Metamath Proof Explorer


Theorem sn-ltaddpos

Description: ltaddpos without ax-mulcom . (Contributed by SN, 13-Feb-2024)

Ref Expression
Assertion sn-ltaddpos ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < 𝐴𝐵 < ( 𝐵 + 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 0re 0 ∈ ℝ
2 ltadd2 ( ( 0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < 𝐴 ↔ ( 𝐵 + 0 ) < ( 𝐵 + 𝐴 ) ) )
3 1 2 mp3an1 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < 𝐴 ↔ ( 𝐵 + 0 ) < ( 𝐵 + 𝐴 ) ) )
4 readdid1 ( 𝐵 ∈ ℝ → ( 𝐵 + 0 ) = 𝐵 )
5 4 adantl ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 + 0 ) = 𝐵 )
6 5 breq1d ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐵 + 0 ) < ( 𝐵 + 𝐴 ) ↔ 𝐵 < ( 𝐵 + 𝐴 ) ) )
7 3 6 bitrd ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < 𝐴𝐵 < ( 𝐵 + 𝐴 ) ) )