Theorem abs3lemd

Description: Lemma involving absolute value of differences. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	abscld.1	|- ( ph -> A e. CC )
	abssubd.2	|- ( ph -> B e. CC )
	abs3difd.3	|- ( ph -> C e. CC )
	abs3lemd.4	|- ( ph -> D e. RR )
	abs3lemd.5	|- ( ph -> ( abs ` ( A - C ) ) < ( D / 2 ) )
	abs3lemd.6	|- ( ph -> ( abs ` ( C - B ) ) < ( D / 2 ) )
Assertion	abs3lemd	|- ( ph -> ( abs ` ( A - B ) ) < D )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	|- ( ph -> A e. CC )
2		abssubd.2	|- ( ph -> B e. CC )
3		abs3difd.3	|- ( ph -> C e. CC )
4		abs3lemd.4	|- ( ph -> D e. RR )
5		abs3lemd.5	|- ( ph -> ( abs ` ( A - C ) ) < ( D / 2 ) )
6		abs3lemd.6	|- ( ph -> ( abs ` ( C - B ) ) < ( D / 2 ) )
7		abs3lem	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) -> ( ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) -> ( abs ` ( A - B ) ) < D ) )
8	1 2 3 4 7	syl22anc	|- ( ph -> ( ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) -> ( abs ` ( A - B ) ) < D ) )
9	5 6 8	mp2and	|- ( ph -> ( abs ` ( A - B ) ) < D )