Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 14720 | . 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 6354 ℂcc 10534 √csqrt 14591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-seq 13369 df-exp 13429 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 |
This theorem is referenced by: msqsqrtd 14799 pythagtriplem12 16162 pythagtriplem14 16164 pythagtriplem16 16166 tcphcphlem1 23837 tcphcph 23839 efif1olem3 25127 efif1olem4 25128 dvcnsqrt 25324 loglesqrt 25338 quad 25417 dcubic 25423 cubic 25426 quartlem2 25435 quartlem3 25436 quartlem4 25437 quart 25438 asinlem 25445 asinlem2 25446 asinlem3a 25447 asinlem3 25448 asinf 25449 asinneg 25463 efiasin 25465 sinasin 25466 asinbnd 25476 cosasin 25481 efiatan2 25494 cosatan 25498 cosatanne0 25499 atans2 25508 addsqnreup 26018 sqsscirc1 31151 divsqrtid 31865 logdivsqrle 31921 dvasin 34977 dvacos 34978 areacirclem1 34981 areacirclem4 34984 areacirc 34986 pell1234qrne0 39448 pell1234qrreccl 39449 pell1234qrmulcl 39450 pell14qrgt0 39454 pell1234qrdich 39456 pell14qrdich 39464 pell1qr1 39466 rmspecsqrtnq 39501 rmxyneg 39515 rmxyadd 39516 rmxy1 39517 rmxy0 39518 jm2.22 39590 stirlinglem3 42360 stirlinglem4 42361 stirlinglem13 42370 stirlinglem14 42371 stirlinglem15 42372 qndenserrnbllem 42578 sqrtnegnre 43506 quad1 43784 requad01 43785 requad1 43786 requad2 43787 itsclc0yqsol 44750 itscnhlc0xyqsol 44751 itschlc0xyqsol1 44752 itschlc0xyqsol 44753 itsclc0xyqsolr 44755 inlinecirc02plem 44772 |
Copyright terms: Public domain | W3C validator |