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Mirrors > Home > MPE Home > Th. List > cjcld | Structured version Visualization version GIF version |
Description: Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
recld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
cjcld | ⊢ (𝜑 → (∗‘𝐴) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | cjcl 15112 | . 2 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (∗‘𝐴) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ‘cfv 6556 ℂcc 11158 ∗ccj 15103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-br 5156 df-opab 5218 df-mpt 5239 df-id 5582 df-po 5596 df-so 5597 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-div 11924 df-cj 15106 |
This theorem is referenced by: absrpcl 15295 absmul 15301 abstri 15337 abs1m 15342 abslem2 15346 sqreulem 15366 gzcjcl 16940 mul4sqlem 16957 gzrngunit 21432 cphipipcj 25222 cphassr 25234 cph2ass 25235 tcphcphlem2 25258 pjthlem1 25459 itgabs 25858 dvcj 25976 dvmptre 25995 dvmptim 25996 tanregt0 26569 logcj 26636 cosargd 26638 root1cj 26787 lawcoslem1 26846 isosctrlem2 26850 asinlem3 26902 atandmcj 26940 atancj 26941 sum2dchr 27306 rpvmasum2 27544 dchrisum0re 27545 pjhthlem1 31327 riesz3i 31998 constrconj 33605 itgabsnc 37392 ftc1cnnclem 37394 ftc2nc 37405 sigarim 46490 sigarac 46491 sigaraf 46492 sigarmf 46493 sigarls 46496 sigardiv 46500 sharhght 46504 |
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