Theorem sltmul1d

Description: Multiplication of both sides of surreal less-than by a positive number. (Contributed by Scott Fenton, 10-Mar-2025)

	Ref	Expression
Hypotheses	sltmul12d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	sltmul12d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	sltmul12d.3	⊢ ( 𝜑 → 𝐶 ∈ No )
	sltmul12d.4	⊢ ( 𝜑 → 0s <s 𝐶 )
Assertion	sltmul1d	⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ( 𝐴 ·s 𝐶 ) <s ( 𝐵 ·s 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sltmul12d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		sltmul12d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		sltmul12d.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		sltmul12d.4	⊢ ( 𝜑 → 0s <s 𝐶 )
5	1 2 3 4	sltmul2d	⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ( 𝐶 ·s 𝐴 ) <s ( 𝐶 ·s 𝐵 ) ) )
6	1 3	mulscomd	⊢ ( 𝜑 → ( 𝐴 ·s 𝐶 ) = ( 𝐶 ·s 𝐴 ) )
7	2 3	mulscomd	⊢ ( 𝜑 → ( 𝐵 ·s 𝐶 ) = ( 𝐶 ·s 𝐵 ) )
8	6 7	breq12d	⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐶 ) <s ( 𝐵 ·s 𝐶 ) ↔ ( 𝐶 ·s 𝐴 ) <s ( 𝐶 ·s 𝐵 ) ) )
9	5 8	bitr4d	⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ( 𝐴 ·s 𝐶 ) <s ( 𝐵 ·s 𝐶 ) ) )