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Theorem divval 11288
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 4711 . . 3 (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
2 eqeq2 2830 . . . . 5 (𝑧 = 𝐴 → ((𝑦 · 𝑥) = 𝑧 ↔ (𝑦 · 𝑥) = 𝐴))
32riotabidv 7105 . . . 4 (𝑧 = 𝐴 → (𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝑧) = (𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝐴))
4 oveq1 7152 . . . . . 6 (𝑦 = 𝐵 → (𝑦 · 𝑥) = (𝐵 · 𝑥))
54eqeq1d 2820 . . . . 5 (𝑦 = 𝐵 → ((𝑦 · 𝑥) = 𝐴 ↔ (𝐵 · 𝑥) = 𝐴))
65riotabidv 7105 . . . 4 (𝑦 = 𝐵 → (𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝐴) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
7 df-div 11286 . . . 4 / = (𝑧 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝑧))
8 riotaex 7107 . . . 4 (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) ∈ V
93, 6, 7, 8ovmpo 7299 . . 3 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ (ℂ ∖ {0})) → (𝐴 / 𝐵) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
101, 9sylan2br 594 . 2 ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
11103impb 1107 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 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  cdif 3930  {csn 4557  crio 7102  (class class class)co 7145  cc 10523  0cc0 10525   · cmul 10530   / cdiv 11285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-div 11286
This theorem is referenced by:  divmul  11289  divcl  11292  cnflddiv  20503  divcn  23403  rexdiv  30529
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