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| Mirrors > Home > MPE Home > Th. List > rphalflt | Structured version Visualization version GIF version | ||
| Description: Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.) |
| Ref | Expression |
|---|---|
| rphalflt | ⊢ (𝐴 ∈ ℝ+ → (𝐴 / 2) < 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elrp 13036 | . 2 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
| 2 | halfpos 12496 | . . 3 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ (𝐴 / 2) < 𝐴)) | |
| 3 | 2 | biimpa 476 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 / 2) < 𝐴) |
| 4 | 1, 3 | sylbi 217 | 1 ⊢ (𝐴 ∈ ℝ+ → (𝐴 / 2) < 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 < clt 11295 / cdiv 11920 2c2 12321 ℝ+crp 13034 |
| 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 df-rp 13035 |
| This theorem is referenced by: rpltrp 13383 rpnnen2lem11 16260 sqrt2irr 16285 metcnpi3 24559 cfilucfil 24572 reperflem 24840 iccntr 24843 icccmplem2 24845 reconnlem2 24849 cnllycmp 24988 bcthlem5 25362 minveclem3 25463 ivthlem2 25487 lhop1lem 26052 dvcnvre 26058 aaliou 26380 aaliou2b 26383 cosordlem 26572 tanord1 26579 argregt0 26652 argrege0 26653 isosctrlem1 26861 asinsin 26935 asin1 26937 atan1 26971 lgamucov 27081 lgsqrlem2 27391 lgsquadlem2 27425 lgsquadlem3 27426 2sqlem8 27470 chebbnd1lem2 27514 pntibnd 27637 pntlem3 27653 ubthlem1 30889 nmcexi 32045 ftc1anc 37708 flt4lem7 42669 isosctrlem1ALT 44954 dstregt0 45293 supxrge 45349 rphalfltd 45466 stoweidlem62 46077 fourierdlem79 46200 |
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