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Metamath Proof Explorer

Theorem relogbval

Description: Value of the general logarithm with integer base. (Contributed by Thierry Arnoux, 27-Sep-2017)

Ref Expression
Assertion relogbval 
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )

Proof

Step Hyp Ref Expression
1 zgt1rpn0n1 
 |-  ( B e. ( ZZ>= ` 2 ) -> ( B e. RR+ /\ B =/= 0 /\ B =/= 1 ) )
2 1 adantr 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> ( B e. RR+ /\ B =/= 0 /\ B =/= 1 ) )
3 2 simp1d 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> B e. RR+ )
4 3 rpcnd 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> B e. CC )
5 2 simp2d 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> B =/= 0 )
6 2 simp3d 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> B =/= 1 )
7 eldifpr 
 |-  ( B e. ( CC \ { 0 , 1 } ) <-> ( B e. CC /\ B =/= 0 /\ B =/= 1 ) )
8 4 5 6 7 syl3anbrc 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> B e. ( CC \ { 0 , 1 } ) )
9 simpr 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> X e. RR+ )
10 9 rpcnne0d 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> ( X e. CC /\ X =/= 0 ) )
11 eldifsn 
 |-  ( X e. ( CC \ { 0 } ) <-> ( X e. CC /\ X =/= 0 ) )
12 10 11 sylibr 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> X e. ( CC \ { 0 } ) )
13 logbval 
 |-  ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )
14 8 12 13 syl2anc 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )