Theorem resubval

Description: Value of real subtraction, which is the (unique) real x such that B + x = A . (Contributed by Steven Nguyen, 7-Jan-2023)

	Ref	Expression
Assertion	resubval	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 −ℝ 𝐵 ) = ( ℩ 𝑥 ∈ ℝ ( 𝐵 + 𝑥 ) = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq2	⊢ ( 𝑦 = 𝐴 → ( ( 𝑧 + 𝑥 ) = 𝑦 ↔ ( 𝑧 + 𝑥 ) = 𝐴 ) )
2	1	riotabidv	⊢ ( 𝑦 = 𝐴 → ( ℩ 𝑥 ∈ ℝ ( 𝑧 + 𝑥 ) = 𝑦 ) = ( ℩ 𝑥 ∈ ℝ ( 𝑧 + 𝑥 ) = 𝐴 ) )
3		oveq1	⊢ ( 𝑧 = 𝐵 → ( 𝑧 + 𝑥 ) = ( 𝐵 + 𝑥 ) )
4	3	eqeq1d	⊢ ( 𝑧 = 𝐵 → ( ( 𝑧 + 𝑥 ) = 𝐴 ↔ ( 𝐵 + 𝑥 ) = 𝐴 ) )
5	4	riotabidv	⊢ ( 𝑧 = 𝐵 → ( ℩ 𝑥 ∈ ℝ ( 𝑧 + 𝑥 ) = 𝐴 ) = ( ℩ 𝑥 ∈ ℝ ( 𝐵 + 𝑥 ) = 𝐴 ) )
6		df-resub	⊢ −ℝ = ( 𝑦 ∈ ℝ , 𝑧 ∈ ℝ ↦ ( ℩ 𝑥 ∈ ℝ ( 𝑧 + 𝑥 ) = 𝑦 ) )
7		riotaex	⊢ ( ℩ 𝑥 ∈ ℝ ( 𝐵 + 𝑥 ) = 𝐴 ) ∈ V
8	2 5 6 7	ovmpo	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 −ℝ 𝐵 ) = ( ℩ 𝑥 ∈ ℝ ( 𝐵 + 𝑥 ) = 𝐴 ) )