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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 13074 | . 2 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ+) |
3 | 2 | rpred 13064 | 1 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 (class class class)co 7416 ℝcr 11148 1c1 11150 / cdiv 11912 ℝ+crp 13022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-rp 13023 |
This theorem is referenced by: xov1plusxeqvd 13523 ltexp2r 14186 expnlbnd2 14246 rlimno1 15653 lebnumii 24980 sca2rab 25529 aalioulem4 26360 aalioulem5 26361 dvradcnv 26447 tanregt0 26563 divlogrlim 26659 logccv 26687 cxplt3 26724 asinlem3 26896 rlimcxp 26999 cxp2lim 27002 divsqrtsumlem 27005 logdiflbnd 27020 lgamgulmlem2 27055 lgamgulmlem3 27056 basellem3 27108 dchrisum0lema 27540 dchrisum0lem1 27542 dchrisum0lem2a 27543 mulog2sumlem1 27560 vmalogdivsum2 27564 pntrlog2bndlem2 27604 pntlemd 27620 pntlemr 27628 ostth3 27664 nmcexi 31956 knoppndvlem18 36245 knoppndvlem20 36247 irrapxlem4 42519 irrapxlem5 42520 ioodvbdlimc1lem2 45589 ioodvbdlimc2lem 45591 stoweidlem14 45671 fourierdlem39 45803 pimrecltpos 46365 smfrec 46446 |
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