Theorem sn-mulgt1d

Description: mulgt1d without ax-mulcom . (Contributed by SN, 26-Jun-2024)

	Ref	Expression
Hypotheses	sn-mulgt1d.a	|- ( ph -> A e. RR )
	sn-mulgt1d.b	|- ( ph -> B e. RR )
	sn-mulgt1d.1	|- ( ph -> 1 < A )
	sn-mulgt1d.2	|- ( ph -> 1 < B )
Assertion	sn-mulgt1d	|- ( ph -> 1 < ( A x. B ) )

Proof

Step	Hyp	Ref	Expression
1		sn-mulgt1d.a	|- ( ph -> A e. RR )
2		sn-mulgt1d.b	|- ( ph -> B e. RR )
3		sn-mulgt1d.1	|- ( ph -> 1 < A )
4		sn-mulgt1d.2	|- ( ph -> 1 < B )
5		1red	|- ( ph -> 1 e. RR )
6	1 2	remulcld	|- ( ph -> ( A x. B ) e. RR )
7		0red	|- ( ph -> 0 e. RR )
8		sn-0lt1	|- 0 < 1
9	8	a1i	|- ( ph -> 0 < 1 )
10	7 5 1 9 3	lttrd	|- ( ph -> 0 < A )
11	2 1 10	sn-ltmulgt11d	|- ( ph -> ( 1 < B <-> A < ( A x. B ) ) )
12	4 11	mpbid	|- ( ph -> A < ( A x. B ) )
13	5 1 6 3 12	lttrd	|- ( ph -> 1 < ( A x. B ) )