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Mirrors > Home > MPE Home > Th. List > rpreccl | Structured version Visualization version GIF version |
Description: Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.) |
Ref | Expression |
---|---|
rpreccl | ⊢ (𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1rp 12734 | . 2 ⊢ 1 ∈ ℝ+ | |
2 | rpdivcl 12755 | . 2 ⊢ ((1 ∈ ℝ+ ∧ 𝐴 ∈ ℝ+) → (1 / 𝐴) ∈ ℝ+) | |
3 | 1, 2 | mpan 687 | 1 ⊢ (𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 (class class class)co 7275 1c1 10872 / cdiv 11632 ℝ+crp 12730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-rp 12731 |
This theorem is referenced by: rpreccld 12782 xlemul1 13024 rpexpcl 13801 rpnnen2lem11 15933 prmreclem6 16622 rpmsubg 20662 lebnumii 24129 nmhmcn 24283 lmnn 24427 advlog 25809 cxprec 25841 dvcxp1 25893 loglesqrt 25911 logrec 25913 rlimcnp 26115 rlimcnp2 26116 rlimcnp3 26117 cxplim 26121 logdifbnd 26143 harmonicbnd4 26160 logfacrlim 26372 dchrmusumlema 26641 mulogsumlem 26679 selberg2lem 26698 pntrsumo1 26713 pntibndlem1 26737 blocnilem 29166 subfacval3 33151 recnnltrp 42916 rpgtrecnn 42919 xrralrecnnle 42922 nnrecrp 42925 sumnnodd 43171 dirkertrigeq 43642 preimageiingt 44257 preimaleiinlt 44258 |
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