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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 15400 | . 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 2108 ‘cfv 6561 ℂcc 11153 √csqrt 15272 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 |
| This theorem is referenced by: msqsqrtd 15479 pythagtriplem12 16864 pythagtriplem14 16866 pythagtriplem16 16868 tcphcphlem1 25269 tcphcph 25271 efif1olem3 26586 efif1olem4 26587 dvcnsqrt 26786 loglesqrt 26804 quad 26883 dcubic 26889 cubic 26892 quartlem2 26901 quartlem3 26902 quartlem4 26903 quart 26904 asinlem 26911 asinlem2 26912 asinlem3a 26913 asinlem3 26914 asinf 26915 asinneg 26929 efiasin 26931 sinasin 26932 asinbnd 26942 cosasin 26947 efiatan2 26960 cosatan 26964 cosatanne0 26965 atans2 26974 addsqnreup 27487 quad3d 32754 sqsscirc1 33907 divsqrtid 34609 logdivsqrle 34665 dvasin 37711 dvacos 37712 areacirclem1 37715 areacirclem4 37718 areacirc 37720 tan3rdpi 42386 pell1234qrne0 42864 pell1234qrreccl 42865 pell1234qrmulcl 42866 pell14qrgt0 42870 pell1234qrdich 42872 pell14qrdich 42880 pell1qr1 42882 rmspecsqrtnq 42917 rmxyneg 42932 rmxyadd 42933 rmxy1 42934 rmxy0 42935 jm2.22 43007 stirlinglem3 46091 stirlinglem4 46092 stirlinglem13 46101 stirlinglem14 46102 stirlinglem15 46103 qndenserrnbllem 46309 sqrtnegnre 47319 quad1 47607 requad01 47608 requad1 47609 requad2 47610 itsclc0yqsol 48685 itscnhlc0xyqsol 48686 itschlc0xyqsol1 48687 itschlc0xyqsol 48688 itsclc0xyqsolr 48690 inlinecirc02plem 48707 |
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