Theorem mulnegs1d

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	|- ( ph -> A e. No )
	mulnegs1d.2	|- ( ph -> B e. No )
Assertion	mulnegs1d	|- ( ph -> ( ( -us ` A ) x.s B ) = ( -us ` ( A x.s B ) ) )

Proof

Step	Hyp	Ref	Expression
1		mulnegs1d.1	|- ( ph -> A e. No )
2		mulnegs1d.2	|- ( ph -> B e. No )
3	1	negsidd	|- ( ph -> ( A +s ( -us ` A ) ) = 0s )
4	3	oveq1d	|- ( ph -> ( ( A +s ( -us ` A ) ) x.s B ) = ( 0s x.s B ) )
5	1	negscld	|- ( ph -> ( -us ` A ) e. No )
6	1 5 2	addsdird	|- ( ph -> ( ( A +s ( -us ` A ) ) x.s B ) = ( ( A x.s B ) +s ( ( -us ` A ) x.s B ) ) )
7		muls02	|- ( B e. No -> ( 0s x.s B ) = 0s )
8	2 7	syl	|- ( ph -> ( 0s x.s B ) = 0s )
9	4 6 8	3eqtr3d	|- ( ph -> ( ( A x.s B ) +s ( ( -us ` A ) x.s B ) ) = 0s )
10	1 2	mulscld	|- ( ph -> ( A x.s B ) e. No )
11	10	negsidd	|- ( ph -> ( ( A x.s B ) +s ( -us ` ( A x.s B ) ) ) = 0s )
12	9 11	eqtr4d	|- ( ph -> ( ( A x.s B ) +s ( ( -us ` A ) x.s B ) ) = ( ( A x.s B ) +s ( -us ` ( A x.s B ) ) ) )
13	5 2	mulscld	|- ( ph -> ( ( -us ` A ) x.s B ) e. No )
14	10	negscld	|- ( ph -> ( -us ` ( A x.s B ) ) e. No )
15	13 14 10	addscan1d	|- ( ph -> ( ( ( A x.s B ) +s ( ( -us ` A ) x.s B ) ) = ( ( A x.s B ) +s ( -us ` ( A x.s B ) ) ) <-> ( ( -us ` A ) x.s B ) = ( -us ` ( A x.s B ) ) ) )
16	12 15	mpbid	|- ( ph -> ( ( -us ` A ) x.s B ) = ( -us ` ( A x.s B ) ) )