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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 11869 | . 2 ⊢ 2 ∈ ℝ | |
2 | 2ne0 11899 | . 2 ⊢ 2 ≠ 0 | |
3 | redivcl 11516 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 2 ≠ 0) → (𝐴 / 2) ∈ ℝ) | |
4 | 1, 2, 3 | mp3an23 1455 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 ≠ wne 2932 (class class class)co 7191 ℝcr 10693 0cc0 10694 / cdiv 11454 2c2 11850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-2 11858 |
This theorem is referenced by: halfpos 12025 lt2halves 12030 nominpos 12032 avgle1 12035 avgle2 12036 rehalfcld 12042 rehalfcli 12044 fldiv4lem1div2uz2 13376 efgt0 15627 sin02gt0 15716 tangtx 25349 sinq12gt0 25351 cosordlem 25373 cxpcn3lem 25587 basellem1 25917 gausslemma2dlem1a 26200 chebbnd1lem2 26305 chebbnd1lem3 26306 sin2h 35453 cos2h 35454 tan2h 35455 itg2addnclem 35514 infleinflem1 42523 fourierdlem57 43322 fourierdlem66 43331 smflimlem4 43924 |
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