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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 11958 | . . 3 ⊢ (1 / 1) = 1 | |
| 2 | 1lt2 12437 | . . 3 ⊢ 1 < 2 | |
| 3 | 1, 2 | eqbrtri 5164 | . 2 ⊢ (1 / 1) < 2 |
| 4 | 1re 11261 | . . 3 ⊢ 1 ∈ ℝ | |
| 5 | 2re 12340 | . . 3 ⊢ 2 ∈ ℝ | |
| 6 | 0lt1 11785 | . . 3 ⊢ 0 < 1 | |
| 7 | 2pos 12369 | . . 3 ⊢ 0 < 2 | |
| 8 | 4, 4, 5, 6, 7 | ltdiv23ii 12195 | . 2 ⊢ ((1 / 1) < 2 ↔ (1 / 2) < 1) |
| 9 | 3, 8 | mpbi 230 | 1 ⊢ (1 / 2) < 1 |
| Colors of variables: wff setvar class |
| Syntax hints: class class class wbr 5143 (class class class)co 7431 1c1 11156 < clt 11295 / cdiv 11920 2c2 12321 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-2 12329 |
| This theorem is referenced by: 2tnp1ge0ge0 13869 absrdbnd 15380 geo2sum 15909 geo2lim 15911 geoihalfsum 15918 efcllem 16113 rpnnen2lem12 16261 ltoddhalfle 16398 halfleoddlt 16399 bitsp1o 16470 elii1 24964 htpycc 25012 pcoval1 25046 pco1 25048 pcocn 25050 pcohtpylem 25052 pcopt 25055 pcopt2 25056 pcoass 25057 pcorevlem 25059 iscmet3lem3 25324 mbfi1fseqlem6 25755 itg2monolem3 25787 aaliou3lem3 26386 cxpcn3lem 26790 lgamgulmlem2 27073 lgsquadlem2 27425 chtppilim 27519 dnizeq0 36476 dnibndlem12 36490 knoppcnlem4 36497 cnndvlem1 36538 iccioo01 37328 cntotbnd 37803 halffl 45308 sumnnodd 45645 stoweidlem5 46020 stoweidlem14 46029 stoweidlem28 46043 dirkertrigeqlem3 46115 dirkercncflem1 46118 dirkercncflem2 46119 zofldiv2ALTV 47649 zofldiv2 48452 sepfsepc 48825 |
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