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Metamath Proof Explorer

Theorem logbrec

Description: Logarithm of a reciprocal changes sign. See logrec . Particular case of Property 3 of Cohen4 p. 361. (Contributed by Thierry Arnoux, 27-Sep-2017)

Ref Expression
Assertion logbrec 
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( B logb ( 1 / A ) ) = -u ( B logb A ) )

Proof

Step Hyp Ref Expression
1 simpr 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> A e. RR+ )
2 1 rpreccld 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( 1 / A ) e. RR+ )
3 relogbval 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ ( 1 / A ) e. RR+ ) -> ( B logb ( 1 / A ) ) = ( ( log ` ( 1 / A ) ) / ( log ` B ) ) )
4 2 3 syldan 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( B logb ( 1 / A ) ) = ( ( log ` ( 1 / A ) ) / ( log ` B ) ) )
5 relogbval 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( B logb A ) = ( ( log ` A ) / ( log ` B ) ) )
6 5 negeqd 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> -u ( B logb A ) = -u ( ( log ` A ) / ( log ` B ) ) )
7 1 rpcnd 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> A e. CC )
8 1 rpne0d 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> A =/= 0 )
9 7 8 logcld 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( log ` A ) e. CC )
10 zgt1rpn0n1 
 |-  ( B e. ( ZZ>= ` 2 ) -> ( B e. RR+ /\ B =/= 0 /\ B =/= 1 ) )
11 10 simp1d 
 |-  ( B e. ( ZZ>= ` 2 ) -> B e. RR+ )
12 11 adantr 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> B e. RR+ )
13 12 relogcld 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( log ` B ) e. RR )
14 13 recnd 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( log ` B ) e. CC )
15 10 simp3d 
 |-  ( B e. ( ZZ>= ` 2 ) -> B =/= 1 )
16 15 adantr 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> B =/= 1 )
17 logne0 
 |-  ( ( B e. RR+ /\ B =/= 1 ) -> ( log ` B ) =/= 0 )
18 12 16 17 syl2anc 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( log ` B ) =/= 0 )
19 9 14 18 divnegd 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> -u ( ( log ` A ) / ( log ` B ) ) = ( -u ( log ` A ) / ( log ` B ) ) )
20 7 8 reccld 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( 1 / A ) e. CC )
21 7 8 recne0d 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( 1 / A ) =/= 0 )
22 20 21 logcld 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( log ` ( 1 / A ) ) e. CC )
23 1 relogcld 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( log ` A ) e. RR )
24 23 reim0d 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( Im ` ( log ` A ) ) = 0 )
25 0re 
 |-  0 e. RR
26 pipos 
 |-  0 < _pi
27 25 26 gtneii 
 |-  _pi =/= 0
28 27 a1i 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> _pi =/= 0 )
29 28 necomd 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> 0 =/= _pi )
30 24 29 eqnetrd 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( Im ` ( log ` A ) ) =/= _pi )
31 logrec 
 |-  ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= _pi ) -> ( log ` A ) = -u ( log ` ( 1 / A ) ) )
32 7 8 30 31 syl3anc 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( log ` A ) = -u ( log ` ( 1 / A ) ) )
33 32 eqcomd 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> -u ( log ` ( 1 / A ) ) = ( log ` A ) )
34 22 33 negcon1ad 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> -u ( log ` A ) = ( log ` ( 1 / A ) ) )
35 34 oveq1d 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( -u ( log ` A ) / ( log ` B ) ) = ( ( log ` ( 1 / A ) ) / ( log ` B ) ) )
36 6 19 35 3eqtrd 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> -u ( B logb A ) = ( ( log ` ( 1 / A ) ) / ( log ` B ) ) )
37 4 36 eqtr4d 
 |-  ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( B logb ( 1 / A ) ) = -u ( B logb A ) )