![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 14240. (Contributed by Mario Carneiro, 7-Jul-2014.) |
Ref | Expression |
---|---|
iexpcyc | ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 12615 | . . . 4 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
2 | 4re 12348 | . . . . 5 ⊢ 4 ∈ ℝ | |
3 | 4pos 12371 | . . . . 5 ⊢ 0 < 4 | |
4 | 2, 3 | elrpii 13035 | . . . 4 ⊢ 4 ∈ ℝ+ |
5 | modval 13908 | . . . 4 ⊢ ((𝐾 ∈ ℝ ∧ 4 ∈ ℝ+) → (𝐾 mod 4) = (𝐾 − (4 · (⌊‘(𝐾 / 4))))) | |
6 | 1, 4, 5 | sylancl 586 | . . 3 ⊢ (𝐾 ∈ ℤ → (𝐾 mod 4) = (𝐾 − (4 · (⌊‘(𝐾 / 4))))) |
7 | 6 | oveq2d 7447 | . 2 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4)))))) |
8 | 4z 12649 | . . . . 5 ⊢ 4 ∈ ℤ | |
9 | 4nn 12347 | . . . . . . 7 ⊢ 4 ∈ ℕ | |
10 | nndivre 12305 | . . . . . . 7 ⊢ ((𝐾 ∈ ℝ ∧ 4 ∈ ℕ) → (𝐾 / 4) ∈ ℝ) | |
11 | 1, 9, 10 | sylancl 586 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → (𝐾 / 4) ∈ ℝ) |
12 | 11 | flcld 13835 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (⌊‘(𝐾 / 4)) ∈ ℤ) |
13 | zmulcl 12664 | . . . . 5 ⊢ ((4 ∈ ℤ ∧ (⌊‘(𝐾 / 4)) ∈ ℤ) → (4 · (⌊‘(𝐾 / 4))) ∈ ℤ) | |
14 | 8, 12, 13 | sylancr 587 | . . . 4 ⊢ (𝐾 ∈ ℤ → (4 · (⌊‘(𝐾 / 4))) ∈ ℤ) |
15 | ax-icn 11212 | . . . . 5 ⊢ i ∈ ℂ | |
16 | ine0 11696 | . . . . 5 ⊢ i ≠ 0 | |
17 | expsub 14148 | . . . . 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 702 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ (4 · (⌊‘(𝐾 / 4))) ∈ ℤ) → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4)))))) |
19 | 14, 18 | mpdan 687 | . . 3 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4)))))) |
20 | expmulz 14146 | . . . . . . . 8 ⊢ (((i ∈ ℂ ∧ i ≠ 0) ∧ (4 ∈ ℤ ∧ (⌊‘(𝐾 / 4)) ∈ ℤ)) → (i↑(4 · (⌊‘(𝐾 / 4)))) = ((i↑4)↑(⌊‘(𝐾 / 4)))) | |
21 | 15, 16, 20 | mpanl12 702 | . . . . . . 7 ⊢ ((4 ∈ ℤ ∧ (⌊‘(𝐾 / 4)) ∈ ℤ) → (i↑(4 · (⌊‘(𝐾 / 4)))) = ((i↑4)↑(⌊‘(𝐾 / 4)))) |
22 | 8, 12, 21 | sylancr 587 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → (i↑(4 · (⌊‘(𝐾 / 4)))) = ((i↑4)↑(⌊‘(𝐾 / 4)))) |
23 | i4 14240 | . . . . . . . 8 ⊢ (i↑4) = 1 | |
24 | 23 | oveq1i 7441 | . . . . . . 7 ⊢ ((i↑4)↑(⌊‘(𝐾 / 4))) = (1↑(⌊‘(𝐾 / 4))) |
25 | 1exp 14129 | . . . . . . . 8 ⊢ ((⌊‘(𝐾 / 4)) ∈ ℤ → (1↑(⌊‘(𝐾 / 4))) = 1) | |
26 | 12, 25 | syl 17 | . . . . . . 7 ⊢ (𝐾 ∈ ℤ → (1↑(⌊‘(𝐾 / 4))) = 1) |
27 | 24, 26 | eqtrid 2787 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → ((i↑4)↑(⌊‘(𝐾 / 4))) = 1) |
28 | 22, 27 | eqtrd 2775 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (i↑(4 · (⌊‘(𝐾 / 4)))) = 1) |
29 | 28 | oveq2d 7447 | . . . 4 ⊢ (𝐾 ∈ ℤ → ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4))))) = ((i↑𝐾) / 1)) |
30 | expclz 14122 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝐾 ∈ ℤ) → (i↑𝐾) ∈ ℂ) | |
31 | 15, 16, 30 | mp3an12 1450 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (i↑𝐾) ∈ ℂ) |
32 | 31 | div1d 12033 | . . . 4 ⊢ (𝐾 ∈ ℤ → ((i↑𝐾) / 1) = (i↑𝐾)) |
33 | 29, 32 | eqtrd 2775 | . . 3 ⊢ (𝐾 ∈ ℤ → ((i↑𝐾) / (i↑(4 · (⌊‘(𝐾 / 4))))) = (i↑𝐾)) |
34 | 19, 33 | eqtrd 2775 | . 2 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 − (4 · (⌊‘(𝐾 / 4))))) = (i↑𝐾)) |
35 | 7, 34 | eqtrd 2775 | 1 ⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 ici 11155 · cmul 11158 − cmin 11490 / cdiv 11918 ℕcn 12264 4c4 12321 ℤcz 12611 ℝ+crp 13032 ⌊cfl 13827 mod cmo 13906 ↑cexp 14099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 |
This theorem is referenced by: iblitg 25818 |
Copyright terms: Public domain | W3C validator |