Metamath Proof Explorer

Theorem ltmuldivs2wd

Description: Surreal less-than relationship between division and multiplication. Weak version. (Contributed by Scott Fenton, 14-Mar-2025)

Ref Expression
Hypotheses ltdivmulswd.1 ⊢ φ → A ∈ No
ltdivmulswd.2 ⊢ φ → B ∈ No
ltdivmulswd.3 ⊢ φ → C ∈ No
ltdivmulswd.4 ⊢ φ → 0 s < s C
ltdivmulswd.5 ⊢ φ → ∃ x ∈ No C ⋅ s x = 1 s
Assertion ltmuldivs2wd ⊢ φ → C ⋅ s A < s B ↔ A < s B / su C

Proof

Step Hyp Ref Expression
1 ltdivmulswd.1 ⊢ φ → A ∈ No
2 ltdivmulswd.2 ⊢ φ → B ∈ No
3 ltdivmulswd.3 ⊢ φ → C ∈ No
4 ltdivmulswd.4 ⊢ φ → 0 s < s C
5 ltdivmulswd.5 ⊢ φ → ∃ x ∈ No C ⋅ s x = 1 s
6 1 3 mulscomd ⊢ φ → A ⋅ s C = C ⋅ s A
7 6 breq1d ⊢ φ → A ⋅ s C < s B ↔ C ⋅ s A < s B
8 1 2 3 4 5 ltmuldivswd ⊢ φ → A ⋅ s C < s B ↔ A < s B / su C
9 7 8 bitr3d ⊢ φ → C ⋅ s A < s B ↔ A < s B / su C