Theorem divval

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. x ) = A . (Contributed by NM, 8-May-1999) (Revised by Mario Carneiro, 17-Feb-2014)

	Ref	Expression
Assertion	divval	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 / 𝐵 ) = ( ℩ 𝑥 ∈ ℂ ( 𝐵 · 𝑥 ) = 𝐴 ) )

Proof

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