Theorem ltmuldivswd

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	ltmuldivswd	⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐶 ) <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 → 0s <s 𝐶 )
5		ltdivmulswd.5	⊢ ( 𝜑 → ∃ 𝑥 ∈ No ( 𝐶 ·s 𝑥 ) = 1s )
6	4	gt0ne0sd	⊢ ( 𝜑 → 𝐶 ≠ 0s )
7	2 3 6 5	divsclwd	⊢ ( 𝜑 → ( 𝐵 /su 𝐶 ) ∈ No )
8	1 7 3 4	ltmuls1d	⊢ ( 𝜑 → ( 𝐴 <s ( 𝐵 /su 𝐶 ) ↔ ( 𝐴 ·s 𝐶 ) <s ( ( 𝐵 /su 𝐶 ) ·s 𝐶 ) ) )
9	2 3 6 5	divscan1wd	⊢ ( 𝜑 → ( ( 𝐵 /su 𝐶 ) ·s 𝐶 ) = 𝐵 )
10	9	breq2d	⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐶 ) <s ( ( 𝐵 /su 𝐶 ) ·s 𝐶 ) ↔ ( 𝐴 ·s 𝐶 ) <s 𝐵 ) )
11	8 10	bitr2d	⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐶 ) <s 𝐵 ↔ 𝐴 <s ( 𝐵 /su 𝐶 ) ) )