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Mirrors > Home > MPE Home > Th. List > negicn | Structured version Visualization version GIF version |
Description: -i is a complex number. (Contributed by David A. Wheeler, 7-Dec-2018.) |
Ref | Expression |
---|---|
negicn | ⊢ -i ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 10311 | . 2 ⊢ i ∈ ℂ | |
2 | negcl 10601 | . 2 ⊢ (i ∈ ℂ → -i ∈ ℂ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ -i ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2166 ℂcc 10250 ici 10254 -cneg 10586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-ltxr 10396 df-sub 10587 df-neg 10588 |
This theorem is referenced by: irec 13258 imcl 14228 absimle 14426 recan 14453 sinf 15226 cosf 15227 tanval2 15235 tanval3 15236 efi4p 15239 sinneg 15248 cosneg 15249 efival 15254 sinhval 15256 coshval 15257 sinadd 15266 cosadd 15267 cphipval2 23409 dvsincos 24143 sincn 24597 coscn 24598 sinperlem 24632 pige3 24669 sineq0 24673 tanregt0 24685 asinlem3a 25010 asinf 25012 asinneg 25026 efiasin 25028 sinasin 25029 asinsinlem 25031 asinsin 25032 asin1 25034 2efiatan 25058 dvatan 25075 atantayl 25077 nvpi 28077 ipval2 28117 4ipval2 28118 ipidsq 28120 dipcj 28124 dip0r 28127 ipasslem10 28249 polid2i 28569 dvasin 34039 areacirclem4 34046 sineq0ALT 39991 |
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