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Mirrors > Home > MPE Home > Th. List > cjcl | Structured version Visualization version GIF version |
Description: The conjugate of a complex number is a complex number (closure law). (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
Ref | Expression |
---|---|
cjcl | ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cjf 14222 | . 2 ⊢ ∗:ℂ⟶ℂ | |
2 | 1 | ffvelrni 6608 | 1 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 ‘cfv 6124 ℂcc 10251 ∗ccj 14214 |
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 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
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 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-po 5264 df-so 5265 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-cj 14217 |
This theorem is referenced by: crre 14232 cjcj 14258 ipcnval 14261 cjmulrcl 14262 addcj 14266 cjsub 14267 cjexp 14268 cjdiv 14282 cjcli 14287 cjcld 14314 absneg 14395 abscj 14397 sqabsadd 14400 sqabssub 14401 recval 14440 sqreulem 14477 cjcn2 14708 efcj 15195 cnsrng 20141 plycjlem 24432 coecj 24434 plyrecj 24435 aacjcl 24482 logcj 24752 argimlt0 24759 atancj 25051 cncph 28230 dipassr2 28258 his52 28500 his35 28501 brafnmul 29366 kbmul 29370 adjmul 29507 cnvbramul 29530 sigarac 41836 sigarid 41842 |
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