Theorem ltmulnegs2d

Description: Multiplication of both sides of surreal less-than by a negative number. (Contributed by Scott Fenton, 14-Mar-2025)

	Ref	Expression
Hypotheses	ltmulnegs.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	ltmulnegs.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	ltmulnegs.3	⊢ ( 𝜑 → 𝐶 ∈ No )
	ltmulnegs.4	⊢ ( 𝜑 → 𝐶 <s 0s )
Assertion	ltmulnegs2d	⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ( 𝐶 ·s 𝐵 ) <s ( 𝐶 ·s 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltmulnegs.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		ltmulnegs.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		ltmulnegs.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		ltmulnegs.4	⊢ ( 𝜑 → 𝐶 <s 0s )
5	1 2 3 4	ltmulnegs1d	⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ( 𝐵 ·s 𝐶 ) <s ( 𝐴 ·s 𝐶 ) ) )
6	2 3	mulscomd	⊢ ( 𝜑 → ( 𝐵 ·s 𝐶 ) = ( 𝐶 ·s 𝐵 ) )
7	1 3	mulscomd	⊢ ( 𝜑 → ( 𝐴 ·s 𝐶 ) = ( 𝐶 ·s 𝐴 ) )
8	6 7	breq12d	⊢ ( 𝜑 → ( ( 𝐵 ·s 𝐶 ) <s ( 𝐴 ·s 𝐶 ) ↔ ( 𝐶 ·s 𝐵 ) <s ( 𝐶 ·s 𝐴 ) ) )
9	5 8	bitrd	⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ( 𝐶 ·s 𝐵 ) <s ( 𝐶 ·s 𝐴 ) ) )