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Mirrors > Home > MPE Home > Th. List > halflt1 | Structured version Visualization version GIF version |
Description: One-half is less than one. (Contributed by NM, 24-Feb-2005.) |
Ref | Expression |
---|---|
halflt1 | ⊢ (1 / 2) < 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1div1e1 11065 | . . 3 ⊢ (1 / 1) = 1 | |
2 | 1lt2 11553 | . . 3 ⊢ 1 < 2 | |
3 | 1, 2 | eqbrtri 4907 | . 2 ⊢ (1 / 1) < 2 |
4 | 1re 10376 | . . 3 ⊢ 1 ∈ ℝ | |
5 | 2re 11449 | . . 3 ⊢ 2 ∈ ℝ | |
6 | 0lt1 10897 | . . 3 ⊢ 0 < 1 | |
7 | 2pos 11485 | . . 3 ⊢ 0 < 2 | |
8 | 4, 4, 5, 6, 7 | ltdiv23ii 11305 | . 2 ⊢ ((1 / 1) < 2 ↔ (1 / 2) < 1) |
9 | 3, 8 | mpbi 222 | 1 ⊢ (1 / 2) < 1 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 4886 (class class class)co 6922 1c1 10273 < clt 10411 / cdiv 11032 2c2 11430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-2 11438 |
This theorem is referenced by: 2tnp1ge0ge0 12949 absrdbnd 14488 geo2sum 15008 geo2lim 15010 geoihalfsum 15017 efcllem 15210 rpnnen2lem12 15358 ltoddhalfle 15489 halfleoddlt 15490 bitsp1o 15561 elii1 23142 htpycc 23187 pcoval1 23220 pco1 23222 pcocn 23224 pcohtpylem 23226 pcopt 23229 pcopt2 23230 pcoass 23231 pcorevlem 23233 iscmet3lem3 23496 mbfi1fseqlem6 23924 itg2monolem3 23956 aaliou3lem3 24536 cxpcn3lem 24928 lgamgulmlem2 25208 lgsquadlem2 25558 chtppilim 25616 dnizeq0 33048 dnibndlem12 33062 knoppcnlem4 33069 cnndvlem1 33110 cntotbnd 34219 halffl 40419 sumnnodd 40770 stoweidlem5 41149 stoweidlem14 41158 stoweidlem28 41172 dirkertrigeqlem3 41244 dirkercncflem1 41247 dirkercncflem2 41248 zofldiv2ALTV 42599 zofldiv2 43340 |
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