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Mirrors > Home > MPE Home > Th. List > resqrtcld | Structured version Visualization version GIF version |
Description: The square root of a nonnegative real is a real. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
resqrcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
resqrcld.2 | ⊢ (𝜑 → 0 ≤ 𝐴) |
Ref | Expression |
---|---|
resqrtcld | ⊢ (𝜑 → (√‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resqrcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | resqrcld.2 | . 2 ⊢ (𝜑 → 0 ≤ 𝐴) | |
3 | resqrtcl 14601 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ ℝ) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (√‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 ℝcr 10524 0cc0 10525 ≤ cle 10664 √csqrt 14580 |
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-cnex 10581 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 ax-pre-sup 10603 |
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-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 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-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 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-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 |
This theorem is referenced by: isprm7 16040 nonsq 16087 ipcau2 23764 tcphcphlem1 23765 tcphcph 23767 rrxcph 23922 trirn 23930 rrxmet 23938 rrxdstprj1 23939 minveclem3b 23958 atans2 25436 chpub 25723 bposlem4 25790 bposlem5 25791 bposlem6 25792 bposlem9 25795 chpchtlim 25982 axsegconlem4 26633 ax5seglem3 26644 normf 28827 normgt0 28831 sqsscirc1 31050 hgt750lemd 31818 hgt750lem 31821 hgt750leme 31828 tgoldbachgtde 31830 sin2h 34763 cos2h 34764 dvasin 34859 areacirclem4 34866 areacirclem5 34867 areacirc 34868 rrnmet 34988 rrndstprj1 34989 rrndstprj2 34990 rrnequiv 34994 rrntotbnd 34995 pellexlem2 39305 pellexlem5 39308 pell14qrgt0 39334 pell1qrge1 39345 stirlingr 42252 rrndistlt 42452 qndenserrnbllem 42456 hoiqssbllem2 42782 sqrtnegnre 43384 sqrtpwpw2p 43577 requad01 43663 requad2 43665 ehl2eudis0lt 44641 inlinecirc02plem 44701 |
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