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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 12325 | . 2 ⊢ 2 ∈ ℂ | |
2 | 2ne0 12354 | . 2 ⊢ 2 ≠ 0 | |
3 | divcl 11916 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (𝐴 / 2) ∈ ℂ) | |
4 | 1, 2, 3 | mp3an23 1449 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 / 2) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ≠ wne 2929 (class class class)co 7419 ℂcc 11143 0cc0 11145 / cdiv 11908 2c2 12305 |
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 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
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-rmo 3363 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 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-2 12313 |
This theorem is referenced by: halfaddsubcl 12482 subhalfhalf 12484 halfcld 12495 geo2sum 15863 efhalfpi 26468 cosq14gt0 26507 cosq14ge0 26508 abssinper 26517 coseq1 26521 efeq1 26524 sqrtcn 26747 1cubr 26839 dquartlem1 26848 acosf 26871 atanf 26877 acosneg 26884 acoscos 26890 acos1 26892 sinacos 26902 atanneg 26904 atancj 26907 efiatan 26909 efiatan2 26914 2efiatan 26915 atantan 26920 atanbndlem 26922 dvatan 26932 atantayl 26934 gausslemma2dlem1a 27363 minvecolem2 30777 sin2h 37234 cos2h 37235 dirkercncflem2 45635 fourierdlem58 45695 |
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