Metamath Proof Explorer


Theorem resubf

Description: Real subtraction is an operation on the real numbers. Based on subf . (Contributed by Steven Nguyen, 7-Jan-2023)

Ref Expression
Assertion resubf : ( ℝ × ℝ ) ⟶ ℝ

Proof

Step Hyp Ref Expression
1 resubval ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 𝑦 ) = ( 𝑧 ∈ ℝ ( 𝑦 + 𝑧 ) = 𝑥 ) )
2 rersubcl ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 𝑦 ) ∈ ℝ )
3 1 2 eqeltrrd ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑧 ∈ ℝ ( 𝑦 + 𝑧 ) = 𝑥 ) ∈ ℝ )
4 3 rgen2 𝑥 ∈ ℝ ∀ 𝑦 ∈ ℝ ( 𝑧 ∈ ℝ ( 𝑦 + 𝑧 ) = 𝑥 ) ∈ ℝ
5 df-resub = ( 𝑥 ∈ ℝ , 𝑦 ∈ ℝ ↦ ( 𝑧 ∈ ℝ ( 𝑦 + 𝑧 ) = 𝑥 ) )
6 5 fmpo ( ∀ 𝑥 ∈ ℝ ∀ 𝑦 ∈ ℝ ( 𝑧 ∈ ℝ ( 𝑦 + 𝑧 ) = 𝑥 ) ∈ ℝ ↔ − : ( ℝ × ℝ ) ⟶ ℝ )
7 4 6 mpbi : ( ℝ × ℝ ) ⟶ ℝ