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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 10617. (Contributed by Mario Carneiro, 7-Jul-2014.) |
Ref | Expression |
---|---|
iexpcyc | ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zq 9622 | . . . 4 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℚ) | |
2 | 4z 9279 | . . . . . 6 ⊢ 4 ∈ ℤ | |
3 | zq 9622 | . . . . . 6 ⊢ (4 ∈ ℤ → 4 ∈ ℚ) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ 4 ∈ ℚ |
5 | 4pos 9012 | . . . . 5 ⊢ 0 < 4 | |
6 | modqval 10319 | . . . . 5 ⊢ ((𝐾 ∈ ℚ ∧ 4 ∈ ℚ ∧ 0 < 4) → (𝐾 mod 4) = (𝐾 − (4 · (⌊‘(𝐾 / 4))))) | |
7 | 4, 5, 6 | mp3an23 1329 | . . . 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 5888 | . 2 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4)))))) |
10 | 4nn 9078 | . . . . . . 7 ⊢ 4 ∈ ℕ | |
11 | znq 9620 | . . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 4 ∈ ℕ) → (𝐾 / 4) ∈ ℚ) | |
12 | 10, 11 | mpan2 425 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → (𝐾 / 4) ∈ ℚ) |
13 | 12 | flqcld 10272 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (⌊‘(𝐾 / 4)) ∈ ℤ) |
14 | zmulcl 9302 | . . . . 5 ⊢ ((4 ∈ ℤ ∧ (⌊‘(𝐾 / 4)) ∈ ℤ) → (4 · (⌊‘(𝐾 / 4))) ∈ ℤ) | |
15 | 2, 13, 14 | sylancr 414 | . . . 4 ⊢ (𝐾 ∈ ℤ → (4 · (⌊‘(𝐾 / 4))) ∈ ℤ) |
16 | ax-icn 7903 | . . . . 5 ⊢ i ∈ ℂ | |
17 | iap0 9138 | . . . . 5 ⊢ i # 0 | |
18 | expsubap 10563 | . . . . 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 436 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ (4 · (⌊‘(𝐾 / 4))) ∈ ℤ) → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4)))))) |
20 | 15, 19 | mpdan 421 | . . 3 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4)))))) |
21 | expmulzap 10561 | . . . . . . . 8 ⊢ (((i ∈ ℂ ∧ i # 0) ∧ (4 ∈ ℤ ∧ (⌊‘(𝐾 / 4)) ∈ ℤ)) → (i↑(4 · (⌊‘(𝐾 / 4)))) = ((i↑4)↑(⌊‘(𝐾 / 4)))) | |
22 | 16, 17, 21 | mpanl12 436 | . . . . . . 7 ⊢ ((4 ∈ ℤ ∧ (⌊‘(𝐾 / 4)) ∈ ℤ) → (i↑(4 · (⌊‘(𝐾 / 4)))) = ((i↑4)↑(⌊‘(𝐾 / 4)))) |
23 | 2, 13, 22 | sylancr 414 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → (i↑(4 · (⌊‘(𝐾 / 4)))) = ((i↑4)↑(⌊‘(𝐾 / 4)))) |
24 | i4 10617 | . . . . . . . 8 ⊢ (i↑4) = 1 | |
25 | 24 | oveq1i 5882 | . . . . . . 7 ⊢ ((i↑4)↑(⌊‘(𝐾 / 4))) = (1↑(⌊‘(𝐾 / 4))) |
26 | 1exp 10544 | . . . . . . . 8 ⊢ ((⌊‘(𝐾 / 4)) ∈ ℤ → (1↑(⌊‘(𝐾 / 4))) = 1) | |
27 | 13, 26 | syl 14 | . . . . . . 7 ⊢ (𝐾 ∈ ℤ → (1↑(⌊‘(𝐾 / 4))) = 1) |
28 | 25, 27 | eqtrid 2222 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → ((i↑4)↑(⌊‘(𝐾 / 4))) = 1) |
29 | 23, 28 | eqtrd 2210 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (i↑(4 · (⌊‘(𝐾 / 4)))) = 1) |
30 | 29 | oveq2d 5888 | . . . 4 ⊢ (𝐾 ∈ ℤ → ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / 1)) |
31 | expclzap 10540 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ i # 0 ∧ 𝐾 ∈ ℤ) → (i↑𝐾) ∈ ℂ) | |
32 | 16, 17, 31 | mp3an12 1327 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (i↑𝐾) ∈ ℂ) |
33 | 32 | div1d 8733 | . . . 4 ⊢ (𝐾 ∈ ℤ → ((i↑𝐾) / 1) = (i↑𝐾)) |
34 | 30, 33 | eqtrd 2210 | . . 3 ⊢ (𝐾 ∈ ℤ → ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4))))) = (i↑𝐾)) |
35 | 20, 34 | eqtrd 2210 | . 2 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = (i↑𝐾)) |
36 | 9, 35 | eqtrd 2210 | 1 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 class class class wbr 4002 ‘cfv 5215 (class class class)co 5872 ℂcc 7806 0cc0 7808 1c1 7809 ici 7810 · cmul 7813 < clt 7988 − cmin 8124 # cap 8534 / cdiv 8625 ℕcn 8915 4c4 8968 ℤcz 9249 ℚcq 9615 ⌊cfl 10263 mod cmo 10317 ↑cexp 10514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-mulrcl 7907 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-mulass 7911 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-1rid 7915 ax-0id 7916 ax-rnegex 7917 ax-precex 7918 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-apti 7923 ax-pre-ltadd 7924 ax-pre-mulgt0 7925 ax-pre-mulext 7926 ax-arch 7927 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-po 4295 df-iso 4296 df-iord 4365 df-on 4367 df-ilim 4368 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-recs 6303 df-frec 6389 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-reap 8528 df-ap 8535 df-div 8626 df-inn 8916 df-2 8974 df-3 8975 df-4 8976 df-n0 9173 df-z 9250 df-uz 9525 df-q 9616 df-rp 9650 df-fl 10265 df-mod 10318 df-seqfrec 10441 df-exp 10515 |
This theorem is referenced by: (None) |
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