Theorem muls12d

Description: Commutative/associative law for surreal multiplication. (Contributed by Scott Fenton, 14-Mar-2025)

	Ref	Expression
Hypotheses	muls12d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	muls12d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	muls12d.3	⊢ ( 𝜑 → 𝐶 ∈ No )
Assertion	muls12d	⊢ ( 𝜑 → ( 𝐴 ·s ( 𝐵 ·s 𝐶 ) ) = ( 𝐵 ·s ( 𝐴 ·s 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		muls12d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		muls12d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		muls12d.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4	1 2	mulscomd	⊢ ( 𝜑 → ( 𝐴 ·s 𝐵 ) = ( 𝐵 ·s 𝐴 ) )
5	4	oveq1d	⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐵 ) ·s 𝐶 ) = ( ( 𝐵 ·s 𝐴 ) ·s 𝐶 ) )
6	1 2 3	mulsassd	⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐵 ) ·s 𝐶 ) = ( 𝐴 ·s ( 𝐵 ·s 𝐶 ) ) )
7	2 1 3	mulsassd	⊢ ( 𝜑 → ( ( 𝐵 ·s 𝐴 ) ·s 𝐶 ) = ( 𝐵 ·s ( 𝐴 ·s 𝐶 ) ) )
8	5 6 7	3eqtr3d	⊢ ( 𝜑 → ( 𝐴 ·s ( 𝐵 ·s 𝐶 ) ) = ( 𝐵 ·s ( 𝐴 ·s 𝐶 ) ) )