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Mirrors > Home > MPE Home > Th. List > cji | Structured version Visualization version GIF version |
Description: The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.) |
Ref | Expression |
---|---|
cji | ⊢ (∗‘i) = -i |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rei 14091 | . . 3 ⊢ (ℜ‘i) = 0 | |
2 | imi 14092 | . . . . 5 ⊢ (ℑ‘i) = 1 | |
3 | 2 | oveq2i 6820 | . . . 4 ⊢ (i · (ℑ‘i)) = (i · 1) |
4 | ax-icn 10183 | . . . . 5 ⊢ i ∈ ℂ | |
5 | 4 | mulid1i 10230 | . . . 4 ⊢ (i · 1) = i |
6 | 3, 5 | eqtri 2778 | . . 3 ⊢ (i · (ℑ‘i)) = i |
7 | 1, 6 | oveq12i 6821 | . 2 ⊢ ((ℜ‘i) − (i · (ℑ‘i))) = (0 − i) |
8 | remim 14052 | . . 3 ⊢ (i ∈ ℂ → (∗‘i) = ((ℜ‘i) − (i · (ℑ‘i)))) | |
9 | 4, 8 | ax-mp 5 | . 2 ⊢ (∗‘i) = ((ℜ‘i) − (i · (ℑ‘i))) |
10 | df-neg 10457 | . 2 ⊢ -i = (0 − i) | |
11 | 7, 9, 10 | 3eqtr4i 2788 | 1 ⊢ (∗‘i) = -i |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1628 ∈ wcel 2135 ‘cfv 6045 (class class class)co 6809 ℂcc 10122 0cc0 10124 1c1 10125 ici 10126 · cmul 10129 − cmin 10454 -cneg 10455 ∗ccj 14031 ℜcre 14032 ℑcim 14033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-8 2137 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 ax-sep 4929 ax-nul 4937 ax-pow 4988 ax-pr 5051 ax-un 7110 ax-resscn 10181 ax-1cn 10182 ax-icn 10183 ax-addcl 10184 ax-addrcl 10185 ax-mulcl 10186 ax-mulrcl 10187 ax-mulcom 10188 ax-addass 10189 ax-mulass 10190 ax-distr 10191 ax-i2m1 10192 ax-1ne0 10193 ax-1rid 10194 ax-rnegex 10195 ax-rrecex 10196 ax-cnre 10197 ax-pre-lttri 10198 ax-pre-lttrn 10199 ax-pre-ltadd 10200 ax-pre-mulgt0 10201 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1631 df-ex 1850 df-nf 1855 df-sb 2043 df-eu 2607 df-mo 2608 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-ne 2929 df-nel 3032 df-ral 3051 df-rex 3052 df-reu 3053 df-rmo 3054 df-rab 3055 df-v 3338 df-sbc 3573 df-csb 3671 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-nul 4055 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4585 df-br 4801 df-opab 4861 df-mpt 4878 df-id 5170 df-po 5183 df-so 5184 df-xp 5268 df-rel 5269 df-cnv 5270 df-co 5271 df-dm 5272 df-rn 5273 df-res 5274 df-ima 5275 df-iota 6008 df-fun 6047 df-fn 6048 df-f 6049 df-f1 6050 df-fo 6051 df-f1o 6052 df-fv 6053 df-riota 6770 df-ov 6812 df-oprab 6813 df-mpt2 6814 df-er 7907 df-en 8118 df-dom 8119 df-sdom 8120 df-pnf 10264 df-mnf 10265 df-xr 10266 df-ltxr 10267 df-le 10268 df-sub 10456 df-neg 10457 df-div 10873 df-2 11267 df-cj 14034 df-re 14035 df-im 14036 |
This theorem is referenced by: cjreim 14095 absi 14221 resinval 15060 recosval 15061 cphassir 23211 cosargd 24549 1cubrlem 24763 atancj 24832 ipasslem10 27999 polid2i 28319 lnophmlem2 29181 |
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