Metamath Proof Explorer


Theorem subresre

Description: Subtraction restricted to the reals. (Contributed by SN, 5-May-2024)

Ref Expression
Assertion subresre = ( − ↾ ( ℝ × ℝ ) )

Proof

Step Hyp Ref Expression
1 resubeqsub ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 𝑦 ) = ( 𝑥𝑦 ) )
2 1 3adant1 ( ( ⊤ ∧ 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 𝑦 ) = ( 𝑥𝑦 ) )
3 ax-resscn ℝ ⊆ ℂ
4 3 a1i ( ⊤ → ℝ ⊆ ℂ )
5 resubf : ( ℝ × ℝ ) ⟶ ℝ
6 5 a1i ( ⊤ → − : ( ℝ × ℝ ) ⟶ ℝ )
7 sn-subf − : ( ℂ × ℂ ) ⟶ ℂ
8 7 a1i ( ⊤ → − : ( ℂ × ℂ ) ⟶ ℂ )
9 2 4 6 8 oprres ( ⊤ → − = ( − ↾ ( ℝ × ℝ ) ) )
10 9 mptru = ( − ↾ ( ℝ × ℝ ) )