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Mirrors > Home > MPE Home > Th. List > rprecred | Structured version Visualization version GIF version |
Description: Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Ref | Expression |
---|---|
rprecred | ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpred.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
2 | 1 | rpreccld 12127 | . 2 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ+) |
3 | 2 | rpred 12117 | 1 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2157 (class class class)co 6878 ℝcr 10223 1c1 10225 / cdiv 10976 ℝ+crp 12074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-po 5233 df-so 5234 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-rp 12075 |
This theorem is referenced by: xov1plusxeqvd 12572 ltexp2r 13171 expnlbnd2 13249 rlimno1 14725 lebnumii 23093 sca2rab 23620 aalioulem4 24431 aalioulem5 24432 dvradcnv 24516 tanregt0 24627 divlogrlim 24722 logccv 24750 cxplt3 24787 asinlem3 24950 rlimcxp 25052 cxp2lim 25055 divsqrtsumlem 25058 logdiflbnd 25073 lgamgulmlem2 25108 lgamgulmlem3 25109 basellem3 25161 dchrisum0lema 25555 dchrisum0lem1 25557 dchrisum0lem2a 25558 mulog2sumlem1 25575 vmalogdivsum2 25579 pntrlog2bndlem2 25619 pntlemd 25635 pntlemr 25643 ostth3 25679 nmcexi 29410 knoppndvlem18 33028 knoppndvlem20 33030 irrapxlem4 38175 irrapxlem5 38176 ioodvbdlimc1lem2 40891 ioodvbdlimc2lem 40893 stoweidlem14 40974 fourierdlem39 41106 pimrecltpos 41665 smfrec 41742 |
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