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Mirrors > Home > MPE Home > Th. List > halfre | Structured version Visualization version GIF version |
Description: One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
halfre | ⊢ (1 / 2) ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 11425 | . 2 ⊢ 2 ∈ ℝ | |
2 | 2ne0 11462 | . 2 ⊢ 2 ≠ 0 | |
3 | 1, 2 | rereccli 11116 | 1 ⊢ (1 / 2) ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2166 (class class class)co 6905 ℝcr 10251 1c1 10253 / cdiv 11009 2c2 11406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-2 11414 |
This theorem is referenced by: halfge0 11575 2tnp1ge0ge0 12925 rddif 14457 absrdbnd 14458 geo2sum 14978 geo2lim 14980 geoihalfsum 14987 efcllem 15180 ege2le3 15192 rpnnen2lem12 15328 oddge22np1 15447 ltoddhalfle 15459 halfleoddlt 15460 bitsp1o 15528 prmreclem5 15995 prmreclem6 15996 iihalf1 23100 iihalf1cn 23101 iihalf2 23102 iihalf2cn 23103 elii1 23104 elii2 23105 htpycc 23149 pcoval1 23182 pco0 23183 pco1 23184 pcoval2 23185 pcocn 23186 pcohtpylem 23188 pcopt 23191 pcopt2 23192 pcoass 23193 pcorevlem 23195 iscmet3lem3 23458 mbfi1fseqlem6 23886 itg2monolem3 23918 aaliou3lem1 24496 aaliou3lem2 24497 aaliou3lem3 24498 cxpsqrtlem 24847 cxpsqrt 24848 logsqrt 24849 sqrt2cxp2logb9e3 24939 ang180lem1 24949 heron 24978 asinsin 25032 birthday 25094 gausslemma2dlem1a 25503 chebbnd1 25574 chtppilim 25577 mulog2sumlem2 25637 opsqrlem4 29557 logdivsqrle 31277 subfacval3 31717 dnicld1 32995 dnizeq0 32998 dnizphlfeqhlf 32999 rddif2 33000 dnibndlem2 33002 dnibndlem3 33003 dnibndlem4 33004 dnibndlem5 33005 dnibndlem6 33006 dnibndlem7 33007 dnibndlem8 33008 dnibndlem9 33009 dnibndlem10 33010 dnibndlem11 33011 dnibndlem12 33012 dnibndlem13 33013 dnibnd 33014 knoppcnlem4 33019 cnndvlem1 33060 cntotbnd 34137 halffl 40308 stoweidlem5 41016 stoweidlem14 41025 stoweidlem28 41039 dirkertrigeqlem2 41110 dirkeritg 41113 dirkercncflem2 41115 fourierdlem18 41136 fourierdlem66 41183 fourierdlem78 41195 fourierdlem83 41200 fourierdlem87 41204 fourierdlem104 41221 zofldiv2ALTV 42404 zofldiv2 43172 |
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