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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 11346 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℝ) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ≠ wne 3013 (class class class)co 7145 ℝcr 10524 0cc0 10525 1c1 10526 / cdiv 11285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 |
This theorem is referenced by: recgt0 11474 prodgt0 11475 ltdiv1 11492 ltrec 11510 lerec 11511 lediv12a 11521 nnrecl 11883 rpnnen1lem5 12368 expnlbnd 13582 cnsubrg 20533 evth 23490 ncvs1 23688 reeff1o 24962 isosctrlem2 25324 chordthmlem2 25338 cxplim 25476 nv1 28379 nmblolbii 28503 norm1 28953 norm1exi 28954 nmbdoplbi 29728 nmcoplbi 29732 nmbdfnlbi 29753 nmcfnlbi 29756 branmfn 29809 strlem1 29954 dya2icoseg 31434 logdivsqrle 31820 rtprmirr 39072 irrapxlem2 39298 irrapxlem5 39301 pell1234qrreccl 39329 pell14qrdich 39344 radcnvrat 40523 hashnzfzclim 40531 reclt0 41539 ltdiv23neg 41542 sumnnodd 41787 ioodvbdlimc1lem2 42093 ioodvbdlimc2lem 42095 stoweidlem7 42169 stoweidlem11 42173 stoweidlem14 42176 stoweidlem25 42187 stoweidlem36 42198 stoweidlem42 42204 stirlinglem10 42245 stirlinglem11 42246 stirlinglem12 42247 fourierdlem40 42309 fourierdlem78 42346 pimrecltpos 42864 pimrecltneg 42878 eenglngeehlnmlem1 44652 |
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