Theorem sn-ltmulgt11d

Description: ltmulgt11d without ax-mulcom . (Contributed by SN, 26-Jun-2024)

	Ref	Expression
Hypotheses	sn-ltmulgt11d.a	$$⊢\phi \to A\in ℝ$$
	sn-ltmulgt11d.b	$$⊢\phi \to B\in ℝ$$
	sn-ltmulgt11d.1	$$⊢\phi \to 0<B$$
Assertion	sn-ltmulgt11d	$$⊢\phi \to \left(1<A\leftrightarrow B<BA\right)$$

Proof

Step	Hyp	Ref	Expression
1		sn-ltmulgt11d.a	$$⊢\phi \to A\in ℝ$$
2		sn-ltmulgt11d.b	$$⊢\phi \to B\in ℝ$$
3		sn-ltmulgt11d.1	$$⊢\phi \to 0<B$$
4		1red	$$⊢\phi \to 1\in ℝ$$
5	4 1 2 3	sn-ltmul2d	$$⊢\phi \to \left(B\cdot 1<BA\leftrightarrow 1<A\right)$$
6		ax-1rid	$$⊢B\in ℝ\to B\cdot 1=B$$
7	2 6	syl	$$⊢\phi \to B\cdot 1=B$$
8	7	breq1d	$$⊢\phi \to \left(B\cdot 1<BA\leftrightarrow B<BA\right)$$
9	5 8	bitr3d	$$⊢\phi \to \left(1<A\leftrightarrow B<BA\right)$$