Theorem cutpos

Description: Reduce the elements of a cut for a positive number. (Contributed by Scott Fenton, 13-Mar-2025)

	Ref	Expression
Hypotheses	cutpos.1	|- ( ph -> A e. No )
	cutpos.2	|- ( ph -> 0s
Assertion	cutpos	|- ( ph -> A = ( ( { 0s } u. { x e. ( _Left ` A ) | 0s

Proof

Step	Hyp	Ref	Expression
1		cutpos.1	|- ( ph -> A e. No )
2		cutpos.2	|- ( ph -> 0s
3		lltropt	|- ( _Left ` A ) <
4	3	a1i	|- ( ph -> ( _Left ` A ) <
5		lrcut	|- ( A e. No -> ( ( _Left ` A ) |s ( _Right ` A ) ) = A )
6	1 5	syl	|- ( ph -> ( ( _Left ` A ) |s ( _Right ` A ) ) = A )
7	6	eqcomd	|- ( ph -> A = ( ( _Left ` A ) |s ( _Right ` A ) ) )
8	1 2	0elleft	|- ( ph -> 0s e. ( _Left ` A ) )
9	4 7 8	cutlt	|- ( ph -> A = ( ( { 0s } u. { x e. ( _Left ` A ) | 0s