Theorem mulnegs2d

Description: Product with negative is negative of product. Part of theorem 7 of Conway p. 19. (Contributed by Scott Fenton, 10-Mar-2025)

	Ref	Expression
Hypotheses	mulnegs1d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	mulnegs1d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
Assertion	mulnegs2d	⊢ ( 𝜑 → ( 𝐴 ·s ( -us ‘ 𝐵 ) ) = ( -us ‘ ( 𝐴 ·s 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mulnegs1d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		mulnegs1d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3	2 1	mulnegs1d	⊢ ( 𝜑 → ( ( -us ‘ 𝐵 ) ·s 𝐴 ) = ( -us ‘ ( 𝐵 ·s 𝐴 ) ) )
4	2	negscld	⊢ ( 𝜑 → ( -us ‘ 𝐵 ) ∈ No )
5	1 4	mulscomd	⊢ ( 𝜑 → ( 𝐴 ·s ( -us ‘ 𝐵 ) ) = ( ( -us ‘ 𝐵 ) ·s 𝐴 ) )
6	1 2	mulscomd	⊢ ( 𝜑 → ( 𝐴 ·s 𝐵 ) = ( 𝐵 ·s 𝐴 ) )
7	6	fveq2d	⊢ ( 𝜑 → ( -us ‘ ( 𝐴 ·s 𝐵 ) ) = ( -us ‘ ( 𝐵 ·s 𝐴 ) ) )
8	3 5 7	3eqtr4d	⊢ ( 𝜑 → ( 𝐴 ·s ( -us ‘ 𝐵 ) ) = ( -us ‘ ( 𝐴 ·s 𝐵 ) ) )