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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 10945 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℝ) | |
4 | 1, 2, 3 | syl2anc 573 | 1 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2145 ≠ wne 2943 (class class class)co 6793 ℝcr 10137 0cc0 10138 1c1 10139 / cdiv 10886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 |
This theorem is referenced by: recgt0 11069 prodgt0 11070 ltdiv1 11089 ltrec 11107 lerec 11108 lediv12a 11118 nnrecl 11492 rpnnen1lem5 12021 rpnnen1lem5OLD 12027 expnlbnd 13201 cnsubrg 20021 evth 22978 ncvs1 23176 reeff1o 24421 isosctrlem2 24770 chordthmlem2 24781 cxplim 24919 nv1 27870 nmblolbii 27994 norm1 28446 norm1exi 28447 nmbdoplbi 29223 nmcoplbi 29227 nmbdfnlbi 29248 nmcfnlbi 29251 branmfn 29304 strlem1 29449 dya2icoseg 30679 logdivsqrle 31068 irrapxlem2 37913 irrapxlem5 37916 pell1234qrreccl 37944 pell14qrdich 37959 radcnvrat 39039 hashnzfzclim 39047 reclt0 40130 ltdiv23neg 40133 sumnnodd 40380 ioodvbdlimc1lem2 40665 ioodvbdlimc2lem 40667 stoweidlem7 40741 stoweidlem11 40745 stoweidlem14 40748 stoweidlem25 40759 stoweidlem36 40770 stoweidlem42 40776 stirlinglem10 40817 stirlinglem11 40818 stirlinglem12 40819 fourierdlem40 40881 fourierdlem78 40918 pimrecltpos 41439 pimrecltneg 41453 |
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