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 id 
 |-  ( ( X e. RR /\ 1 <_ X ) -> ( X e. RR /\ 1 <_ X ) )
5 3 4 sylbi 
 |-  ( X e. ( 1 [,) +oo ) -> ( X e. RR /\ 1 <_ X ) )
6 5 adantl 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> ( X e. RR /\ 1 <_ X ) )
7 logge0 
 |-  ( ( X e. RR /\ 1 <_ X ) -> 0 <_ ( log ` X ) )
8 6 7 syl 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> 0 <_ ( log ` X ) )
9 simpl 
 |-  ( ( X e. RR /\ 1 <_ X ) -> X e. RR )
10 0lt1 
 |-  0 < 1
11 0red 
 |-  ( X e. RR -> 0 e. RR )
12 1red 
 |-  ( X e. RR -> 1 e. RR )
13 id 
 |-  ( X e. RR -> X e. RR )
14 ltletr 
 |-  ( ( 0 e. RR /\ 1 e. RR /\ X e. RR ) -> ( ( 0 < 1 /\ 1 <_ X ) -> 0 < X ) )
15 11 12 13 14 syl3anc 
 |-  ( X e. RR -> ( ( 0 < 1 /\ 1 <_ X ) -> 0 < X ) )
16 10 15 mpani 
 |-  ( X e. RR -> ( 1 <_ X -> 0 < X ) )
17 16 imp 
 |-  ( ( X e. RR /\ 1 <_ X ) -> 0 < X )
18 9 17 elrpd 
 |-  ( ( X e. RR /\ 1 <_ X ) -> X e. RR+ )
19 3 18 sylbi 
 |-  ( X e. ( 1 [,) +oo ) -> X e. RR+ )
20 19 relogcld 
 |-  ( X e. ( 1 [,) +oo ) -> ( log ` X ) e. RR )
21 20 adantl 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> ( log ` X ) e. RR )
22 1xr 
 |-  1 e. RR*
23 elioopnf 
 |-  ( 1 e. RR* -> ( B e. ( 1 (,) +oo ) <-> ( B e. RR /\ 1 < B ) ) )
24 22 23 ax-mp 
 |-  ( B e. ( 1 (,) +oo ) <-> ( B e. RR /\ 1 < B ) )
25 simpl 
 |-  ( ( B e. RR /\ 1 < B ) -> B e. RR )
26 0red 
 |-  ( B e. RR -> 0 e. RR )
27 1red 
 |-  ( B e. RR -> 1 e. RR )
28 id 
 |-  ( B e. RR -> B e. RR )
29 lttr 
 |-  ( ( 0 e. RR /\ 1 e. RR /\ B e. RR ) -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
30 26 27 28 29 syl3anc 
 |-  ( B e. RR -> ( ( 0 < 1 /\ 1 < B ) -> 0 < B ) )
31 10 30 mpani 
 |-  ( B e. RR -> ( 1 < B -> 0 < B ) )
32 31 imp 
 |-  ( ( B e. RR /\ 1 < B ) -> 0 < B )
33 25 32 elrpd 
 |-  ( ( B e. RR /\ 1 < B ) -> B e. RR+ )
34 24 33 sylbi 
 |-  ( B e. ( 1 (,) +oo ) -> B e. RR+ )
35 34 relogcld 
 |-  ( B e. ( 1 (,) +oo ) -> ( log ` B ) e. RR )
36 35 adantr 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> ( log ` B ) e. RR )
37 regt1loggt0 
 |-  ( B e. ( 1 (,) +oo ) -> 0 < ( log ` B ) )
38 37 adantr 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> 0 < ( log ` B ) )
39 ge0div 
 |-  ( ( ( log ` X ) e. RR /\ ( log ` B ) e. RR /\ 0 < ( log ` B ) ) -> ( 0 <_ ( log ` X ) <-> 0 <_ ( ( log ` X ) / ( log ` B ) ) ) )
40 21 36 38 39 syl3anc 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> ( 0 <_ ( log ` X ) <-> 0 <_ ( ( log ` X ) / ( log ` B ) ) ) )
41 8 40 mpbid 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> 0 <_ ( ( log ` X ) / ( log ` B ) ) )
42 recn 
 |-  ( B e. RR -> B e. CC )
43 42 adantr 
 |-  ( ( B e. RR /\ 1 < B ) -> B e. CC )
44 32 gt0ne0d 
 |-  ( ( B e. RR /\ 1 < B ) -> B =/= 0 )
45 27 28 ltlend 
 |-  ( B e. RR -> ( 1 < B <-> ( 1 <_ B /\ B =/= 1 ) ) )
46 45 simplbda 
 |-  ( ( B e. RR /\ 1 < B ) -> B =/= 1 )
47 43 44 46 3jca 
 |-  ( ( B e. RR /\ 1 < B ) -> ( B e. CC /\ B =/= 0 /\ B =/= 1 ) )
48 eldifpr 
 |-  ( B e. ( CC \ { 0 , 1 } ) <-> ( B e. CC /\ B =/= 0 /\ B =/= 1 ) )
49 47 24 48 3imtr4i 
 |-  ( B e. ( 1 (,) +oo ) -> B e. ( CC \ { 0 , 1 } ) )
50 recn 
 |-  ( X e. RR -> X e. CC )
51 50 adantr 
 |-  ( ( X e. RR /\ 1 <_ X ) -> X e. CC )
52 17 gt0ne0d 
 |-  ( ( X e. RR /\ 1 <_ X ) -> X =/= 0 )
53 51 52 jca 
 |-  ( ( X e. RR /\ 1 <_ X ) -> ( X e. CC /\ X =/= 0 ) )
54 eldifsn 
 |-  ( X e. ( CC \ { 0 } ) <-> ( X e. CC /\ X =/= 0 ) )
55 53 3 54 3imtr4i 
 |-  ( X e. ( 1 [,) +oo ) -> X e. ( CC \ { 0 } ) )
56 logbval 
 |-  ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )
57 49 55 56 syl2an 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )
58 41 57 breqtrrd 
 |-  ( ( B e. ( 1 (,) +oo ) /\ X e. ( 1 [,) +oo ) ) -> 0 <_ ( B logb X ) )