Theorem zgt1rpn0n1

Description: An integer greater than 1 is a positive real number not equal to 0 or 1. Useful for working with integer logarithm bases (which is a common case, e.g., base 2, base 3, or base 10). (Contributed by Thierry Arnoux, 26-Sep-2017) (Proof shortened by AV, 9-Jul-2022)

	Ref	Expression
Assertion	zgt1rpn0n1	|- ( B e. ( ZZ>= ` 2 ) -> ( B e. RR+ /\ B =/= 0 /\ B =/= 1 ) )

Proof

Step	Hyp	Ref	Expression
1		eluz2nn	|- ( B e. ( ZZ>= ` 2 ) -> B e. NN )
2	1	nnrpd	|- ( B e. ( ZZ>= ` 2 ) -> B e. RR+ )
3		eluz2n0	|- ( B e. ( ZZ>= ` 2 ) -> B =/= 0 )
4		1nuz2	|- -. 1 e. ( ZZ>= ` 2 )
5		nelne2	|- ( ( B e. ( ZZ>= ` 2 ) /\ -. 1 e. ( ZZ>= ` 2 ) ) -> B =/= 1 )
6	4 5	mpan2	|- ( B e. ( ZZ>= ` 2 ) -> B =/= 1 )
7	2 3 6	3jca	|- ( B e. ( ZZ>= ` 2 ) -> ( B e. RR+ /\ B =/= 0 /\ B =/= 1 ) )