Theorem dnibndlem1

Description: Lemma for dnibnd . (Contributed by Asger C. Ipsen, 4-Apr-2021)

	Ref	Expression
Hypotheses	dnibndlem1.1	|- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
	dnibndlem1.2	|- ( ph -> A e. RR )
	dnibndlem1.3	|- ( ph -> B e. RR )
Assertion	dnibndlem1	|- ( ph -> ( ( abs ` ( ( T ` B ) - ( T ` A ) ) ) <_ S <-> ( abs ` ( ( abs ` ( ( |_ ` ( B + ( 1 / 2 ) ) ) - B ) ) - ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) ) ) <_ S ) )

Proof

Step	Hyp	Ref	Expression
1		dnibndlem1.1	|- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
2		dnibndlem1.2	|- ( ph -> A e. RR )
3		dnibndlem1.3	|- ( ph -> B e. RR )
4	1	dnival	|- ( B e. RR -> ( T ` B ) = ( abs ` ( ( |_ ` ( B + ( 1 / 2 ) ) ) - B ) ) )
5	3 4	syl	|- ( ph -> ( T ` B ) = ( abs ` ( ( |_ ` ( B + ( 1 / 2 ) ) ) - B ) ) )
6	1	dnival	|- ( A e. RR -> ( T ` A ) = ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) )
7	2 6	syl	|- ( ph -> ( T ` A ) = ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) )
8	5 7	oveq12d	|- ( ph -> ( ( T ` B ) - ( T ` A ) ) = ( ( abs ` ( ( |_ ` ( B + ( 1 / 2 ) ) ) - B ) ) - ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) ) )
9	8	fveq2d	|- ( ph -> ( abs ` ( ( T ` B ) - ( T ` A ) ) ) = ( abs ` ( ( abs ` ( ( |_ ` ( B + ( 1 / 2 ) ) ) - B ) ) - ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) ) ) )
10	9	breq1d	|- ( ph -> ( ( abs ` ( ( T ` B ) - ( T ` A ) ) ) <_ S <-> ( abs ` ( ( abs ` ( ( |_ ` ( B + ( 1 / 2 ) ) ) - B ) ) - ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) ) ) <_ S ) )