Metamath Proof Explorer

Theorem ltmuls1d

Description: Multiplication of both sides of surreal less-than by a positive number. (Contributed by Scott Fenton, 10-Mar-2025)

Ref Expression
Hypotheses ltmuls12d.1 ⊢ φ → A ∈ No
ltmuls12d.2 ⊢ φ → B ∈ No
ltmuls12d.3 ⊢ φ → C ∈ No
ltmuls12d.4 ⊢ φ → 0 s < s C
Assertion ltmuls1d ⊢ φ → A < s B ↔ A ⋅ s C < s B ⋅ s C

Proof

Step Hyp Ref Expression
1 ltmuls12d.1 ⊢ φ → A ∈ No
2 ltmuls12d.2 ⊢ φ → B ∈ No
3 ltmuls12d.3 ⊢ φ → C ∈ No
4 ltmuls12d.4 ⊢ φ → 0 s < s C
5 1 2 3 4 ltmuls2d ⊢ φ → A < s B ↔ C ⋅ s A < s C ⋅ s B
6 1 3 mulscomd ⊢ φ → A ⋅ s C = C ⋅ s A
7 2 3 mulscomd ⊢ φ → B ⋅ s C = C ⋅ s B
8 6 7 breq12d ⊢ φ → A ⋅ s C < s B ⋅ s C ↔ C ⋅ s A < s C ⋅ s B
9 5 8 bitr4d ⊢ φ → A < s B ↔ A ⋅ s C < s B ⋅ s C