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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 12476 | . 2 ⊢ 1 ∈ ℝ+ | |
2 | rpdivcl 12497 | . 2 ⊢ ((1 ∈ ℝ+ ∧ 𝐴 ∈ ℝ+) → (1 / 𝐴) ∈ ℝ+) | |
3 | 1, 2 | mpan 690 | 1 ⊢ (𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 (class class class)co 7170 1c1 10616 / cdiv 11375 ℝ+crp 12472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-po 5442 df-so 5443 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-rp 12473 |
This theorem is referenced by: rpreccld 12524 xlemul1 12766 rpexpcl 13540 rpnnen2lem11 15669 prmreclem6 16357 rpmsubg 20281 lebnumii 23718 nmhmcn 23872 lmnn 24015 advlog 25397 cxprec 25429 dvcxp1 25481 loglesqrt 25499 logrec 25501 rlimcnp 25703 rlimcnp2 25704 rlimcnp3 25705 cxplim 25709 logdifbnd 25731 harmonicbnd4 25748 logfacrlim 25960 dchrmusumlema 26229 mulogsumlem 26267 selberg2lem 26286 pntrsumo1 26301 pntibndlem1 26325 blocnilem 28739 subfacval3 32722 recnnltrp 42454 rpgtrecnn 42457 xrralrecnnle 42460 nnrecrp 42463 sumnnodd 42713 dirkertrigeq 43184 preimageiingt 43796 preimaleiinlt 43797 |
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