Theorem xdivval

Description: Value of division: the (unique) element x such that ( B x. x ) = A . This is meaningful only when B is nonzero. (Contributed by Thierry Arnoux, 17-Dec-2016)

	Ref	Expression
Assertion	xdivval	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 𝐴 /𝑒 𝐵 ) = ( ℩ 𝑥 ∈ ℝ* ( 𝐵 ·e 𝑥 ) = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		eldifsn	⊢ ( 𝐵 ∈ ( ℝ ∖ { 0 } ) ↔ ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) )
2		simpl	⊢ ( ( 𝑦 = 𝐴 ∧ 𝑥 ∈ ℝ* ) → 𝑦 = 𝐴 )
3	2	eqeq2d	⊢ ( ( 𝑦 = 𝐴 ∧ 𝑥 ∈ ℝ* ) → ( ( 𝑧 ·e 𝑥 ) = 𝑦 ↔ ( 𝑧 ·e 𝑥 ) = 𝐴 ) )
4	3	riotabidva	⊢ ( 𝑦 = 𝐴 → ( ℩ 𝑥 ∈ ℝ* ( 𝑧 ·e 𝑥 ) = 𝑦 ) = ( ℩ 𝑥 ∈ ℝ* ( 𝑧 ·e 𝑥 ) = 𝐴 ) )
5		simpl	⊢ ( ( 𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ* ) → 𝑧 = 𝐵 )
6	5	oveq1d	⊢ ( ( 𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ* ) → ( 𝑧 ·e 𝑥 ) = ( 𝐵 ·e 𝑥 ) )
7	6	eqeq1d	⊢ ( ( 𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ* ) → ( ( 𝑧 ·e 𝑥 ) = 𝐴 ↔ ( 𝐵 ·e 𝑥 ) = 𝐴 ) )
8	7	riotabidva	⊢ ( 𝑧 = 𝐵 → ( ℩ 𝑥 ∈ ℝ* ( 𝑧 ·e 𝑥 ) = 𝐴 ) = ( ℩ 𝑥 ∈ ℝ* ( 𝐵 ·e 𝑥 ) = 𝐴 ) )
9		df-xdiv	⊢ /𝑒 = ( 𝑦 ∈ ℝ* , 𝑧 ∈ ( ℝ ∖ { 0 } ) ↦ ( ℩ 𝑥 ∈ ℝ* ( 𝑧 ·e 𝑥 ) = 𝑦 ) )
10		riotaex	⊢ ( ℩ 𝑥 ∈ ℝ* ( 𝐵 ·e 𝑥 ) = 𝐴 ) ∈ V
11	4 8 9 10	ovmpo	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ( ℝ ∖ { 0 } ) ) → ( 𝐴 /𝑒 𝐵 ) = ( ℩ 𝑥 ∈ ℝ* ( 𝐵 ·e 𝑥 ) = 𝐴 ) )
12	1 11	sylan2br	⊢ ( ( 𝐴 ∈ ℝ* ∧ ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ) → ( 𝐴 /𝑒 𝐵 ) = ( ℩ 𝑥 ∈ ℝ* ( 𝐵 ·e 𝑥 ) = 𝐴 ) )
13	12	3impb	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 𝐴 /𝑒 𝐵 ) = ( ℩ 𝑥 ∈ ℝ* ( 𝐵 ·e 𝑥 ) = 𝐴 ) )