Description: ltaddneg without ax-mulcom . (Contributed by SN, 25-Jan-2025)
Ref | Expression | ||
---|---|---|---|
Assertion | sn-ltaddneg | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 0 ↔ ( 𝐵 + 𝐴 ) < 𝐵 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re | ⊢ 0 ∈ ℝ | |
2 | ltadd2 | ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 0 ↔ ( 𝐵 + 𝐴 ) < ( 𝐵 + 0 ) ) ) | |
3 | 1 2 | mp3an2 | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 0 ↔ ( 𝐵 + 𝐴 ) < ( 𝐵 + 0 ) ) ) |
4 | readdrid | ⊢ ( 𝐵 ∈ ℝ → ( 𝐵 + 0 ) = 𝐵 ) | |
5 | 4 | adantl | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 + 0 ) = 𝐵 ) |
6 | 5 | breq2d | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐵 + 𝐴 ) < ( 𝐵 + 0 ) ↔ ( 𝐵 + 𝐴 ) < 𝐵 ) ) |
7 | 3 6 | bitrd | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 0 ↔ ( 𝐵 + 𝐴 ) < 𝐵 ) ) |