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Mirrors > Home > MPE Home > Th. List > sqsqrtd | Structured version Visualization version GIF version |
Description: Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
abscld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
sqsqrtd | ⊢ (𝜑 → ((√‘𝐴)↑2) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abscld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | sqrtth 14585 | . 2 ⊢ (𝐴 ∈ ℂ → ((√‘𝐴)↑2) = 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → ((√‘𝐴)↑2) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ‘cfv 6188 (class class class)co 6976 ℂcc 10333 2c2 11495 ↑cexp 13244 √csqrt 14453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-sup 8701 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-n0 11708 df-z 11794 df-uz 12059 df-rp 12205 df-seq 13185 df-exp 13245 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 |
This theorem is referenced by: msqsqrtd 14661 sqr00d 14662 sqrt2irrlem 15461 zsqrtelqelz 15954 nonsq 15955 prmreclem3 16110 nmsq 23501 cphipipcj 23507 ipcau2 23540 tcphcphlem1 23541 tcphcph 23543 minveclem3b 23734 efif1olem3 24829 efif1olem4 24830 cxpsqrt 24987 loglesqrt 25040 quad 25119 cubic 25128 quartlem4 25139 quart 25140 asinlem 25147 asinlem2 25148 efiatan2 25196 cosatan 25200 cosatanne0 25201 atans2 25210 chpub 25498 addsqnreup 25721 chtppilim 25753 rplogsumlem1 25762 dchrisum0flblem1 25786 dchrisum0flblem2 25787 dchrisum0fno1 25789 sin2h 34320 cos2h 34321 areacirclem1 34420 areacirclem5 34424 pell1234qrne0 38843 pell1234qrreccl 38844 pell1234qrmulcl 38845 pell14qrgt0 38849 pell14qrdich 38859 pell1qrgaplem 38863 pell14qrgapw 38866 pellqrex 38869 rmxyneg 38910 jm2.22 38985 |
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