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Mirrors > Home > MPE Home > Th. List > rereccld | Structured version Visualization version GIF version |
Description: Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
redivcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
rereccld.2 | ⊢ (𝜑 → 𝐴 ≠ 0) |
Ref | Expression |
---|---|
rereccld | ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | redivcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | rereccld.2 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) | |
3 | rereccl 11069 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℝ) | |
4 | 1, 2, 3 | syl2anc 579 | 1 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2164 ≠ wne 2999 (class class class)co 6905 ℝcr 10251 0cc0 10252 1c1 10253 / cdiv 11009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 |
This theorem is referenced by: recgt0 11197 prodgt0 11198 ltdiv1 11217 ltrec 11235 lerec 11236 lediv12a 11246 nnrecl 11616 rpnnen1lem5 12103 expnlbnd 13288 cnsubrg 20166 evth 23128 ncvs1 23326 reeff1o 24600 isosctrlem2 24959 chordthmlem2 24973 cxplim 25111 nv1 28074 nmblolbii 28198 norm1 28650 norm1exi 28651 nmbdoplbi 29427 nmcoplbi 29431 nmbdfnlbi 29452 nmcfnlbi 29455 branmfn 29508 strlem1 29653 dya2icoseg 30873 logdivsqrle 31266 irrapxlem2 38224 irrapxlem5 38227 pell1234qrreccl 38255 pell14qrdich 38270 radcnvrat 39346 hashnzfzclim 39354 reclt0 40402 ltdiv23neg 40405 sumnnodd 40650 ioodvbdlimc1lem2 40935 ioodvbdlimc2lem 40937 stoweidlem7 41011 stoweidlem11 41015 stoweidlem14 41018 stoweidlem25 41029 stoweidlem36 41040 stoweidlem42 41046 stirlinglem10 41087 stirlinglem11 41088 stirlinglem12 41089 fourierdlem40 41151 fourierdlem78 41188 pimrecltpos 41706 pimrecltneg 41720 eenglngeehlnmlem1 43281 |
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