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Theorem cjval 14049
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 6802 . . . . 5 (𝑦 = 𝐴 → (𝑦 + 𝑥) = (𝐴 + 𝑥))
21eleq1d 2835 . . . 4 (𝑦 = 𝐴 → ((𝑦 + 𝑥) ∈ ℝ ↔ (𝐴 + 𝑥) ∈ ℝ))
3 oveq1 6802 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑥) = (𝐴𝑥))
43oveq2d 6811 . . . . 5 (𝑦 = 𝐴 → (i · (𝑦𝑥)) = (i · (𝐴𝑥)))
54eleq1d 2835 . . . 4 (𝑦 = 𝐴 → ((i · (𝑦𝑥)) ∈ ℝ ↔ (i · (𝐴𝑥)) ∈ ℝ))
62, 5anbi12d 616 . . 3 (𝑦 = 𝐴 → (((𝑦 + 𝑥) ∈ ℝ ∧ (i · (𝑦𝑥)) ∈ ℝ) ↔ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)))
76riotabidv 6758 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℂ ((𝑦 + 𝑥) ∈ ℝ ∧ (i · (𝑦𝑥)) ∈ ℝ)) = (𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)))
8 df-cj 14046 . 2 ∗ = (𝑦 ∈ ℂ ↦ (𝑥 ∈ ℂ ((𝑦 + 𝑥) ∈ ℝ ∧ (i · (𝑦𝑥)) ∈ ℝ)))
9 riotaex 6760 . 2 (𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴𝑥)) ∈ ℝ)) ∈ V
107, 8, 9fvmpt 6426 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 1631  wcel 2145  cfv 6030  crio 6755  (class class class)co 6795  cc 10139  cr 10140  ici 10143   + caddc 10144   · cmul 10146  cmin 10471  ccj 14043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-riota 6756  df-ov 6798  df-cj 14046
This theorem is referenced by:  cjth  14050  remim  14064
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