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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 14865 | . . 3 ⊢ (ℜ‘i) = 0 | |
2 | imi 14866 | . . . . 5 ⊢ (ℑ‘i) = 1 | |
3 | 2 | oveq2i 7282 | . . . 4 ⊢ (i · (ℑ‘i)) = (i · 1) |
4 | ax-icn 10931 | . . . . 5 ⊢ i ∈ ℂ | |
5 | 4 | mulid1i 10980 | . . . 4 ⊢ (i · 1) = i |
6 | 3, 5 | eqtri 2768 | . . 3 ⊢ (i · (ℑ‘i)) = i |
7 | 1, 6 | oveq12i 7283 | . 2 ⊢ ((ℜ‘i) − (i · (ℑ‘i))) = (0 − i) |
8 | remim 14826 | . . 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 11208 | . 2 ⊢ -i = (0 − i) | |
11 | 7, 9, 10 | 3eqtr4i 2778 | 1 ⊢ (∗‘i) = -i |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2110 ‘cfv 6432 (class class class)co 7271 ℂcc 10870 0cc0 10872 1c1 10873 ici 10874 · cmul 10877 − cmin 11205 -cneg 11206 ∗ccj 14805 ℜcre 14806 ℑcim 14807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-po 5504 df-so 5505 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-2 12036 df-cj 14808 df-re 14809 df-im 14810 |
This theorem is referenced by: cjreim 14869 absi 14996 resinval 15842 recosval 15843 cphassir 24377 cosargd 25761 1cubrlem 25989 atancj 26058 ipasslem10 29197 polid2i 29515 lnophmlem2 30375 |
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