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Theorem divval 10566
Description: Value of division: if 𝐴 and 𝐵 are complex numbers with 𝐵 nonzero, then (𝐴 / 𝐵) is the (unique) complex number such that (𝐵 · 𝑥) = 𝐴. (Contributed by NM, 8-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)
Assertion
Ref Expression
divval ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem divval
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifsn 4260 . . 3 (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
2 eqeq2 2621 . . . . 5 (𝑧 = 𝐴 → ((𝑦 · 𝑥) = 𝑧 ↔ (𝑦 · 𝑥) = 𝐴))
32riotabidv 6513 . . . 4 (𝑧 = 𝐴 → (𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝑧) = (𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝐴))
4 oveq1 6556 . . . . . 6 (𝑦 = 𝐵 → (𝑦 · 𝑥) = (𝐵 · 𝑥))
54eqeq1d 2612 . . . . 5 (𝑦 = 𝐵 → ((𝑦 · 𝑥) = 𝐴 ↔ (𝐵 · 𝑥) = 𝐴))
65riotabidv 6513 . . . 4 (𝑦 = 𝐵 → (𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝐴) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
7 df-div 10564 . . . 4 / = (𝑧 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝑧))
8 riotaex 6515 . . . 4 (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) ∈ V
93, 6, 7, 8ovmpt2 6694 . . 3 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ (ℂ ∖ {0})) → (𝐴 / 𝐵) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
101, 9sylan2br 492 . 2 ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
11103impb 1252 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  cdif 3537  {csn 4125  crio 6510  (class class class)co 6549  cc 9813  0cc0 9815   · cmul 9820   / cdiv 10563
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-ne 2782  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-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-oprab 6553  df-mpt2 6554  df-div 10564
This theorem is referenced by:  divmul  10567  divcl  10570  cnflddiv  19595  divcn  22479  rexdiv  28965
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