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	$$⊢\phi \to A\in \mathrm{No}$$
	mulnegs1d.2	$$⊢\phi \to B\in \mathrm{No}$$
Assertion	mulnegs2d	Could not format assertion : No typesetting found for |- ( ph -> ( A x.s ( -us ` 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	2 1	mulnegs1d	Could not format ( ph -> ( ( -us ` B ) x.s A ) = ( -us ` ( B x.s A ) ) ) : No typesetting found for |- ( ph -> ( ( -us ` B ) x.s A ) = ( -us ` ( B x.s A ) ) ) with typecode |-
4	2	negscld	Could not format ( ph -> ( -us ` B ) e. No ) : No typesetting found for |- ( ph -> ( -us ` B ) e. No ) with typecode |-
5	1 4	mulscomd	Could not format ( ph -> ( A x.s ( -us ` B ) ) = ( ( -us ` B ) x.s A ) ) : No typesetting found for |- ( ph -> ( A x.s ( -us ` B ) ) = ( ( -us ` B ) x.s A ) ) with typecode |-
6	1 2	mulscomd	Could not format ( ph -> ( A x.s B ) = ( B x.s A ) ) : No typesetting found for |- ( ph -> ( A x.s B ) = ( B x.s A ) ) with typecode |-
7	6	fveq2d	Could not format ( ph -> ( -us ` ( A x.s B ) ) = ( -us ` ( B x.s A ) ) ) : No typesetting found for |- ( ph -> ( -us ` ( A x.s B ) ) = ( -us ` ( B x.s A ) ) ) with typecode |-
8	3 5 7	3eqtr4d	Could not format ( ph -> ( A x.s ( -us ` B ) ) = ( -us ` ( A x.s B ) ) ) : No typesetting found for |- ( ph -> ( A x.s ( -us ` B ) ) = ( -us ` ( A x.s B ) ) ) with typecode |-