Theorem int-sqgeq0d

Description: SquareGEQZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020)

	Ref	Expression
Hypotheses	int-sqgeq0d.1	|- ( ph -> A e. RR )
	int-sqgeq0d.2	|- ( ph -> B e. RR )
	int-sqgeq0d.3	|- ( ph -> A = B )
Assertion	int-sqgeq0d	|- ( ph -> 0 <_ ( A x. B ) )

Proof

Step	Hyp	Ref	Expression
1		int-sqgeq0d.1	|- ( ph -> A e. RR )
2		int-sqgeq0d.2	|- ( ph -> B e. RR )
3		int-sqgeq0d.3	|- ( ph -> A = B )
4	1	sqge0d	|- ( ph -> 0 <_ ( A ^ 2 ) )
5	3	oveq1d	|- ( ph -> ( A ^ 2 ) = ( B ^ 2 ) )
6	2	recnd	|- ( ph -> B e. CC )
7	6	sqvald	|- ( ph -> ( B ^ 2 ) = ( B x. B ) )
8		eqcom	|- ( A = B <-> B = A )
9	8	imbi2i	|- ( ( ph -> A = B ) <-> ( ph -> B = A ) )
10	3 9	mpbi	|- ( ph -> B = A )
11	10	oveq1d	|- ( ph -> ( B x. B ) = ( A x. B ) )
12	7 11	eqtrd	|- ( ph -> ( B ^ 2 ) = ( A x. B ) )
13	5 12	eqtrd	|- ( ph -> ( A ^ 2 ) = ( A x. B ) )
14	4 13	breqtrd	|- ( ph -> 0 <_ ( A x. B ) )