Metamath Proof Explorer

Theorem regt1loggt0

Description: The natural logarithm for a real number greater than 1 is greater than 0. (Contributed by AV, 25-May-2020)

Ref Expression
Assertion regt1loggt0 
|- ( B e. ( 1 (,) +oo ) -> 0 < ( log ` B ) )

Proof

Step Hyp Ref Expression
1 1xr 
 |-  1 e. RR*
2 elioopnf 
 |-  ( 1 e. RR* -> ( B e. ( 1 (,) +oo ) <-> ( B e. RR /\ 1 < B ) ) )
3 1 2 ax-mp 
 |-  ( B e. ( 1 (,) +oo ) <-> ( B e. RR /\ 1 < B ) )
4 3 simprbi 
 |-  ( B e. ( 1 (,) +oo ) -> 1 < B )
5 log1 
 |-  ( log ` 1 ) = 0
6 5 eqcomi 
 |-  0 = ( log ` 1 )
7 6 a1i 
 |-  ( B e. ( 1 (,) +oo ) -> 0 = ( log ` 1 ) )
8 7 breq1d 
 |-  ( B e. ( 1 (,) +oo ) -> ( 0 < ( log ` B ) <-> ( log ` 1 ) < ( log ` B ) ) )
9 1rp 
 |-  1 e. RR+
10 0lt1 
 |-  0 < 1
11 0red 
 |-  ( B e. RR -> 0 e. RR )
12 1red 
 |-  ( B e. RR -> 1 e. RR )
13 id 
 |-  ( B e. RR -> B e. RR )
14 lttr 
 |-  ( ( 0 e. RR /\ 1 e. RR /\ B e. RR ) -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
15 11 12 13 14 syl3anc 
 |-  ( B e. RR -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
16 10 15 mpani 
 |-  ( B e. RR -> ( 1 < B -> 0 < B ) )
17 16 imdistani 
 |-  ( ( B e. RR /\ 1 < B ) -> ( B e. RR /\ 0 < B ) )
18 elrp 
 |-  ( B e. RR+ <-> ( B e. RR /\ 0 < B ) )
19 17 3 18 3imtr4i 
 |-  ( B e. ( 1 (,) +oo ) -> B e. RR+ )
20 logltb 
 |-  ( ( 1 e. RR+ /\ B e. RR+ ) -> ( 1 < B <-> ( log ` 1 ) < ( log ` B ) ) )
21 9 19 20 sylancr 
 |-  ( B e. ( 1 (,) +oo ) -> ( 1 < B <-> ( log ` 1 ) < ( log ` B ) ) )
22 8 21 bitr4d 
 |-  ( B e. ( 1 (,) +oo ) -> ( 0 < ( log ` B ) <-> 1 < B ) )
23 4 22 mpbird 
 |-  ( B e. ( 1 (,) +oo ) -> 0 < ( log ` B ) )