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Mirrors > Home > MPE Home > Th. List > root1id | Structured version Visualization version GIF version |
Description: Property of an 𝑁-th root of unity. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Ref | Expression |
---|---|
root1id | ⊢ (𝑁 ∈ ℕ → ((-1↑𝑐(2 / 𝑁))↑𝑁) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1cn 12097 | . . . 4 ⊢ -1 ∈ ℂ | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ → -1 ∈ ℂ) |
3 | 2re 12057 | . . . . 5 ⊢ 2 ∈ ℝ | |
4 | nndivre 12024 | . . . . 5 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (2 / 𝑁) ∈ ℝ) | |
5 | 3, 4 | mpan 687 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2 / 𝑁) ∈ ℝ) |
6 | 5 | recnd 11013 | . . 3 ⊢ (𝑁 ∈ ℕ → (2 / 𝑁) ∈ ℂ) |
7 | nnnn0 12250 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
8 | 2, 6, 7 | cxpmul2d 25874 | . 2 ⊢ (𝑁 ∈ ℕ → (-1↑𝑐((2 / 𝑁) · 𝑁)) = ((-1↑𝑐(2 / 𝑁))↑𝑁)) |
9 | 2cnd 12061 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ) | |
10 | nncn 11991 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
11 | nnne0 12017 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
12 | 9, 10, 11 | divcan1d 11762 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((2 / 𝑁) · 𝑁) = 2) |
13 | 12 | oveq2d 7283 | . . 3 ⊢ (𝑁 ∈ ℕ → (-1↑𝑐((2 / 𝑁) · 𝑁)) = (-1↑𝑐2)) |
14 | 2nn0 12260 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
15 | cxpexp 25833 | . . . . 5 ⊢ ((-1 ∈ ℂ ∧ 2 ∈ ℕ0) → (-1↑𝑐2) = (-1↑2)) | |
16 | 1, 14, 15 | mp2an 689 | . . . 4 ⊢ (-1↑𝑐2) = (-1↑2) |
17 | ax-1cn 10939 | . . . . 5 ⊢ 1 ∈ ℂ | |
18 | sqneg 13846 | . . . . 5 ⊢ (1 ∈ ℂ → (-1↑2) = (1↑2)) | |
19 | 17, 18 | ax-mp 5 | . . . 4 ⊢ (-1↑2) = (1↑2) |
20 | sq1 13922 | . . . 4 ⊢ (1↑2) = 1 | |
21 | 16, 19, 20 | 3eqtri 2770 | . . 3 ⊢ (-1↑𝑐2) = 1 |
22 | 13, 21 | eqtrdi 2794 | . 2 ⊢ (𝑁 ∈ ℕ → (-1↑𝑐((2 / 𝑁) · 𝑁)) = 1) |
23 | 8, 22 | eqtr3d 2780 | 1 ⊢ (𝑁 ∈ ℕ → ((-1↑𝑐(2 / 𝑁))↑𝑁) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 (class class class)co 7267 ℂcc 10879 ℝcr 10880 1c1 10882 · cmul 10886 -cneg 11216 / cdiv 11642 ℕcn 11983 2c2 12038 ℕ0cn0 12243 ↑cexp 13792 ↑𝑐ccxp 25721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-inf2 9386 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-pre-sup 10959 ax-addf 10960 ax-mulf 10961 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-se 5540 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-isom 6435 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-of 7523 df-om 7703 df-1st 7820 df-2nd 7821 df-supp 7965 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-2o 8285 df-er 8485 df-map 8604 df-pm 8605 df-ixp 8673 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-fsupp 9116 df-fi 9157 df-sup 9188 df-inf 9189 df-oi 9256 df-card 9707 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-q 12699 df-rp 12741 df-xneg 12858 df-xadd 12859 df-xmul 12860 df-ioo 13093 df-ioc 13094 df-ico 13095 df-icc 13096 df-fz 13250 df-fzo 13393 df-fl 13522 df-mod 13600 df-seq 13732 df-exp 13793 df-fac 13998 df-bc 14027 df-hash 14055 df-shft 14788 df-cj 14820 df-re 14821 df-im 14822 df-sqrt 14956 df-abs 14957 df-limsup 15190 df-clim 15207 df-rlim 15208 df-sum 15408 df-ef 15787 df-sin 15789 df-cos 15790 df-pi 15792 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-starv 16987 df-sca 16988 df-vsca 16989 df-ip 16990 df-tset 16991 df-ple 16992 df-ds 16994 df-unif 16995 df-hom 16996 df-cco 16997 df-rest 17143 df-topn 17144 df-0g 17162 df-gsum 17163 df-topgen 17164 df-pt 17165 df-prds 17168 df-xrs 17223 df-qtop 17228 df-imas 17229 df-xps 17231 df-mre 17305 df-mrc 17306 df-acs 17308 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-submnd 18441 df-mulg 18711 df-cntz 18933 df-cmn 19398 df-psmet 20599 df-xmet 20600 df-met 20601 df-bl 20602 df-mopn 20603 df-fbas 20604 df-fg 20605 df-cnfld 20608 df-top 22053 df-topon 22070 df-topsp 22092 df-bases 22106 df-cld 22180 df-ntr 22181 df-cls 22182 df-nei 22259 df-lp 22297 df-perf 22298 df-cn 22388 df-cnp 22389 df-haus 22476 df-tx 22723 df-hmeo 22916 df-fil 23007 df-fm 23099 df-flim 23100 df-flf 23101 df-xms 23483 df-ms 23484 df-tms 23485 df-cncf 24051 df-limc 25040 df-dv 25041 df-log 25722 df-cxp 25723 |
This theorem is referenced by: root1cj 25919 cxpeq 25920 |
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