Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sqrtval Structured version   Visualization version   GIF version

Theorem sqrtval 14168
 Description: Value of square root function. (Contributed by Mario Carneiro, 8-Jul-2013.)
Assertion
Ref Expression
sqrtval (𝐴 ∈ ℂ → (√‘𝐴) = (𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem sqrtval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2763 . . . 4 (𝑦 = 𝐴 → ((𝑥↑2) = 𝑦 ↔ (𝑥↑2) = 𝐴))
213anbi1d 1544 . . 3 (𝑦 = 𝐴 → (((𝑥↑2) = 𝑦 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+) ↔ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)))
32riotabidv 6768 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℂ ((𝑥↑2) = 𝑦 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) = (𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)))
4 df-sqrt 14166 . 2 √ = (𝑦 ∈ ℂ ↦ (𝑥 ∈ ℂ ((𝑥↑2) = 𝑦 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)))
5 riotaex 6770 . 2 (𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) ∈ V
63, 4, 5fvmpt 6436 1 (𝐴 ∈ ℂ → (√‘𝐴) = (𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1072   = wceq 1624   ∈ wcel 2131   ∉ wnel 3027   class class class wbr 4796  ‘cfv 6041  ℩crio 6765  (class class class)co 6805  ℂcc 10118  0cc0 10120  ici 10122   · cmul 10125   ≤ cle 10259  2c2 11254  ℝ+crp 12017  ↑cexp 13046  ℜcre 14028  √csqrt 14164 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fv 6049  df-riota 6766  df-sqrt 14166 This theorem is referenced by:  sqrt0  14173  resqrtcl  14185  resqrtthlem  14186  sqrtneg  14199  sqrtcl  14292  sqrtthlem  14293
 Copyright terms: Public domain W3C validator