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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 15192 | . . 3 ⊢ (ℜ‘i) = 0 | |
2 | imi 15193 | . . . . 5 ⊢ (ℑ‘i) = 1 | |
3 | 2 | oveq2i 7442 | . . . 4 ⊢ (i · (ℑ‘i)) = (i · 1) |
4 | ax-icn 11212 | . . . . 5 ⊢ i ∈ ℂ | |
5 | 4 | mulridi 11263 | . . . 4 ⊢ (i · 1) = i |
6 | 3, 5 | eqtri 2763 | . . 3 ⊢ (i · (ℑ‘i)) = i |
7 | 1, 6 | oveq12i 7443 | . 2 ⊢ ((ℜ‘i) − (i · (ℑ‘i))) = (0 − i) |
8 | remim 15153 | . . 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 11493 | . 2 ⊢ -i = (0 − i) | |
11 | 7, 9, 10 | 3eqtr4i 2773 | 1 ⊢ (∗‘i) = -i |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 ici 11155 · cmul 11158 − cmin 11490 -cneg 11491 ∗ccj 15132 ℜcre 15133 ℑcim 15134 |
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-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 |
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-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 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-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-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 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-2 12327 df-cj 15135 df-re 15136 df-im 15137 |
This theorem is referenced by: cjreim 15196 absi 15322 resinval 16168 recosval 16169 cphassir 25263 cosargd 26665 1cubrlem 26899 atancj 26968 ipasslem10 30868 polid2i 31186 lnophmlem2 32046 |
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