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Theorem absval 14022
Description: The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of [Gleason] p. 133. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
Assertion
Ref Expression
absval (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴))))

Proof of Theorem absval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6229 . . . 4 (𝑥 = 𝐴 → (∗‘𝑥) = (∗‘𝐴))
2 oveq12 6699 . . . 4 ((𝑥 = 𝐴 ∧ (∗‘𝑥) = (∗‘𝐴)) → (𝑥 · (∗‘𝑥)) = (𝐴 · (∗‘𝐴)))
31, 2mpdan 703 . . 3 (𝑥 = 𝐴 → (𝑥 · (∗‘𝑥)) = (𝐴 · (∗‘𝐴)))
43fveq2d 6233 . 2 (𝑥 = 𝐴 → (√‘(𝑥 · (∗‘𝑥))) = (√‘(𝐴 · (∗‘𝐴))))
5 df-abs 14020 . 2 abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))
6 fvex 6239 . 2 (√‘(𝐴 · (∗‘𝐴))) ∈ V
74, 5, 6fvmpt 6321 1 (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  cfv 5926  (class class class)co 6690  cc 9972   · cmul 9979  ccj 13880  csqrt 14017  abscabs 14018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-abs 14020
This theorem is referenced by:  absneg  14061  abscl  14062  abscj  14063  absvalsq  14064  absval2  14068  abs0  14069  absi  14070  absge0  14071  absrpcl  14072  absmul  14078  absid  14080  absre  14085  absf  14121  cphabscl  23031  cphipipcj  23046  tchcphlem2  23081  siii  27836  norm-iii-i  28124  absfico  39724
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