Description: When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019)
Ref | Expression | ||
---|---|---|---|
Hypotheses | sublt0d.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℝ ) | |
sublt0d.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℝ ) | ||
Assertion | sublt0d | ⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) < 0 ↔ 𝐴 < 𝐵 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sublt0d.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℝ ) | |
2 | sublt0d.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℝ ) | |
3 | 0red | ⊢ ( 𝜑 → 0 ∈ ℝ ) | |
4 | 1 2 3 | ltsubaddd | ⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) < 0 ↔ 𝐴 < ( 0 + 𝐵 ) ) ) |
5 | 2 | recnd | ⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
6 | 5 | addid2d | ⊢ ( 𝜑 → ( 0 + 𝐵 ) = 𝐵 ) |
7 | 6 | breq2d | ⊢ ( 𝜑 → ( 𝐴 < ( 0 + 𝐵 ) ↔ 𝐴 < 𝐵 ) ) |
8 | 4 7 | bitrd | ⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) < 0 ↔ 𝐴 < 𝐵 ) ) |