Theorem sltsubposd

Description: Subtraction of a positive number decreases the sum. (Contributed by Scott Fenton, 15-Apr-2025)

	Ref	Expression
Hypotheses	sltsubpos.1	|- ( ph -> A e. No )
	sltsubpos.2	|- ( ph -> B e. No )
Assertion	sltsubposd	|- ( ph -> ( 0s ( B -s A )

Proof

Step	Hyp	Ref	Expression
1		sltsubpos.1	|- ( ph -> A e. No )
2		sltsubpos.2	|- ( ph -> B e. No )
3		0sno	|- 0s e. No
4	3	a1i	|- ( ph -> 0s e. No )
5	4 1 2	sltsub2d	|- ( ph -> ( 0s ( B -s A )
6		subsid1	|- ( B e. No -> ( B -s 0s ) = B )
7	2 6	syl	|- ( ph -> ( B -s 0s ) = B )
8	7	breq2d	|- ( ph -> ( ( B -s A ) ( B -s A )
9	5 8	bitrd	|- ( ph -> ( 0s ( B -s A )