Theorem divval 11892

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.

Step	Hyp	Ref	Expression
1		eldifsn 4758	. . 3 ⊢ (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
2		eqeq2 2778	. . . . 5 ⊢ (𝑧 = 𝐴 → ((𝑦 · 𝑥) = 𝑧 ↔ (𝑦 · 𝑥) = 𝐴))
3	2	riotabidv 7382	. . . 4 ⊢ (𝑧 = 𝐴 → (℩𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝑧) = (℩𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝐴))
4		oveq1 7430	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 · 𝑥) = (𝐵 · 𝑥))
5	4	eqeq1d 2768	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑦 · 𝑥) = 𝐴 ↔ (𝐵 · 𝑥) = 𝐴))
6	5	riotabidv 7382	. . . 4 ⊢ (𝑦 = 𝐵 → (℩𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝐴) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
7		df-div 11890	. . . 4 ⊢ / = (𝑧 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝑧))
8		riotaex 7384	. . . 4 ⊢ (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) ∈ V
9	3, 6, 7, 8	ovmpo 7583	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ (ℂ ∖ {0})) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
10	1, 9	sylan2br 607	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
11	10	3impb 1132	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2961   ∖ cdif 3905  {csn 4594  ℩crio 7379  (class class class)co 7423  ℂcc 11116  0cc0 11118   · cmul 11123   / cdiv 11889

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-div 11890

This theorem is used by:  divmul  11893  divcl  11896  cnflddiv  21589  divcn  25064  rexdiv  33282