Theorem sn-ltmul2d

Description: ltmul2d without ax-mulcom . (Contributed by SN, 26-Jun-2024)

	Ref	Expression
Hypotheses	sn-ltmul2d.a	|- ( ph -> A e. RR )
	sn-ltmul2d.b	|- ( ph -> B e. RR )
	sn-ltmul2d.c	|- ( ph -> C e. RR )
	sn-ltmul2d.1	|- ( ph -> 0 < C )
Assertion	sn-ltmul2d	|- ( ph -> ( ( C x. A ) < ( C x. B ) <-> A < B ) )

Proof

Step	Hyp	Ref	Expression
1		sn-ltmul2d.a	|- ( ph -> A e. RR )
2		sn-ltmul2d.b	|- ( ph -> B e. RR )
3		sn-ltmul2d.c	|- ( ph -> C e. RR )
4		sn-ltmul2d.1	|- ( ph -> 0 < C )
5		rersubcl	|- ( ( B e. RR /\ A e. RR ) -> ( B -R A ) e. RR )
6	2 1 5	syl2anc	|- ( ph -> ( B -R A ) e. RR )
7	3 6 4	mulgt0b2d	|- ( ph -> ( 0 < ( B -R A ) <-> 0 < ( C x. ( B -R A ) ) ) )
8		resubdi	|- ( ( C e. RR /\ B e. RR /\ A e. RR ) -> ( C x. ( B -R A ) ) = ( ( C x. B ) -R ( C x. A ) ) )
9	3 2 1 8	syl3anc	|- ( ph -> ( C x. ( B -R A ) ) = ( ( C x. B ) -R ( C x. A ) ) )
10	9	breq2d	|- ( ph -> ( 0 < ( C x. ( B -R A ) ) <-> 0 < ( ( C x. B ) -R ( C x. A ) ) ) )
11	7 10	bitr2d	|- ( ph -> ( 0 < ( ( C x. B ) -R ( C x. A ) ) <-> 0 < ( B -R A ) ) )
12	3 1	remulcld	|- ( ph -> ( C x. A ) e. RR )
13	3 2	remulcld	|- ( ph -> ( C x. B ) e. RR )
14		reposdif	|- ( ( ( C x. A ) e. RR /\ ( C x. B ) e. RR ) -> ( ( C x. A ) < ( C x. B ) <-> 0 < ( ( C x. B ) -R ( C x. A ) ) ) )
15	12 13 14	syl2anc	|- ( ph -> ( ( C x. A ) < ( C x. B ) <-> 0 < ( ( C x. B ) -R ( C x. A ) ) ) )
16		reposdif	|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> 0 < ( B -R A ) ) )
17	1 2 16	syl2anc	|- ( ph -> ( A < B <-> 0 < ( B -R A ) ) )
18	11 15 17	3bitr4d	|- ( ph -> ( ( C x. A ) < ( C x. B ) <-> A < B ) )