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Theorem cjval 13636
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 6534 . . . . 5 (𝑦 = 𝐴 → (𝑦 + 𝑥) = (𝐴 + 𝑥))
21eleq1d 2671 . . . 4 (𝑦 = 𝐴 → ((𝑦 + 𝑥) ∈ ℝ ↔ (𝐴 + 𝑥) ∈ ℝ))
3 oveq1 6534 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑥) = (𝐴𝑥))
43oveq2d 6543 . . . . 5 (𝑦 = 𝐴 → (i · (𝑦𝑥)) = (i · (𝐴𝑥)))
54eleq1d 2671 . . . 4 (𝑦 = 𝐴 → ((i · (𝑦𝑥)) ∈ ℝ ↔ (i · (𝐴𝑥)) ∈ ℝ))
62, 5anbi12d 742 . . 3 (𝑦 = 𝐴 → (((𝑦 + 𝑥) ∈ ℝ ∧ (i · (𝑦𝑥)) ∈ ℝ) ↔ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)))
76riotabidv 6491 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℂ ((𝑦 + 𝑥) ∈ ℝ ∧ (i · (𝑦𝑥)) ∈ ℝ)) = (𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)))
8 df-cj 13633 . 2 ∗ = (𝑦 ∈ ℂ ↦ (𝑥 ∈ ℂ ((𝑦 + 𝑥) ∈ ℝ ∧ (i · (𝑦𝑥)) ∈ ℝ)))
9 riotaex 6493 . 2 (𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)) ∈ V
107, 8, 9fvmpt 6176 1 (𝐴 ∈ ℂ → (∗‘𝐴) = (𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cfv 5790  crio 6488  (class class class)co 6527  cc 9790  cr 9791  ici 9794   + caddc 9795   · cmul 9797  cmin 10117  ccj 13630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-riota 6489  df-ov 6530  df-cj 13633
This theorem is referenced by:  cjth  13637  remim  13651
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