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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 11199 | . 2 ⊢ i ∈ ℂ | |
2 | negcl 11492 | . 2 ⊢ (i ∈ ℂ → -i ∈ ℂ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ -i ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ℂcc 11138 ici 11142 -cneg 11477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-ltxr 11285 df-sub 11478 df-neg 11479 |
This theorem is referenced by: irec 14200 imcl 15094 absimle 15292 recan 15319 sinf 16104 cosf 16105 tanval2 16113 tanval3 16114 efi4p 16117 sinneg 16126 cosneg 16127 efival 16132 sinhval 16134 coshval 16135 sinadd 16144 cosadd 16145 cphipval2 25213 dvsincos 25957 sincn 26426 coscn 26427 sinperlem 26460 pige3ALT 26499 sineq0 26503 tanregt0 26518 asinlem3a 26847 asinf 26849 asinneg 26863 efiasin 26865 sinasin 26866 asinsinlem 26868 asinsin 26869 asin1 26871 2efiatan 26895 dvatan 26912 atantayl 26914 nvpi 30549 ipval2 30589 4ipval2 30590 ipidsq 30592 dipcj 30596 dip0r 30599 ipasslem10 30721 polid2i 31039 dvasin 37308 areacirclem4 37315 sineq0ALT 44518 |
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