . . 4 ⊢ (𝐾 ∈ ℤ → ((i↑𝐾) / 1) = (i↑𝐾)) |
34 | 30, 33 | eqtrd 2197 | . . 3 ⊢ (𝐾 ∈ ℤ → ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4))))) = (i↑𝐾)) |
35 | 20, 34 | eqtrd 2197 | . 2 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = (i↑𝐾)) |
36 | 9, 35 | eqtrd 2197 | 1 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ℂcc 7742 0cc0 7744 1c1 7745 ici 7746 · cmul 7749 < clt 7924 − cmin 8060 # cap 8470 / cdiv 8559 ℕcn 8848 4c4 8901 ℤcz 9182 ℚcq 9548 ⌊cfl 10193 mod cmo 10247 ↑cexp 10444 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fl 10195 df-mod 10248 df-seqfrec 10371 df-exp 10445 |
This theorem is referenced by: (None) |
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