Theorem negsubsdi2d

Description: Distribution of negative over subtraction. (Contributed by Scott Fenton, 5-Feb-2025)

	Ref	Expression
Hypotheses	negsubsdi2d.1	|- ( ph -> A e. No )
	negsubsdi2d.2	|- ( ph -> B e. No )
Assertion	negsubsdi2d	|- ( ph -> ( -us ` ( A -s B ) ) = ( B -s A ) )

Proof

Step	Hyp	Ref	Expression
1		negsubsdi2d.1	|- ( ph -> A e. No )
2		negsubsdi2d.2	|- ( ph -> B e. No )
3	2	negscld	|- ( ph -> ( -us ` B ) e. No )
4		negsdi	|- ( ( A e. No /\ ( -us ` B ) e. No ) -> ( -us ` ( A +s ( -us ` B ) ) ) = ( ( -us ` A ) +s ( -us ` ( -us ` B ) ) ) )
5	1 3 4	syl2anc	|- ( ph -> ( -us ` ( A +s ( -us ` B ) ) ) = ( ( -us ` A ) +s ( -us ` ( -us ` B ) ) ) )
6		negnegs	|- ( B e. No -> ( -us ` ( -us ` B ) ) = B )
7	2 6	syl	|- ( ph -> ( -us ` ( -us ` B ) ) = B )
8	7	oveq2d	|- ( ph -> ( ( -us ` A ) +s ( -us ` ( -us ` B ) ) ) = ( ( -us ` A ) +s B ) )
9	1	negscld	|- ( ph -> ( -us ` A ) e. No )
10	9 2	addscomd	|- ( ph -> ( ( -us ` A ) +s B ) = ( B +s ( -us ` A ) ) )
11	5 8 10	3eqtrd	|- ( ph -> ( -us ` ( A +s ( -us ` B ) ) ) = ( B +s ( -us ` A ) ) )
12	1 2	subsvald	|- ( ph -> ( A -s B ) = ( A +s ( -us ` B ) ) )
13	12	fveq2d	|- ( ph -> ( -us ` ( A -s B ) ) = ( -us ` ( A +s ( -us ` B ) ) ) )
14	2 1	subsvald	|- ( ph -> ( B -s A ) = ( B +s ( -us ` A ) ) )
15	11 13 14	3eqtr4d	|- ( ph -> ( -us ` ( A -s B ) ) = ( B -s A ) )