Description: Subtraction restricted to the reals. (Contributed by SN, 5-May-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | subresre | ⊢ −ℝ = ( − ↾ ( ℝ × ℝ ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resubeqsub | ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 −ℝ 𝑦 ) = ( 𝑥 − 𝑦 ) ) | |
2 | 1 | 3adant1 | ⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 −ℝ 𝑦 ) = ( 𝑥 − 𝑦 ) ) |
3 | ax-resscn | ⊢ ℝ ⊆ ℂ | |
4 | 3 | a1i | ⊢ ( ⊤ → ℝ ⊆ ℂ ) |
5 | resubf | ⊢ −ℝ : ( ℝ × ℝ ) ⟶ ℝ | |
6 | 5 | a1i | ⊢ ( ⊤ → −ℝ : ( ℝ × ℝ ) ⟶ ℝ ) |
7 | sn-subf | ⊢ − : ( ℂ × ℂ ) ⟶ ℂ | |
8 | 7 | a1i | ⊢ ( ⊤ → − : ( ℂ × ℂ ) ⟶ ℂ ) |
9 | 2 4 6 8 | oprres | ⊢ ( ⊤ → −ℝ = ( − ↾ ( ℝ × ℝ ) ) ) |
10 | 9 | mptru | ⊢ −ℝ = ( − ↾ ( ℝ × ℝ ) ) |