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Mirrors > Home > MPE Home > Th. List > iexpcyc | Structured version Visualization version GIF version |
Description: Taking i to the 𝐾-th power is the same as using the 𝐾 mod 4 -th power instead, by i4 12829. (Contributed by Mario Carneiro, 7-Jul-2014.) |
Ref | Expression |
---|---|
iexpcyc | ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 11258 | . . . 4 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
2 | 4re 10974 | . . . . 5 ⊢ 4 ∈ ℝ | |
3 | 4pos 10993 | . . . . 5 ⊢ 0 < 4 | |
4 | 2, 3 | elrpii 11711 | . . . 4 ⊢ 4 ∈ ℝ+ |
5 | modval 12532 | . . . 4 ⊢ ((𝐾 ∈ ℝ ∧ 4 ∈ ℝ+) → (𝐾 mod 4) = (𝐾 − (4 · (⌊‘(𝐾 / 4))))) | |
6 | 1, 4, 5 | sylancl 693 | . . 3 ⊢ (𝐾 ∈ ℤ → (𝐾 mod 4) = (𝐾 − (4 · (⌊‘(𝐾 / 4))))) |
7 | 6 | oveq2d 6565 | . 2 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4)))))) |
8 | 4z 11288 | . . . . 5 ⊢ 4 ∈ ℤ | |
9 | 4nn 11064 | . . . . . . 7 ⊢ 4 ∈ ℕ | |
10 | nndivre 10933 | . . . . . . 7 ⊢ ((𝐾 ∈ ℝ ∧ 4 ∈ ℕ) → (𝐾 / 4) ∈ ℝ) | |
11 | 1, 9, 10 | sylancl 693 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → (𝐾 / 4) ∈ ℝ) |
12 | 11 | flcld 12461 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (⌊‘(𝐾 / 4)) ∈ ℤ) |
13 | zmulcl 11303 | . . . . 5 ⊢ ((4 ∈ ℤ ∧ (⌊‘(𝐾 / 4)) ∈ ℤ) → (4 · (⌊‘(𝐾 / 4))) ∈ ℤ) | |
14 | 8, 12, 13 | sylancr 694 | . . . 4 ⊢ (𝐾 ∈ ℤ → (4 · (⌊‘(𝐾 / 4))) ∈ ℤ) |
15 | ax-icn 9874 | . . . . 5 ⊢ i ∈ ℂ | |
16 | ine0 10344 | . . . . 5 ⊢ i ≠ 0 | |
17 | expsub 12770 | . . . . 5 ⊢ (((i ∈ ℂ ∧ i ≠ 0) ∧ (𝐾 ∈ ℤ ∧ (4 · (⌊‘(𝐾 / 4))) ∈ ℤ)) → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4)))))) | |
18 | 15, 16, 17 | mpanl12 714 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ (4 · (⌊‘(𝐾 / 4))) ∈ ℤ) → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4)))))) |
19 | 14, 18 | mpdan 699 | . . 3 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4)))))) |
20 | expmulz 12768 | . . . . . . . 8 ⊢ (((i ∈ ℂ ∧ i ≠ 0) ∧ (4 ∈ ℤ ∧ (⌊‘(𝐾 / 4)) ∈ ℤ)) → (i↑(4 · (⌊‘(𝐾 / 4)))) = ((i↑4)↑(⌊‘(𝐾 / 4)))) | |
21 | 15, 16, 20 | mpanl12 714 | . . . . . . 7 ⊢ ((4 ∈ ℤ ∧ (⌊‘(𝐾 / 4)) ∈ ℤ) → (i↑(4 · (⌊‘(𝐾 / 4)))) = ((i↑4)↑(⌊‘(𝐾 / 4)))) |
22 | 8, 12, 21 | sylancr 694 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → (i↑(4 · (⌊‘(𝐾 / 4)))) = ((i↑4)↑(⌊‘(𝐾 / 4)))) |
23 | i4 12829 | . . . . . . . 8 ⊢ (i↑4) = 1 | |
24 | 23 | oveq1i 6559 | . . . . . . 7 ⊢ ((i↑4)↑(⌊‘(𝐾 / 4))) = (1↑(⌊‘(𝐾 / 4))) |
25 | 1exp 12751 | . . . . . . . 8 ⊢ ((⌊‘(𝐾 / 4)) ∈ ℤ → (1↑(⌊‘(𝐾 / 4))) = 1) | |
26 | 12, 25 | syl 17 | . . . . . . 7 ⊢ (𝐾 ∈ ℤ → (1↑(⌊‘(𝐾 / 4))) = 1) |
27 | 24, 26 | syl5eq 2656 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → ((i↑4)↑(⌊‘(𝐾 / 4))) = 1) |
28 | 22, 27 | eqtrd 2644 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (i↑(4 · (⌊‘(𝐾 / 4)))) = 1) |
29 | 28 | oveq2d 6565 | . . . 4 ⊢ (𝐾 ∈ ℤ → ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / 1)) |
30 | expclz 12747 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝐾 ∈ ℤ) → (i↑𝐾) ∈ ℂ) | |
31 | 15, 16, 30 | mp3an12 1406 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (i↑𝐾) ∈ ℂ) |
32 | 31 | div1d 10672 | . . . 4 ⊢ (𝐾 ∈ ℤ → ((i↑𝐾) / 1) = (i↑𝐾)) |
33 | 29, 32 | eqtrd 2644 | . . 3 ⊢ (𝐾 ∈ ℤ → ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4))))) = (i↑𝐾)) |
34 | 19, 33 | eqtrd 2644 | . 2 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = (i↑𝐾)) |
35 | 7, 34 | eqtrd 2644 | 1 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 ici 9817 · cmul 9820 − cmin 10145 / cdiv 10563 ℕcn 10897 4c4 10949 ℤcz 11254 ℝ+crp 11708 ⌊cfl 12453 mod cmo 12530 ↑cexp 12722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 |
This theorem is referenced by: iblitg 23341 |
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