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Theorem cjval 13690
Description: The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
Assertion
Ref Expression
cjval (𝐴 ∈ ℂ → (∗‘𝐴) = (𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem cjval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 oveq1 6556 . . . . 5 (𝑦 = 𝐴 → (𝑦 + 𝑥) = (𝐴 + 𝑥))
21eleq1d 2672 . . . 4 (𝑦 = 𝐴 → ((𝑦 + 𝑥) ∈ ℝ ↔ (𝐴 + 𝑥) ∈ ℝ))
3 oveq1 6556 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑥) = (𝐴𝑥))
43oveq2d 6565 . . . . 5 (𝑦 = 𝐴 → (i · (𝑦𝑥)) = (i · (𝐴𝑥)))
54eleq1d 2672 . . . 4 (𝑦 = 𝐴 → ((i · (𝑦𝑥)) ∈ ℝ ↔ (i · (𝐴𝑥)) ∈ ℝ))
62, 5anbi12d 743 . . 3 (𝑦 = 𝐴 → (((𝑦 + 𝑥) ∈ ℝ ∧ (i · (𝑦𝑥)) ∈ ℝ) ↔ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)))
76riotabidv 6513 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℂ ((𝑦 + 𝑥) ∈ ℝ ∧ (i · (𝑦𝑥)) ∈ ℝ)) = (𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)))
8 df-cj 13687 . 2 ∗ = (𝑦 ∈ ℂ ↦ (𝑥 ∈ ℂ ((𝑦 + 𝑥) ∈ ℝ ∧ (i · (𝑦𝑥)) ∈ ℝ)))
9 riotaex 6515 . 2 (𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)) ∈ V
107, 8, 9fvmpt 6191 1 (𝐴 ∈ ℂ → (∗‘𝐴) = (𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  cfv 5804  crio 6510  (class class class)co 6549  cc 9813  cr 9814  ici 9817   + caddc 9818   · cmul 9820  cmin 10145  ccj 13684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-ov 6552  df-cj 13687
This theorem is referenced by:  cjth  13691  remim  13705
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