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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 11927 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℝ) | |
4 | 1, 2, 3 | syl2anc 585 | 1 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ≠ wne 2941 (class class class)co 7403 ℝcr 11104 0cc0 11105 1c1 11106 / cdiv 11866 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-mpt 5230 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 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-div 11867 |
This theorem is referenced by: recgt0 12055 prodgt0 12056 ltdiv1 12073 ltrec 12091 lerec 12092 lediv12a 12102 nnrecl 12465 rpnnen1lem5 12960 expnlbnd 14191 cnsubrg 20989 evth 24456 ncvs1 24655 reeff1o 25940 isosctrlem2 26303 chordthmlem2 26317 cxplim 26455 nv1 29905 nmblolbii 30029 norm1 30479 norm1exi 30480 nmbdoplbi 31254 nmcoplbi 31258 nmbdfnlbi 31279 nmcfnlbi 31282 branmfn 31335 strlem1 31480 dya2icoseg 33213 logdivsqrle 33599 rtprmirr 41180 irrapxlem2 41493 irrapxlem5 41496 pell1234qrreccl 41524 pell14qrdich 41539 radcnvrat 43005 hashnzfzclim 43013 reclt0 44035 ltdiv23neg 44038 sumnnodd 44280 ioodvbdlimc1lem2 44582 ioodvbdlimc2lem 44584 stoweidlem7 44657 stoweidlem11 44661 stoweidlem14 44664 stoweidlem25 44675 stoweidlem36 44686 stoweidlem42 44692 stirlinglem10 44733 stirlinglem11 44734 stirlinglem12 44735 fourierdlem40 44797 fourierdlem78 44834 pimrecltpos 45358 pimrecltneg 45374 eenglngeehlnmlem1 47324 |
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