Theorem mulscan1d

Description: Cancellation of surreal multiplication when the left term is non-zero. (Contributed by Scott Fenton, 10-Mar-2025)

	Ref	Expression
Hypotheses	mulscan2d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	mulscan2d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	mulscan2d.3	⊢ ( 𝜑 → 𝐶 ∈ No )
	mulscan2d.4	⊢ ( 𝜑 → 𝐶 ≠ 0s )
Assertion	mulscan1d	⊢ ( 𝜑 → ( ( 𝐶 ·s 𝐴 ) = ( 𝐶 ·s 𝐵 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		mulscan2d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		mulscan2d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		mulscan2d.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		mulscan2d.4	⊢ ( 𝜑 → 𝐶 ≠ 0s )
5	1 3	mulscomd	⊢ ( 𝜑 → ( 𝐴 ·s 𝐶 ) = ( 𝐶 ·s 𝐴 ) )
6	2 3	mulscomd	⊢ ( 𝜑 → ( 𝐵 ·s 𝐶 ) = ( 𝐶 ·s 𝐵 ) )
7	5 6	eqeq12d	⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐶 ) = ( 𝐵 ·s 𝐶 ) ↔ ( 𝐶 ·s 𝐴 ) = ( 𝐶 ·s 𝐵 ) ) )
8	1 2 3 4	mulscan2d	⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐶 ) = ( 𝐵 ·s 𝐶 ) ↔ 𝐴 = 𝐵 ) )
9	7 8	bitr3d	⊢ ( 𝜑 → ( ( 𝐶 ·s 𝐴 ) = ( 𝐶 ·s 𝐵 ) ↔ 𝐴 = 𝐵 ) )