Theorem ltdivmulswd

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 → 0s <s 𝐶 )
	ltdivmulswd.5	⊢ ( 𝜑 → ∃ 𝑥 ∈ No ( 𝐶 ·s 𝑥 ) = 1s )
Assertion	ltdivmulswd	⊢ ( 𝜑 → ( ( 𝐴 /su 𝐶 ) <s 𝐵 ↔ 𝐴 <s ( 𝐶 ·s 𝐵 ) ) )

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 → 0s <s 𝐶 )
5		ltdivmulswd.5	⊢ ( 𝜑 → ∃ 𝑥 ∈ No ( 𝐶 ·s 𝑥 ) = 1s )
6	4	gt0ne0sd	⊢ ( 𝜑 → 𝐶 ≠ 0s )
7	1 3 6 5	divsclwd	⊢ ( 𝜑 → ( 𝐴 /su 𝐶 ) ∈ No )
8	7 2 3 4	ltmuls2d	⊢ ( 𝜑 → ( ( 𝐴 /su 𝐶 ) <s 𝐵 ↔ ( 𝐶 ·s ( 𝐴 /su 𝐶 ) ) <s ( 𝐶 ·s 𝐵 ) ) )
9	1 3 6 5	divscan2wd	⊢ ( 𝜑 → ( 𝐶 ·s ( 𝐴 /su 𝐶 ) ) = 𝐴 )
10	9	breq1d	⊢ ( 𝜑 → ( ( 𝐶 ·s ( 𝐴 /su 𝐶 ) ) <s ( 𝐶 ·s 𝐵 ) ↔ 𝐴 <s ( 𝐶 ·s 𝐵 ) ) )
11	8 10	bitrd	⊢ ( 𝜑 → ( ( 𝐴 /su 𝐶 ) <s 𝐵 ↔ 𝐴 <s ( 𝐶 ·s 𝐵 ) ) )