Metamath Proof Explorer

Theorem rege1logbrege0

Description: The general logarithm, with a real base greater than 1, for a real number greater than or equal to 1 is greater than or equal to 0. (Contributed by AV, 25-May-2020)

Ref Expression
Assertion rege1logbrege0 
|- ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> 0 <_ ( B logb X ) )

Proof

Step Hyp Ref Expression
1 1re 
 |-  1 e. RR
2 elicopnf 
 |-  ( 1 e. RR -> ( X e. ( 1 [,) +oo ) <-> ( X e. RR /\ 1 <_ X ) ) )
3 1 2 ax-mp 
 |-  ( X e. ( 1 [,) +oo ) <-> ( X e. RR /\ 1 <_ X ) )
4 3 bilani 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> ( X e. RR /\ 1 <_ X ) )
5 logge0 
 |-  ( ( X e. RR /\ 1 <_ X ) -> 0 <_ ( log ` X ) )
6 4 5 syl 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> 0 <_ ( log ` X ) )
7 simpl 
 |-  ( ( X e. RR /\ 1 <_ X ) -> X e. RR )
8 0lt1 
 |-  0 < 1
9 0red 
 |-  ( X e. RR -> 0 e. RR )
10 1red 
 |-  ( X e. RR -> 1 e. RR )
11 id 
 |-  ( X e. RR -> X e. RR )
12 ltletr 
 |-  ( ( 0 e. RR /\ 1 e. RR /\ X e. RR ) -> ( ( 0 < 1 /\ 1 <_ X ) -> 0 < X ) )
13 9 10 11 12 syl3anc 
 |-  ( X e. RR -> ( ( 0 < 1 /\ 1 <_ X ) -> 0 < X ) )
14 8 13 mpani 
 |-  ( X e. RR -> ( 1 <_ X -> 0 < X ) )
15 14 imp 
 |-  ( ( X e. RR /\ 1 <_ X ) -> 0 < X )
16 7 15 elrpd 
 |-  ( ( X e. RR /\ 1 <_ X ) -> X e. RR+ )
17 3 16 sylbi 
 |-  ( X e. ( 1 [,) +oo ) -> X e. RR+ )
18 17 relogcld 
 |-  ( X e. ( 1 [,) +oo ) -> ( log ` X ) e. RR )
19 18 adantl 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> ( log ` X ) e. RR )
20 1xr 
 |-  1 e. RR*
21 elioopnf 
 |-  ( 1 e. RR* -> ( B e. ( 1 (,) +oo ) <-> ( B e. RR /\ 1 < B ) ) )
22 20 21 ax-mp 
 |-  ( B e. ( 1 (,) +oo ) <-> ( B e. RR /\ 1 < B ) )
23 simpl 
 |-  ( ( B e. RR /\ 1 < B ) -> B e. RR )
24 0red 
 |-  ( B e. RR -> 0 e. RR )
25 1red 
 |-  ( B e. RR -> 1 e. RR )
26 id 
 |-  ( B e. RR -> B e. RR )
27 lttr 
 |-  ( ( 0 e. RR /\ 1 e. RR /\ B e. RR ) -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
28 24 25 26 27 syl3anc 
 |-  ( B e. RR -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
29 8 28 mpani 
 |-  ( B e. RR -> ( 1 < B -> 0 < B ) )
30 29 imp 
 |-  ( ( B e. RR /\ 1 < B ) -> 0 < B )
31 23 30 elrpd 
 |-  ( ( B e. RR /\ 1 < B ) -> B e. RR+ )
32 22 31 sylbi 
 |-  ( B e. ( 1 (,) +oo ) -> B e. RR+ )
33 32 relogcld 
 |-  ( B e. ( 1 (,) +oo ) -> ( log ` B ) e. RR )
34 33 adantr 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> ( log ` B ) e. RR )
35 regt1loggt0 
 |-  ( B e. ( 1 (,) +oo ) -> 0 < ( log ` B ) )
36 35 adantr 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> 0 < ( log ` B ) )
37 ge0div 
 |-  ( ( ( log ` X ) e. RR /\ ( log ` B ) e. RR /\ 0 < ( log ` B ) ) -> ( 0 <_ ( log ` X ) <-> 0 <_ ( ( log ` X ) / ( log ` B ) ) ) )
38 19 34 36 37 syl3anc 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> ( 0 <_ ( log ` X ) <-> 0 <_ ( ( log ` X ) / ( log ` B ) ) ) )
39 6 38 mpbid 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> 0 <_ ( ( log ` X ) / ( log ` B ) ) )
40 recn 
 |-  ( B e. RR -> B e. CC )
41 40 adantr 
 |-  ( ( B e. RR /\ 1 < B ) -> B e. CC )
42 30 gt0ne0d 
 |-  ( ( B e. RR /\ 1 < B ) -> B =/= 0 )
43 25 26 ltlend 
 |-  ( B e. RR -> ( 1 < B <-> ( 1 <_ B /\ B =/= 1 ) ) )
44 43 simplbda 
 |-  ( ( B e. RR /\ 1 < B ) -> B =/= 1 )
45 41 42 44 3jca 
 |-  ( ( B e. RR /\ 1 < B ) -> ( B e. CC /\ B =/= 0 /\ B =/= 1 ) )
46 eldifpr 
 |-  ( B e. ( CC \ { 0 , 1 } ) <-> ( B e. CC /\ B =/= 0 /\ B =/= 1 ) )
47 45 22 46 3imtr4i 
 |-  ( B e. ( 1 (,) +oo ) -> B e. ( CC \ { 0 , 1 } ) )
48 recn 
 |-  ( X e. RR -> X e. CC )
49 48 adantr 
 |-  ( ( X e. RR /\ 1 <_ X ) -> X e. CC )
50 15 gt0ne0d 
 |-  ( ( X e. RR /\ 1 <_ X ) -> X =/= 0 )
51 49 50 jca 
 |-  ( ( X e. RR /\ 1 <_ X ) -> ( X e. CC /\ X =/= 0 ) )
52 eldifsn 
 |-  ( X e. ( CC \ { 0 } ) <-> ( X e. CC /\ X =/= 0 ) )
53 51 3 52 3imtr4i 
 |-  ( X e. ( 1 [,) +oo ) -> X e. ( CC \ { 0 } ) )
54 logbval 
 |-  ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )
55 47 53 54 syl2an 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )
56 39 55 breqtrrd 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> 0 <_ ( B logb X ) )