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 ) ⟶ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resubval | ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 −ℝ 𝑦 ) = ( ℩ 𝑧 ∈ ℝ ( 𝑦 + 𝑧 ) = 𝑥 ) ) | |
2 | rersubcl | ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 −ℝ 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 | ⊢ −ℝ = ( 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 ) ⟶ ℝ ) |
7 | 4 6 | mpbi | ⊢ −ℝ : ( ℝ × ℝ ) ⟶ ℝ |