Theorem relogbcld

Description: Closure of the general logarithm with a positive real base on positive reals, a deduction version. (Contributed by metakunt, 22-May-2024)

	Ref	Expression
Hypotheses	relogbcld.1	$$⊢\phi \to B\in ℝ$$
	relogbcld.2	$$⊢\phi \to 0<B$$
	relogbcld.3	$$⊢\phi \to X\in ℝ$$
	relogbcld.4	$$⊢\phi \to 0<X$$
	relogbcld.5	$$⊢\phi \to B\ne 1$$
Assertion	relogbcld	$$⊢\phi \to {\log }_{B}X\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		relogbcld.1	$$⊢\phi \to B\in ℝ$$
2		relogbcld.2	$$⊢\phi \to 0<B$$
3		relogbcld.3	$$⊢\phi \to X\in ℝ$$
4		relogbcld.4	$$⊢\phi \to 0<X$$
5		relogbcld.5	$$⊢\phi \to B\ne 1$$
6	1 2	elrpd	$$⊢\phi \to B\in {ℝ}^{+}$$
7	3 4	elrpd	$$⊢\phi \to X\in {ℝ}^{+}$$
8	6 7 5	3jca	$$⊢\phi \to \left(B\in {ℝ}^{+}\wedge X\in {ℝ}^{+}\wedge B\ne 1\right)$$
9		relogbcl	$$⊢\left(B\in {ℝ}^{+}\wedge X\in {ℝ}^{+}\wedge B\ne 1\right)\to {\log }_{B}X\in ℝ$$
10	8 9	syl	$$⊢\phi \to {\log }_{B}X\in ℝ$$