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	$$⊢\phi \to A\in \mathrm{No}$$
	mulnegs1d.2	$$⊢\phi \to B\in \mathrm{No}$$
Assertion	mulnegs1d	Could not format assertion : No typesetting found for |- ( ph -> ( ( -us ` A ) x.s B ) = ( -us ` ( A x.s B ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		mulnegs1d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		mulnegs1d.2	$$⊢\phi \to B\in \mathrm{No}$$
3	1	negsidd	Could not format ( ph -> ( A +s ( -us ` A ) ) = 0s ) : No typesetting found for |- ( ph -> ( A +s ( -us ` A ) ) = 0s ) with typecode |-
4	3	oveq1d	Could not format ( ph -> ( ( A +s ( -us ` A ) ) x.s B ) = ( 0s x.s B ) ) : No typesetting found for |- ( ph -> ( ( A +s ( -us ` A ) ) x.s B ) = ( 0s x.s B ) ) with typecode |-
5	1	negscld	Could not format ( ph -> ( -us ` A ) e. No ) : No typesetting found for |- ( ph -> ( -us ` A ) e. No ) with typecode |-
6	1 5 2	addsdird	Could not format ( ph -> ( ( A +s ( -us ` A ) ) x.s B ) = ( ( A x.s B ) +s ( ( -us ` A ) x.s B ) ) ) : No typesetting found for |- ( ph -> ( ( A +s ( -us ` A ) ) x.s B ) = ( ( A x.s B ) +s ( ( -us ` A ) x.s B ) ) ) with typecode |-
7		muls02	Could not format ( B e. No -> ( 0s x.s B ) = 0s ) : No typesetting found for |- ( B e. No -> ( 0s x.s B ) = 0s ) with typecode |-
8	2 7	syl	Could not format ( ph -> ( 0s x.s B ) = 0s ) : No typesetting found for |- ( ph -> ( 0s x.s B ) = 0s ) with typecode |-
9	4 6 8	3eqtr3d	Could not format ( ph -> ( ( A x.s B ) +s ( ( -us ` A ) x.s B ) ) = 0s ) : No typesetting found for |- ( ph -> ( ( A x.s B ) +s ( ( -us ` A ) x.s B ) ) = 0s ) with typecode |-
10	1 2	mulscld	Could not format ( ph -> ( A x.s B ) e. No ) : No typesetting found for |- ( ph -> ( A x.s B ) e. No ) with typecode |-
11	10	negsidd	Could not format ( ph -> ( ( A x.s B ) +s ( -us ` ( A x.s B ) ) ) = 0s ) : No typesetting found for |- ( ph -> ( ( A x.s B ) +s ( -us ` ( A x.s B ) ) ) = 0s ) with typecode |-
12	9 11	eqtr4d	Could not format ( ph -> ( ( A x.s B ) +s ( ( -us ` A ) x.s B ) ) = ( ( A x.s B ) +s ( -us ` ( A x.s B ) ) ) ) : No typesetting found for |- ( ph -> ( ( A x.s B ) +s ( ( -us ` A ) x.s B ) ) = ( ( A x.s B ) +s ( -us ` ( A x.s B ) ) ) ) with typecode |-
13	5 2	mulscld	Could not format ( ph -> ( ( -us ` A ) x.s B ) e. No ) : No typesetting found for |- ( ph -> ( ( -us ` A ) x.s B ) e. No ) with typecode |-
14	10	negscld	Could not format ( ph -> ( -us ` ( A x.s B ) ) e. No ) : No typesetting found for |- ( ph -> ( -us ` ( A x.s B ) ) e. No ) with typecode |-
15	13 14 10	addscan1d	Could not format ( 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 ) ) ) ) : No typesetting found for |- ( 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 ) ) ) ) with typecode |-
16	12 15	mpbid	Could not format ( ph -> ( ( -us ` A ) x.s B ) = ( -us ` ( A x.s B ) ) ) : No typesetting found for |- ( ph -> ( ( -us ` A ) x.s B ) = ( -us ` ( A x.s B ) ) ) with typecode |-