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Mirrors > Home > MPE Home > Th. List > halfcl | Structured version Visualization version GIF version |
Description: Closure of half of a number. (Contributed by NM, 1-Jan-2006.) |
Ref | Expression |
---|---|
halfcl | ⊢ (𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cn 11515 | . 2 ⊢ 2 ∈ ℂ | |
2 | 2ne0 11551 | . 2 ⊢ 2 ≠ 0 | |
3 | divcl 11105 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (𝐴 / 2) ∈ ℂ) | |
4 | 1, 2, 3 | mp3an23 1432 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2050 ≠ wne 2967 (class class class)co 6976 ℂcc 10333 0cc0 10335 / cdiv 11098 2c2 11495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-po 5326 df-so 5327 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-2 11503 |
This theorem is referenced by: halfaddsubcl 11679 subhalfhalf 11681 halfcld 11692 geo2sum 15089 efhalfpi 24760 cosq14gt0 24799 cosq14ge0 24800 abssinper 24809 coseq1 24813 efeq1 24814 sqrtcn 25032 1cubr 25121 dquartlem1 25130 acosf 25153 atanf 25159 acosneg 25166 acoscos 25172 acos1 25174 sinacos 25184 atanneg 25186 atancj 25189 efiatan 25191 efiatan2 25196 2efiatan 25197 atantan 25202 atanbndlem 25204 dvatan 25214 atantayl 25216 gausslemma2dlem1a 25643 minvecolem2 28430 sin2h 34329 cos2h 34330 dirkercncflem2 41826 fourierdlem58 41886 |
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