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Theorem cjval 13792
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 6622 . . . . 5 (𝑦 = 𝐴 → (𝑦 + 𝑥) = (𝐴 + 𝑥))
21eleq1d 2683 . . . 4 (𝑦 = 𝐴 → ((𝑦 + 𝑥) ∈ ℝ ↔ (𝐴 + 𝑥) ∈ ℝ))
3 oveq1 6622 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑥) = (𝐴𝑥))
43oveq2d 6631 . . . . 5 (𝑦 = 𝐴 → (i · (𝑦𝑥)) = (i · (𝐴𝑥)))
54eleq1d 2683 . . . 4 (𝑦 = 𝐴 → ((i · (𝑦𝑥)) ∈ ℝ ↔ (i · (𝐴𝑥)) ∈ ℝ))
62, 5anbi12d 746 . . 3 (𝑦 = 𝐴 → (((𝑦 + 𝑥) ∈ ℝ ∧ (i · (𝑦𝑥)) ∈ ℝ) ↔ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)))
76riotabidv 6578 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℂ ((𝑦 + 𝑥) ∈ ℝ ∧ (i · (𝑦𝑥)) ∈ ℝ)) = (𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)))
8 df-cj 13789 . 2 ∗ = (𝑦 ∈ ℂ ↦ (𝑥 ∈ ℂ ((𝑦 + 𝑥) ∈ ℝ ∧ (i · (𝑦𝑥)) ∈ ℝ)))
9 riotaex 6580 . 2 (𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)) ∈ V
107, 8, 9fvmpt 6249 1 (𝐴 ∈ ℂ → (∗‘𝐴) = (𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cfv 5857  crio 6575  (class class class)co 6615  cc 9894  cr 9895  ici 9898   + caddc 9899   · cmul 9901  cmin 10226  ccj 13786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-riota 6576  df-ov 6618  df-cj 13789
This theorem is referenced by:  cjth  13793  remim  13807
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