Metamath Proof Explorer

Theorem sltdivmuld

Description: Surreal less-than relationship between division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025)

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

Proof

Step Hyp Ref Expression
1 sltdivmuld.1 ⊢ φ → A ∈ No
2 sltdivmuld.2 ⊢ φ → B ∈ No
3 sltdivmuld.3 ⊢ φ → C ∈ No
4 sltdivmuld.4 ⊢ φ → 0 s < s C
5 4 sgt0ne0d ⊢ φ → C ≠ 0 s
6 3 5 recsexd ⊢ φ → ∃ x ∈ No C ⋅ s x = 1 s
7 1 2 3 4 6 sltdivmulwd ⊢ φ → A / su C < s B ↔ A < s C ⋅ s B