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Mirrors > Home > MPE Home > Th. List > rehalfcl | Structured version Visualization version GIF version |
Description: Real closure of half. (Contributed by NM, 1-Jan-2006.) |
Ref | Expression |
---|---|
rehalfcl | ⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 12319 | . 2 ⊢ 2 ∈ ℝ | |
2 | 2ne0 12349 | . 2 ⊢ 2 ≠ 0 | |
3 | redivcl 11966 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 2 ≠ 0) → (𝐴 / 2) ∈ ℝ) | |
4 | 1, 2, 3 | mp3an23 1449 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ≠ wne 2929 (class class class)co 7419 ℝcr 11139 0cc0 11140 / cdiv 11903 2c2 12300 |
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 ax-pre-mulgt0 11217 |
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 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-2 12308 |
This theorem is referenced by: halfpos 12475 lt2halves 12480 nominpos 12482 avgle1 12485 avgle2 12486 rehalfcld 12492 rehalfcli 12494 fldiv4lem1div2uz2 13837 efgt0 16083 sin02gt0 16172 tangtx 26485 sinq12gt0 26487 cosordlem 26509 cxpcn3lem 26727 basellem1 27058 gausslemma2dlem1a 27343 chebbnd1lem2 27448 chebbnd1lem3 27449 sin2h 37214 cos2h 37215 tan2h 37216 itg2addnclem 37275 infleinflem1 44890 fourierdlem57 45689 fourierdlem66 45698 smflimlem4 46300 |
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