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| Mirrors > Home > MPE Home > Th. List > root1cj | Structured version Visualization version GIF version | ||
| Description: Within the 𝑁-th roots of unity, the conjugate of the 𝐾-th root is the 𝑁 − 𝐾-th root. (Contributed by Mario Carneiro, 23-Apr-2015.) |
| Ref | Expression |
|---|---|
| root1cj | ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (∗‘((-1↑𝑐(2 / 𝑁))↑𝐾)) = ((-1↑𝑐(2 / 𝑁))↑(𝑁 − 𝐾))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | neg1cn 12380 | . . . 4 ⊢ -1 ∈ ℂ | |
| 2 | 2re 12340 | . . . . . 6 ⊢ 2 ∈ ℝ | |
| 3 | simpl 482 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → 𝑁 ∈ ℕ) | |
| 4 | nndivre 12307 | . . . . . 6 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (2 / 𝑁) ∈ ℝ) | |
| 5 | 2, 3, 4 | sylancr 587 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (2 / 𝑁) ∈ ℝ) |
| 6 | 5 | recnd 11289 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (2 / 𝑁) ∈ ℂ) |
| 7 | cxpcl 26716 | . . . 4 ⊢ ((-1 ∈ ℂ ∧ (2 / 𝑁) ∈ ℂ) → (-1↑𝑐(2 / 𝑁)) ∈ ℂ) | |
| 8 | 1, 6, 7 | sylancr 587 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (-1↑𝑐(2 / 𝑁)) ∈ ℂ) |
| 9 | 1 | a1i 11 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → -1 ∈ ℂ) |
| 10 | neg1ne0 12382 | . . . . 5 ⊢ -1 ≠ 0 | |
| 11 | 10 | a1i 11 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → -1 ≠ 0) |
| 12 | 9, 11, 6 | cxpne0d 26755 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (-1↑𝑐(2 / 𝑁)) ≠ 0) |
| 13 | simpr 484 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → 𝐾 ∈ ℤ) | |
| 14 | nnz 12634 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
| 15 | 14 | adantr 480 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → 𝑁 ∈ ℤ) |
| 16 | 8, 12, 13, 15 | expsubd 14197 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → ((-1↑𝑐(2 / 𝑁))↑(𝑁 − 𝐾)) = (((-1↑𝑐(2 / 𝑁))↑𝑁) / ((-1↑𝑐(2 / 𝑁))↑𝐾))) |
| 17 | root1id 26797 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((-1↑𝑐(2 / 𝑁))↑𝑁) = 1) | |
| 18 | 17 | adantr 480 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → ((-1↑𝑐(2 / 𝑁))↑𝑁) = 1) |
| 19 | 18 | oveq1d 7446 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (((-1↑𝑐(2 / 𝑁))↑𝑁) / ((-1↑𝑐(2 / 𝑁))↑𝐾)) = (1 / ((-1↑𝑐(2 / 𝑁))↑𝐾))) |
| 20 | 8, 12, 13 | expclzd 14191 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → ((-1↑𝑐(2 / 𝑁))↑𝐾) ∈ ℂ) |
| 21 | 8, 12, 13 | expne0d 14192 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → ((-1↑𝑐(2 / 𝑁))↑𝐾) ≠ 0) |
| 22 | recval 15361 | . . . 4 ⊢ ((((-1↑𝑐(2 / 𝑁))↑𝐾) ∈ ℂ ∧ ((-1↑𝑐(2 / 𝑁))↑𝐾) ≠ 0) → (1 / ((-1↑𝑐(2 / 𝑁))↑𝐾)) = ((∗‘((-1↑𝑐(2 / 𝑁))↑𝐾)) / ((abs‘((-1↑𝑐(2 / 𝑁))↑𝐾))↑2))) | |
| 23 | 20, 21, 22 | syl2anc 584 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (1 / ((-1↑𝑐(2 / 𝑁))↑𝐾)) = ((∗‘((-1↑𝑐(2 / 𝑁))↑𝐾)) / ((abs‘((-1↑𝑐(2 / 𝑁))↑𝐾))↑2))) |
| 24 | absexpz 15344 | . . . . . . . 8 ⊢ (((-1↑𝑐(2 / 𝑁)) ∈ ℂ ∧ (-1↑𝑐(2 / 𝑁)) ≠ 0 ∧ 𝐾 ∈ ℤ) → (abs‘((-1↑𝑐(2 / 𝑁))↑𝐾)) = ((abs‘(-1↑𝑐(2 / 𝑁)))↑𝐾)) | |
| 25 | 8, 12, 13, 24 | syl3anc 1373 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (abs‘((-1↑𝑐(2 / 𝑁))↑𝐾)) = ((abs‘(-1↑𝑐(2 / 𝑁)))↑𝐾)) |
| 26 | abscxp2 26735 | . . . . . . . . . . 11 ⊢ ((-1 ∈ ℂ ∧ (2 / 𝑁) ∈ ℝ) → (abs‘(-1↑𝑐(2 / 𝑁))) = ((abs‘-1)↑𝑐(2 / 𝑁))) | |
| 27 | 1, 5, 26 | sylancr 587 | . . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (abs‘(-1↑𝑐(2 / 𝑁))) = ((abs‘-1)↑𝑐(2 / 𝑁))) |
| 28 | ax-1cn 11213 | . . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ | |
| 29 | 28 | absnegi 15439 | . . . . . . . . . . . 12 ⊢ (abs‘-1) = (abs‘1) |
| 30 | abs1 15336 | . . . . . . . . . . . 12 ⊢ (abs‘1) = 1 | |
| 31 | 29, 30 | eqtri 2765 | . . . . . . . . . . 11 ⊢ (abs‘-1) = 1 |
| 32 | 31 | oveq1i 7441 | . . . . . . . . . 10 ⊢ ((abs‘-1)↑𝑐(2 / 𝑁)) = (1↑𝑐(2 / 𝑁)) |
| 33 | 27, 32 | eqtrdi 2793 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (abs‘(-1↑𝑐(2 / 𝑁))) = (1↑𝑐(2 / 𝑁))) |
| 34 | 6 | 1cxpd 26749 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (1↑𝑐(2 / 𝑁)) = 1) |
| 35 | 33, 34 | eqtrd 2777 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (abs‘(-1↑𝑐(2 / 𝑁))) = 1) |
| 36 | 35 | oveq1d 7446 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → ((abs‘(-1↑𝑐(2 / 𝑁)))↑𝐾) = (1↑𝐾)) |
| 37 | 1exp 14132 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → (1↑𝐾) = 1) | |
| 38 | 37 | adantl 481 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (1↑𝐾) = 1) |
| 39 | 25, 36, 38 | 3eqtrd 2781 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (abs‘((-1↑𝑐(2 / 𝑁))↑𝐾)) = 1) |
| 40 | 39 | oveq1d 7446 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → ((abs‘((-1↑𝑐(2 / 𝑁))↑𝐾))↑2) = (1↑2)) |
| 41 | sq1 14234 | . . . . 5 ⊢ (1↑2) = 1 | |
| 42 | 40, 41 | eqtrdi 2793 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → ((abs‘((-1↑𝑐(2 / 𝑁))↑𝐾))↑2) = 1) |
| 43 | 42 | oveq2d 7447 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → ((∗‘((-1↑𝑐(2 / 𝑁))↑𝐾)) / ((abs‘((-1↑𝑐(2 / 𝑁))↑𝐾))↑2)) = ((∗‘((-1↑𝑐(2 / 𝑁))↑𝐾)) / 1)) |
| 44 | 20 | cjcld 15235 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (∗‘((-1↑𝑐(2 / 𝑁))↑𝐾)) ∈ ℂ) |
| 45 | 44 | div1d 12035 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → ((∗‘((-1↑𝑐(2 / 𝑁))↑𝐾)) / 1) = (∗‘((-1↑𝑐(2 / 𝑁))↑𝐾))) |
| 46 | 23, 43, 45 | 3eqtrd 2781 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (1 / ((-1↑𝑐(2 / 𝑁))↑𝐾)) = (∗‘((-1↑𝑐(2 / 𝑁))↑𝐾))) |
| 47 | 16, 19, 46 | 3eqtrrd 2782 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (∗‘((-1↑𝑐(2 / 𝑁))↑𝐾)) = ((-1↑𝑐(2 / 𝑁))↑(𝑁 − 𝐾))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 − cmin 11492 -cneg 11493 / cdiv 11920 ℕcn 12266 2c2 12321 ℤcz 12613 ↑cexp 14102 ∗ccj 15135 abscabs 15273 ↑𝑐ccxp 26597 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 df-cxp 26599 |
| This theorem is referenced by: 1cubrlem 26884 |
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