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Mirrors > Home > MPE Home > Th. List > sqrtcld | Structured version Visualization version GIF version |
Description: Closure of the square root function over the complex numbers. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
abscld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
sqrtcld | ⊢ (𝜑 → (√‘𝐴) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abscld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | sqrtcl 14925 | . 2 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (√‘𝐴) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ‘cfv 6380 ℂcc 10727 √csqrt 14796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-sup 9058 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-seq 13575 df-exp 13636 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 |
This theorem is referenced by: msqsqrtd 15004 pythagtriplem12 16379 pythagtriplem14 16381 pythagtriplem16 16383 tcphcphlem1 24132 tcphcph 24134 efif1olem3 25433 efif1olem4 25434 dvcnsqrt 25630 loglesqrt 25644 quad 25723 dcubic 25729 cubic 25732 quartlem2 25741 quartlem3 25742 quartlem4 25743 quart 25744 asinlem 25751 asinlem2 25752 asinlem3a 25753 asinlem3 25754 asinf 25755 asinneg 25769 efiasin 25771 sinasin 25772 asinbnd 25782 cosasin 25787 efiatan2 25800 cosatan 25804 cosatanne0 25805 atans2 25814 addsqnreup 26324 sqsscirc1 31572 divsqrtid 32286 logdivsqrle 32342 dvasin 35598 dvacos 35599 areacirclem1 35602 areacirclem4 35605 areacirc 35607 pell1234qrne0 40378 pell1234qrreccl 40379 pell1234qrmulcl 40380 pell14qrgt0 40384 pell1234qrdich 40386 pell14qrdich 40394 pell1qr1 40396 rmspecsqrtnq 40431 rmxyneg 40445 rmxyadd 40446 rmxy1 40447 rmxy0 40448 jm2.22 40520 stirlinglem3 43292 stirlinglem4 43293 stirlinglem13 43302 stirlinglem14 43303 stirlinglem15 43304 qndenserrnbllem 43510 sqrtnegnre 44472 quad1 44745 requad01 44746 requad1 44747 requad2 44748 itsclc0yqsol 45783 itscnhlc0xyqsol 45784 itschlc0xyqsol1 45785 itschlc0xyqsol 45786 itsclc0xyqsolr 45788 inlinecirc02plem 45805 |
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