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Theorem divval 10899
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 4462 . . 3 (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
2 eqeq2 2771 . . . . 5 (𝑧 = 𝐴 → ((𝑦 · 𝑥) = 𝑧 ↔ (𝑦 · 𝑥) = 𝐴))
32riotabidv 6777 . . . 4 (𝑧 = 𝐴 → (𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝑧) = (𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝐴))
4 oveq1 6821 . . . . . 6 (𝑦 = 𝐵 → (𝑦 · 𝑥) = (𝐵 · 𝑥))
54eqeq1d 2762 . . . . 5 (𝑦 = 𝐵 → ((𝑦 · 𝑥) = 𝐴 ↔ (𝐵 · 𝑥) = 𝐴))
65riotabidv 6777 . . . 4 (𝑦 = 𝐵 → (𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝐴) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
7 df-div 10897 . . . 4 / = (𝑧 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝑧))
8 riotaex 6779 . . . 4 (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) ∈ V
93, 6, 7, 8ovmpt2 6962 . . 3 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ (ℂ ∖ {0})) → (𝐴 / 𝐵) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
101, 9sylan2br 494 . 2 ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
11103impb 1108 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 1072   = wceq 1632  wcel 2139  wne 2932  cdif 3712  {csn 4321  crio 6774  (class class class)co 6814  cc 10146  0cc0 10148   · cmul 10153   / cdiv 10896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-div 10897
This theorem is referenced by:  divmul  10900  divcl  10903  cnflddiv  19998  divcn  22892  rexdiv  29964
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