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| Mirrors > Home > MPE Home > Th. List > rphalfcl | Structured version Visualization version GIF version | ||
| Description: Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.) |
| Ref | Expression |
|---|---|
| rphalfcl | ⊢ (𝐴 ∈ ℝ+ → (𝐴 / 2) ∈ ℝ+) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 2rp 13039 | . 2 ⊢ 2 ∈ ℝ+ | |
| 2 | rpdivcl 13060 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 2 ∈ ℝ+) → (𝐴 / 2) ∈ ℝ+) | |
| 3 | 1, 2 | mpan2 691 | 1 ⊢ (𝐴 ∈ ℝ+ → (𝐴 / 2) ∈ ℝ+) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 (class class class)co 7431 / 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: rphalfcld 13089 rpltrp 13383 cau3lem 15393 2clim 15608 addcn2 15630 mulcn2 15632 climcau 15707 metcnpi3 24559 ngptgp 24649 iccntr 24843 reconnlem2 24849 opnreen 24853 xmetdcn2 24859 cnllycmp 24988 iscfil3 25307 cfilfcls 25308 iscmet3lem3 25324 iscmet3lem1 25325 iscmet3lem2 25326 iscmet3 25327 lmcau 25347 bcthlem5 25362 ivthlem2 25487 uniioombl 25624 dvcnvre 26058 aaliou 26380 ulmcaulem 26437 ulmcau 26438 ulmcn 26442 ulmdvlem3 26445 tanregt0 26581 argregt0 26652 argrege0 26653 logimul 26656 resqrtcn 26792 asin1 26937 reasinsin 26939 atanbnd 26969 atan1 26971 sqrtlim 27016 basellem4 27127 chpchtlim 27523 mulog2sumlem2 27579 pntlem3 27653 vacn 30713 ubthlem1 30889 nmcexi 32045 poimirlem29 37656 heicant 37662 ftc1anclem6 37705 ftc1anclem7 37706 ftc1anc 37708 heibor1lem 37816 heiborlem8 37825 bfplem2 37830 supxrge 45349 suplesup 45350 infleinflem1 45381 infleinf 45383 addlimc 45663 fourierdlem103 46224 fourierdlem104 46225 sge0xaddlem2 46449 smflimlem4 46789 |
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