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Theorem log2ub 23737
Description:  log 2 is less than  2 5 3  / 
3 6 5. If written in decimal, this is because  log 2  = 0.693147... is less than 253/365 = 0.693151... , so this is a very tight bound, at five decimal places. (Contributed by Mario Carneiro, 7-Apr-2015.)
Assertion
Ref Expression
log2ub  |-  ( log `  2 )  < 
(;; 2 5 3  / ;; 3 6 5 )

Proof of Theorem log2ub
StepHypRef Expression
1 df-4 10659 . . . . . . . . . . 11  |-  4  =  ( 3  +  1 )
21oveq1i 6306 . . . . . . . . . 10  |-  ( 4  -  1 )  =  ( ( 3  +  1 )  -  1 )
3 3cn 10673 . . . . . . . . . . 11  |-  3  e.  CC
4 ax-1cn 9586 . . . . . . . . . . 11  |-  1  e.  CC
53, 4pncan3oi 9880 . . . . . . . . . 10  |-  ( ( 3  +  1 )  -  1 )  =  3
62, 5eqtri 2449 . . . . . . . . 9  |-  ( 4  -  1 )  =  3
76oveq2i 6307 . . . . . . . 8  |-  ( 0 ... ( 4  -  1 ) )  =  ( 0 ... 3
)
87sumeq1i 13731 . . . . . . 7  |-  sum_ n  e.  ( 0 ... (
4  -  1 ) ) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  =  sum_ n  e.  ( 0 ... 3 ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n
) ) )
98oveq2i 6307 . . . . . 6  |-  ( ( log `  2 )  -  sum_ n  e.  ( 0 ... ( 4  -  1 ) ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) ) )  =  ( ( log `  2
)  -  sum_ n  e.  ( 0 ... 3
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) ) )
10 4nn0 10877 . . . . . . 7  |-  4  e.  NN0
11 log2tlbnd 23733 . . . . . . 7  |-  ( 4  e.  NN0  ->  ( ( log `  2 )  -  sum_ n  e.  ( 0 ... ( 4  -  1 ) ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) ) )  e.  ( 0 [,] ( 3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) ) )
1210, 11ax-mp 5 . . . . . 6  |-  ( ( log `  2 )  -  sum_ n  e.  ( 0 ... ( 4  -  1 ) ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) ) )  e.  ( 0 [,] ( 3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) )
139, 12eqeltrri 2505 . . . . 5  |-  ( ( log `  2 )  -  sum_ n  e.  ( 0 ... 3 ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) ) )  e.  ( 0 [,] ( 3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) )
14 0re 9632 . . . . . 6  |-  0  e.  RR
15 3re 10672 . . . . . . 7  |-  3  e.  RR
16 4nn 10758 . . . . . . . . 9  |-  4  e.  NN
17 2nn0 10875 . . . . . . . . . 10  |-  2  e.  NN0
18 1nn 10609 . . . . . . . . . 10  |-  1  e.  NN
1917, 10, 18numnncl 11048 . . . . . . . . 9  |-  ( ( 2  x.  4 )  +  1 )  e.  NN
2016, 19nnmulcli 10622 . . . . . . . 8  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  e.  NN
21 9nn 10763 . . . . . . . . 9  |-  9  e.  NN
22 nnexpcl 12271 . . . . . . . . 9  |-  ( ( 9  e.  NN  /\  4  e.  NN0 )  -> 
( 9 ^ 4 )  e.  NN )
2321, 10, 22mp2an 676 . . . . . . . 8  |-  ( 9 ^ 4 )  e.  NN
2420, 23nnmulcli 10622 . . . . . . 7  |-  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) )  e.  NN
25 nndivre 10634 . . . . . . 7  |-  ( ( 3  e.  RR  /\  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  (
9 ^ 4 ) )  e.  NN )  ->  ( 3  / 
( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  (
9 ^ 4 ) ) )  e.  RR )
2615, 24, 25mp2an 676 . . . . . 6  |-  ( 3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) )  e.  RR
2714, 26elicc2i 11689 . . . . 5  |-  ( ( ( log `  2
)  -  sum_ n  e.  ( 0 ... 3
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) ) )  e.  ( 0 [,] (
3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) )  <->  ( ( ( log `  2 )  -  sum_ n  e.  ( 0 ... 3 ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) ) )  e.  RR  /\  0  <_  ( ( log `  2 )  -  sum_ n  e.  ( 0 ... 3 ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) ) )  /\  ( ( log `  2 )  -  sum_ n  e.  ( 0 ... 3 ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) ) )  <_  (
3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) ) )
2813, 27mpbi 211 . . . 4  |-  ( ( ( log `  2
)  -  sum_ n  e.  ( 0 ... 3
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) ) )  e.  RR  /\  0  <_ 
( ( log `  2
)  -  sum_ n  e.  ( 0 ... 3
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) ) )  /\  ( ( log `  2
)  -  sum_ n  e.  ( 0 ... 3
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) ) )  <_ 
( 3  /  (
( 4  x.  (
( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) )
2928simp3i 1016 . . 3  |-  ( ( log `  2 )  -  sum_ n  e.  ( 0 ... 3 ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) ) )  <_  (
3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) )
30 2rp 11296 . . . . 5  |-  2  e.  RR+
31 relogcl 23387 . . . . 5  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
3230, 31ax-mp 5 . . . 4  |-  ( log `  2 )  e.  RR
33 fzfid 12172 . . . . . 6  |-  ( T. 
->  ( 0 ... 3
)  e.  Fin )
34 2re 10668 . . . . . . 7  |-  2  e.  RR
35 3nn 10757 . . . . . . . . 9  |-  3  e.  NN
36 elfznn0 11874 . . . . . . . . . . . 12  |-  ( n  e.  ( 0 ... 3 )  ->  n  e.  NN0 )
3736adantl 467 . . . . . . . . . . 11  |-  ( ( T.  /\  n  e.  ( 0 ... 3
) )  ->  n  e.  NN0 )
38 nn0mulcl 10895 . . . . . . . . . . 11  |-  ( ( 2  e.  NN0  /\  n  e.  NN0 )  -> 
( 2  x.  n
)  e.  NN0 )
3917, 37, 38sylancr 667 . . . . . . . . . 10  |-  ( ( T.  /\  n  e.  ( 0 ... 3
) )  ->  (
2  x.  n )  e.  NN0 )
40 nn0p1nn 10898 . . . . . . . . . 10  |-  ( ( 2  x.  n )  e.  NN0  ->  ( ( 2  x.  n )  +  1 )  e.  NN )
4139, 40syl 17 . . . . . . . . 9  |-  ( ( T.  /\  n  e.  ( 0 ... 3
) )  ->  (
( 2  x.  n
)  +  1 )  e.  NN )
42 nnmulcl 10621 . . . . . . . . 9  |-  ( ( 3  e.  NN  /\  ( ( 2  x.  n )  +  1 )  e.  NN )  ->  ( 3  x.  ( ( 2  x.  n )  +  1 ) )  e.  NN )
4335, 41, 42sylancr 667 . . . . . . . 8  |-  ( ( T.  /\  n  e.  ( 0 ... 3
) )  ->  (
3  x.  ( ( 2  x.  n )  +  1 ) )  e.  NN )
44 nnexpcl 12271 . . . . . . . . 9  |-  ( ( 9  e.  NN  /\  n  e.  NN0 )  -> 
( 9 ^ n
)  e.  NN )
4521, 37, 44sylancr 667 . . . . . . . 8  |-  ( ( T.  /\  n  e.  ( 0 ... 3
) )  ->  (
9 ^ n )  e.  NN )
4643, 45nnmulcld 10646 . . . . . . 7  |-  ( ( T.  /\  n  e.  ( 0 ... 3
) )  ->  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) )  e.  NN )
47 nndivre 10634 . . . . . . 7  |-  ( ( 2  e.  RR  /\  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) )  e.  NN )  ->  ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  e.  RR )
4834, 46, 47sylancr 667 . . . . . 6  |-  ( ( T.  /\  n  e.  ( 0 ... 3
) )  ->  (
2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) )  e.  RR )
4933, 48fsumrecl 13767 . . . . 5  |-  ( T. 
->  sum_ n  e.  ( 0 ... 3 ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) )  e.  RR )
5049trud 1446 . . . 4  |-  sum_ n  e.  ( 0 ... 3
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  e.  RR
5132, 50, 26lesubadd2i 10163 . . 3  |-  ( ( ( log `  2
)  -  sum_ n  e.  ( 0 ... 3
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) ) )  <_ 
( 3  /  (
( 4  x.  (
( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) )  <->  ( log `  2
)  <_  ( sum_ n  e.  ( 0 ... 3 ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n
) ) )  +  ( 3  /  (
( 4  x.  (
( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) ) )
5229, 51mpbi 211 . 2  |-  ( log `  2 )  <_ 
( sum_ n  e.  ( 0 ... 3 ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) )  +  ( 3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) )
53 log2ublem3 23736 . . . . 5  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  sum_ n  e.  ( 0 ... 3 ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) ) )  <_ ;;;; 5 3 0 5 6
54 3nn0 10876 . . . . 5  |-  3  e.  NN0
55 5nn0 10878 . . . . . . . . 9  |-  5  e.  NN0
5655, 54deccl 11054 . . . . . . . 8  |- ; 5 3  e.  NN0
57 0nn0 10873 . . . . . . . 8  |-  0  e.  NN0
5856, 57deccl 11054 . . . . . . 7  |- ;; 5 3 0  e.  NN0
5958, 55deccl 11054 . . . . . 6  |- ;;; 5 3 0 5  e.  NN0
60 6nn0 10879 . . . . . 6  |-  6  e.  NN0
6159, 60deccl 11054 . . . . 5  |- ;;;; 5 3 0 5 6  e.  NN0
62 1nn0 10874 . . . . 5  |-  1  e.  NN0
63 eqid 2420 . . . . 5  |-  ( sum_ n  e.  ( 0 ... 3 ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n
) ) )  +  ( 3  /  (
( 4  x.  (
( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) )  =  (
sum_ n  e.  (
0 ... 3 ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) )  +  ( 3  / 
( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  (
9 ^ 4 ) ) ) )
64 6p1e7 10727 . . . . . 6  |-  ( 6  +  1 )  =  7
65 eqid 2420 . . . . . 6  |- ;;;; 5 3 0 5 6  = ;;;; 5 3 0 5 6
6659, 60, 64, 65decsuc 11063 . . . . 5  |-  (;;;; 5 3 0 5 6  +  1 )  = ;;;; 5 3 0 5 7
67 5nn 10759 . . . . . . . . . 10  |-  5  e.  NN
68 7nn 10761 . . . . . . . . . 10  |-  7  e.  NN
6967, 68nnmulcli 10622 . . . . . . . . 9  |-  ( 5  x.  7 )  e.  NN
7069nnrei 10607 . . . . . . . 8  |-  ( 5  x.  7 )  e.  RR
7120nnrei 10607 . . . . . . . 8  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  e.  RR
72 6nn 10760 . . . . . . . . . 10  |-  6  e.  NN
73 5lt6 10775 . . . . . . . . . 10  |-  5  <  6
7454, 55, 72, 73declt 11061 . . . . . . . . 9  |- ; 3 5  < ; 3 6
75 7cn 10682 . . . . . . . . . 10  |-  7  e.  CC
76 5cn 10678 . . . . . . . . . 10  |-  5  e.  CC
77 7t5e35 11125 . . . . . . . . . 10  |-  ( 7  x.  5 )  = ; 3
5
7875, 76, 77mulcomli 9639 . . . . . . . . 9  |-  ( 5  x.  7 )  = ; 3
5
79 4cn 10676 . . . . . . . . . . . . . 14  |-  4  e.  CC
80 2cn 10669 . . . . . . . . . . . . . 14  |-  2  e.  CC
81 4t2e8 10752 . . . . . . . . . . . . . 14  |-  ( 4  x.  2 )  =  8
8279, 80, 81mulcomli 9639 . . . . . . . . . . . . 13  |-  ( 2  x.  4 )  =  8
8382oveq1i 6306 . . . . . . . . . . . 12  |-  ( ( 2  x.  4 )  +  1 )  =  ( 8  +  1 )
84 8p1e9 10729 . . . . . . . . . . . 12  |-  ( 8  +  1 )  =  9
8583, 84eqtri 2449 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  +  1 )  =  9
8685oveq2i 6307 . . . . . . . . . 10  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  =  ( 4  x.  9 )
87 9cn 10686 . . . . . . . . . . 11  |-  9  e.  CC
88 9t4e36 11137 . . . . . . . . . . 11  |-  ( 9  x.  4 )  = ; 3
6
8987, 79, 88mulcomli 9639 . . . . . . . . . 10  |-  ( 4  x.  9 )  = ; 3
6
9086, 89eqtri 2449 . . . . . . . . 9  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  = ; 3
6
9174, 78, 903brtr4i 4445 . . . . . . . 8  |-  ( 5  x.  7 )  < 
( 4  x.  (
( 2  x.  4 )  +  1 ) )
9270, 71, 91ltleii 9746 . . . . . . 7  |-  ( 5  x.  7 )  <_ 
( 4  x.  (
( 2  x.  4 )  +  1 ) )
9323nngt0i 10632 . . . . . . . 8  |-  0  <  ( 9 ^ 4 )
9423nnrei 10607 . . . . . . . . 9  |-  ( 9 ^ 4 )  e.  RR
9570, 71, 94lemul2i 10519 . . . . . . . 8  |-  ( 0  <  ( 9 ^ 4 )  ->  (
( 5  x.  7 )  <_  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  <->  ( (
9 ^ 4 )  x.  ( 5  x.  7 ) )  <_ 
( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) ) ) )
9693, 95ax-mp 5 . . . . . . 7  |-  ( ( 5  x.  7 )  <_  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  <->  ( (
9 ^ 4 )  x.  ( 5  x.  7 ) )  <_ 
( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) ) )
9792, 96mpbi 211 . . . . . 6  |-  ( ( 9 ^ 4 )  x.  ( 5  x.  7 ) )  <_ 
( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) )
98 7nn0 10880 . . . . . . . . . 10  |-  7  e.  NN0
99 nnexpcl 12271 . . . . . . . . . 10  |-  ( ( 3  e.  NN  /\  7  e.  NN0 )  -> 
( 3 ^ 7 )  e.  NN )
10035, 98, 99mp2an 676 . . . . . . . . 9  |-  ( 3 ^ 7 )  e.  NN
101100nncni 10608 . . . . . . . 8  |-  ( 3 ^ 7 )  e.  CC
10269nncni 10608 . . . . . . . 8  |-  ( 5  x.  7 )  e.  CC
103101, 102, 3mul32i 9818 . . . . . . 7  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  3 )  =  ( ( ( 3 ^ 7 )  x.  3 )  x.  (
5  x.  7 ) )
10479, 80mulcomi 9638 . . . . . . . . . . . 12  |-  ( 4  x.  2 )  =  ( 2  x.  4 )
105 df-8 10663 . . . . . . . . . . . 12  |-  8  =  ( 7  +  1 )
10681, 104, 1053eqtr3i 2457 . . . . . . . . . . 11  |-  ( 2  x.  4 )  =  ( 7  +  1 )
107106oveq2i 6307 . . . . . . . . . 10  |-  ( 3 ^ ( 2  x.  4 ) )  =  ( 3 ^ (
7  +  1 ) )
108 expmul 12303 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  2  e.  NN0  /\  4  e.  NN0 )  ->  (
3 ^ ( 2  x.  4 ) )  =  ( ( 3 ^ 2 ) ^
4 ) )
1093, 17, 10, 108mp3an 1360 . . . . . . . . . 10  |-  ( 3 ^ ( 2  x.  4 ) )  =  ( ( 3 ^ 2 ) ^ 4 )
110107, 109eqtr3i 2451 . . . . . . . . 9  |-  ( 3 ^ ( 7  +  1 ) )  =  ( ( 3 ^ 2 ) ^ 4 )
111 expp1 12265 . . . . . . . . . 10  |-  ( ( 3  e.  CC  /\  7  e.  NN0 )  -> 
( 3 ^ (
7  +  1 ) )  =  ( ( 3 ^ 7 )  x.  3 ) )
1123, 98, 111mp2an 676 . . . . . . . . 9  |-  ( 3 ^ ( 7  +  1 ) )  =  ( ( 3 ^ 7 )  x.  3 )
113 sq3 12358 . . . . . . . . . 10  |-  ( 3 ^ 2 )  =  9
114113oveq1i 6306 . . . . . . . . 9  |-  ( ( 3 ^ 2 ) ^ 4 )  =  ( 9 ^ 4 )
115110, 112, 1143eqtr3i 2457 . . . . . . . 8  |-  ( ( 3 ^ 7 )  x.  3 )  =  ( 9 ^ 4 )
116115oveq1i 6306 . . . . . . 7  |-  ( ( ( 3 ^ 7 )  x.  3 )  x.  ( 5  x.  7 ) )  =  ( ( 9 ^ 4 )  x.  (
5  x.  7 ) )
117103, 116eqtri 2449 . . . . . 6  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  3 )  =  ( ( 9 ^ 4 )  x.  (
5  x.  7 ) )
11820nncni 10608 . . . . . . . . 9  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  e.  CC
11923nncni 10608 . . . . . . . . 9  |-  ( 9 ^ 4 )  e.  CC
120118, 119mulcomi 9638 . . . . . . . 8  |-  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) )  =  ( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) )
121120oveq1i 6306 . . . . . . 7  |-  ( ( ( 4  x.  (
( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) )  x.  1 )  =  ( ( ( 9 ^ 4 )  x.  ( 4  x.  (
( 2  x.  4 )  +  1 ) ) )  x.  1 )
122119, 118mulcli 9637 . . . . . . . 8  |-  ( ( 9 ^ 4 )  x.  ( 4  x.  ( ( 2  x.  4 )  +  1 ) ) )  e.  CC
123122mulid1i 9634 . . . . . . 7  |-  ( ( ( 9 ^ 4 )  x.  ( 4  x.  ( ( 2  x.  4 )  +  1 ) ) )  x.  1 )  =  ( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) )
124121, 123eqtri 2449 . . . . . 6  |-  ( ( ( 4  x.  (
( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) )  x.  1 )  =  ( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) )
12597, 117, 1243brtr4i 4445 . . . . 5  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  3 )  <_ 
( ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) )  x.  1 )
12653, 50, 54, 24, 61, 62, 63, 66, 125log2ublem1 23734 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( sum_ n  e.  ( 0 ... 3
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  +  ( 3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) ) )  <_ ;;;; 5 3 0 5 7
12750, 26readdcli 9645 . . . . 5  |-  ( sum_ n  e.  ( 0 ... 3 ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n
) ) )  +  ( 3  /  (
( 4  x.  (
( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) )  e.  RR
12859, 98deccl 11054 . . . . . 6  |- ;;;; 5 3 0 5 7  e.  NN0
129128nn0rei 10869 . . . . 5  |- ;;;; 5 3 0 5 7  e.  RR
130100, 69nnmulcli 10622 . . . . . . 7  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  NN
131130nnrei 10607 . . . . . 6  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR
132130nngt0i 10632 . . . . . 6  |-  0  <  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) )
133131, 132pm3.2i 456 . . . . 5  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR  /\  0  <  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) ) )
134 lemuldiv2 10476 . . . . 5  |-  ( ( ( sum_ n  e.  ( 0 ... 3 ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) )  +  ( 3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) )  e.  RR  /\ ;;;; 5 3 0 5 7  e.  RR  /\  (
( ( 3 ^ 7 )  x.  (
5  x.  7 ) )  e.  RR  /\  0  <  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) ) )  ->  ( ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  ( sum_ n  e.  ( 0 ... 3
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  +  ( 3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) ) )  <_ ;;;; 5 3 0 5 7  <-> 
( sum_ n  e.  ( 0 ... 3 ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) )  +  ( 3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) )  <_  (;;;; 5 3 0 5 7  /  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) ) ) )
135127, 129, 133, 134mp3an 1360 . . . 4  |-  ( ( ( ( 3 ^ 7 )  x.  (
5  x.  7 ) )  x.  ( sum_ n  e.  ( 0 ... 3 ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n
) ) )  +  ( 3  /  (
( 4  x.  (
( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) ) )  <_ ;;;; 5 3 0 5 7  <->  ( sum_ n  e.  ( 0 ... 3
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  +  ( 3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) )  <_  (;;;; 5 3 0 5 7  /  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) ) )
136126, 135mpbi 211 . . 3  |-  ( sum_ n  e.  ( 0 ... 3 ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n
) ) )  +  ( 3  /  (
( 4  x.  (
( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) )  <_  (;;;; 5 3 0 5 7  /  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )
137 8nn0 10881 . . . . . . . . . . . . 13  |-  8  e.  NN0
13854, 137deccl 11054 . . . . . . . . . . . 12  |- ; 3 8  e.  NN0
139138, 98deccl 11054 . . . . . . . . . . 11  |- ;; 3 8 7  e.  NN0
140139, 54deccl 11054 . . . . . . . . . 10  |- ;;; 3 8 7 3  e.  NN0
141140, 62deccl 11054 . . . . . . . . 9  |- ;;;; 3 8 7 3 1  e.  NN0
142141, 60deccl 11054 . . . . . . . 8  |- ;;;;; 3 8 7 3 1 6  e.  NN0
143141, 98deccl 11054 . . . . . . . 8  |- ;;;;; 3 8 7 3 1 7  e.  NN0
144 1lt10 10809 . . . . . . . 8  |-  1  <  10
145 6lt7 10780 . . . . . . . . 9  |-  6  <  7
146141, 60, 68, 145declt 11061 . . . . . . . 8  |- ;;;;; 3 8 7 3 1 6  < ;;;;; 3 8 7 3 1 7
147142, 143, 62, 98, 144, 146decltc 11062 . . . . . . 7  |- ;;;;;; 3 8 7 3 1 6 1  < ;;;;;; 3 8 7 3 1 7 7
148 eqid 2420 . . . . . . . 8  |- ; 7 3  = ; 7 3
14962, 55deccl 11054 . . . . . . . . . . 11  |- ; 1 5  e.  NN0
150 9nn0 10882 . . . . . . . . . . 11  |-  9  e.  NN0
151149, 150deccl 11054 . . . . . . . . . 10  |- ;; 1 5 9  e.  NN0
152151, 62deccl 11054 . . . . . . . . 9  |- ;;; 1 5 9 1  e.  NN0
153152, 98deccl 11054 . . . . . . . 8  |- ;;;; 1 5 9 1 7  e.  NN0
154 eqid 2420 . . . . . . . . 9  |- ;;;; 5 3 0 5 7  = ;;;; 5 3 0 5 7
155 eqid 2420 . . . . . . . . 9  |- ;;;; 1 5 9 1 7  = ;;;; 1 5 9 1 7
156 eqid 2420 . . . . . . . . . 10  |- ;;; 5 3 0 5  = ;;; 5 3 0 5
157 eqid 2420 . . . . . . . . . . 11  |- ;;; 1 5 9 1  = ;;; 1 5 9 1
158 5p1e6 10726 . . . . . . . . . . . 12  |-  ( 5  +  1 )  =  6
15976, 4, 158addcomli 9814 . . . . . . . . . . 11  |-  ( 1  +  5 )  =  6
160151, 62, 55, 157, 159decaddi 11084 . . . . . . . . . 10  |-  (;;; 1 5 9 1  +  5 )  = ;;; 1 5 9 6
16162, 60deccl 11054 . . . . . . . . . . 11  |- ; 1 6  e.  NN0
162 eqid 2420 . . . . . . . . . . 11  |- ;; 5 3 0  = ;; 5 3 0
163 eqid 2420 . . . . . . . . . . . 12  |- ;; 1 5 9  = ;; 1 5 9
164 eqid 2420 . . . . . . . . . . . . 13  |- ; 1 5  = ; 1 5
16562, 55, 158, 164decsuc 11063 . . . . . . . . . . . 12  |-  (; 1 5  +  1 )  = ; 1 6
166 9p4e13 11104 . . . . . . . . . . . 12  |-  ( 9  +  4 )  = ; 1
3
167149, 150, 10, 163, 165, 54, 166decaddci 11085 . . . . . . . . . . 11  |-  (;; 1 5 9  +  4 )  = ;; 1 6 3
168 eqid 2420 . . . . . . . . . . . 12  |- ; 5 3  = ; 5 3
169161nn0cni 10870 . . . . . . . . . . . . 13  |- ; 1 6  e.  CC
170169addid1i 9809 . . . . . . . . . . . 12  |-  (; 1 6  +  0 )  = ; 1 6
171 1p2e3 10723 . . . . . . . . . . . . . 14  |-  ( 1  +  2 )  =  3
172171oveq2i 6307 . . . . . . . . . . . . 13  |-  ( ( 5  x.  7 )  +  ( 1  +  2 ) )  =  ( ( 5  x.  7 )  +  3 )
173 5p3e8 10737 . . . . . . . . . . . . . 14  |-  ( 5  +  3 )  =  8
17454, 55, 54, 78, 173decaddi 11084 . . . . . . . . . . . . 13  |-  ( ( 5  x.  7 )  +  3 )  = ; 3
8
175172, 174eqtri 2449 . . . . . . . . . . . 12  |-  ( ( 5  x.  7 )  +  ( 1  +  2 ) )  = ; 3
8
176 7t3e21 11123 . . . . . . . . . . . . . 14  |-  ( 7  x.  3 )  = ; 2
1
17775, 3, 176mulcomli 9639 . . . . . . . . . . . . 13  |-  ( 3  x.  7 )  = ; 2
1
178 6cn 10680 . . . . . . . . . . . . . 14  |-  6  e.  CC
179178, 4, 64addcomli 9814 . . . . . . . . . . . . 13  |-  ( 1  +  6 )  =  7
18017, 62, 60, 177, 179decaddi 11084 . . . . . . . . . . . 12  |-  ( ( 3  x.  7 )  +  6 )  = ; 2
7
18155, 54, 62, 60, 168, 170, 98, 98, 17, 175, 180decmac 11079 . . . . . . . . . . 11  |-  ( (; 5
3  x.  7 )  +  (; 1 6  +  0 ) )  = ;; 3 8 7
18275mul02i 9811 . . . . . . . . . . . . 13  |-  ( 0  x.  7 )  =  0
183182oveq1i 6306 . . . . . . . . . . . 12  |-  ( ( 0  x.  7 )  +  3 )  =  ( 0  +  3 )
1843addid2i 9810 . . . . . . . . . . . . 13  |-  ( 0  +  3 )  =  3
18554dec0h 11056 . . . . . . . . . . . . 13  |-  3  = ; 0 3
186184, 185eqtri 2449 . . . . . . . . . . . 12  |-  ( 0  +  3 )  = ; 0
3
187183, 186eqtri 2449 . . . . . . . . . . 11  |-  ( ( 0  x.  7 )  +  3 )  = ; 0
3
18856, 57, 161, 54, 162, 167, 98, 54, 57, 181, 187decmac 11079 . . . . . . . . . 10  |-  ( (;; 5 3 0  x.  7 )  +  (;; 1 5 9  +  4 ) )  = ;;; 3 8 7 3
189 3p1e4 10724 . . . . . . . . . . 11  |-  ( 3  +  1 )  =  4
190 6p5e11 11090 . . . . . . . . . . . 12  |-  ( 6  +  5 )  = ; 1
1
191178, 76, 190addcomli 9814 . . . . . . . . . . 11  |-  ( 5  +  6 )  = ; 1
1
19254, 55, 60, 78, 189, 62, 191decaddci 11085 . . . . . . . . . 10  |-  ( ( 5  x.  7 )  +  6 )  = ; 4
1
19358, 55, 151, 60, 156, 160, 98, 62, 10, 188, 192decmac 11079 . . . . . . . . 9  |-  ( (;;; 5 3 0 5  x.  7 )  +  (;;; 1 5 9 1  +  5 ) )  = ;;;; 3 8 7 3 1
194 7t7e49 11127 . . . . . . . . . 10  |-  ( 7  x.  7 )  = ; 4
9
195 4p1e5 10725 . . . . . . . . . 10  |-  ( 4  +  1 )  =  5
196 9p7e16 11107 . . . . . . . . . 10  |-  ( 9  +  7 )  = ; 1
6
19710, 150, 98, 194, 195, 60, 196decaddci 11085 . . . . . . . . 9  |-  ( ( 7  x.  7 )  +  7 )  = ; 5
6
19859, 98, 152, 98, 154, 155, 98, 60, 55, 193, 197decmac 11079 . . . . . . . 8  |-  ( (;;;; 5 3 0 5 7  x.  7 )  + ;;;; 1 5 9 1 7 )  = ;;;;; 3 8 7 3 1 6
19917dec0h 11056 . . . . . . . . . 10  |-  2  = ; 0 2
2004addid2i 9810 . . . . . . . . . . . 12  |-  ( 0  +  1 )  =  1
20162dec0h 11056 . . . . . . . . . . . 12  |-  1  = ; 0 1
202200, 201eqtri 2449 . . . . . . . . . . 11  |-  ( 0  +  1 )  = ; 0
1
203 00id 9797 . . . . . . . . . . . . 13  |-  ( 0  +  0 )  =  0
20457dec0h 11056 . . . . . . . . . . . . 13  |-  0  = ; 0 0
205203, 204eqtri 2449 . . . . . . . . . . . 12  |-  ( 0  +  0 )  = ; 0
0
206 5t3e15 11114 . . . . . . . . . . . . . 14  |-  ( 5  x.  3 )  = ; 1
5
207206oveq1i 6306 . . . . . . . . . . . . 13  |-  ( ( 5  x.  3 )  +  0 )  =  (; 1 5  +  0 )
208149nn0cni 10870 . . . . . . . . . . . . . 14  |- ; 1 5  e.  CC
209208addid1i 9809 . . . . . . . . . . . . 13  |-  (; 1 5  +  0 )  = ; 1 5
210207, 209eqtri 2449 . . . . . . . . . . . 12  |-  ( ( 5  x.  3 )  +  0 )  = ; 1
5
211 3t3e9 10751 . . . . . . . . . . . . . 14  |-  ( 3  x.  3 )  =  9
212211oveq1i 6306 . . . . . . . . . . . . 13  |-  ( ( 3  x.  3 )  +  0 )  =  ( 9  +  0 )
21387addid1i 9809 . . . . . . . . . . . . 13  |-  ( 9  +  0 )  =  9
214212, 213eqtri 2449 . . . . . . . . . . . 12  |-  ( ( 3  x.  3 )  +  0 )  =  9
21555, 54, 57, 57, 168, 205, 54, 210, 214decma 11078 . . . . . . . . . . 11  |-  ( (; 5
3  x.  3 )  +  ( 0  +  0 ) )  = ;; 1 5 9
2163mul02i 9811 . . . . . . . . . . . . 13  |-  ( 0  x.  3 )  =  0
217216oveq1i 6306 . . . . . . . . . . . 12  |-  ( ( 0  x.  3 )  +  1 )  =  ( 0  +  1 )
218217, 202eqtri 2449 . . . . . . . . . . 11  |-  ( ( 0  x.  3 )  +  1 )  = ; 0
1
21956, 57, 57, 62, 162, 202, 54, 62, 57, 215, 218decmac 11079 . . . . . . . . . 10  |-  ( (;; 5 3 0  x.  3 )  +  ( 0  +  1 ) )  = ;;; 1 5 9 1
220 5p2e7 10736 . . . . . . . . . . 11  |-  ( 5  +  2 )  =  7
22162, 55, 17, 206, 220decaddi 11084 . . . . . . . . . 10  |-  ( ( 5  x.  3 )  +  2 )  = ; 1
7
22258, 55, 57, 17, 156, 199, 54, 98, 62, 219, 221decmac 11079 . . . . . . . . 9  |-  ( (;;; 5 3 0 5  x.  3 )  +  2 )  = ;;;; 1 5 9 1 7
22354, 59, 98, 154, 62, 17, 222, 176decmul1c 11087 . . . . . . . 8  |-  (;;;; 5 3 0 5 7  x.  3 )  = ;;;;; 1 5 9 1 7 1
224128, 98, 54, 148, 62, 153, 198, 223decmul2c 11088 . . . . . . 7  |-  (;;;; 5 3 0 5 7  x. ; 7 3 )  = ;;;;;; 3 8 7 3 1 6 1
22555, 55deccl 11054 . . . . . . . . . . 11  |- ; 5 5  e.  NN0
226225, 54deccl 11054 . . . . . . . . . 10  |- ;; 5 5 3  e.  NN0
227226, 54deccl 11054 . . . . . . . . 9  |- ;;; 5 5 3 3  e.  NN0
228227, 62deccl 11054 . . . . . . . 8  |- ;;;; 5 5 3 3 1  e.  NN0
22917, 55deccl 11054 . . . . . . . . . 10  |- ; 2 5  e.  NN0
230229, 54deccl 11054 . . . . . . . . 9  |- ;; 2 5 3  e.  NN0
23117, 62deccl 11054 . . . . . . . . . 10  |- ; 2 1  e.  NN0
232231, 137deccl 11054 . . . . . . . . 9  |- ;; 2 1 8  e.  NN0
23398, 17deccl 11054 . . . . . . . . . . 11  |- ; 7 2  e.  NN0
234 3t2e6 10750 . . . . . . . . . . . . 13  |-  ( 3  x.  2 )  =  6
2353, 80, 234mulcomli 9639 . . . . . . . . . . . 12  |-  ( 2  x.  3 )  =  6
236 3exp3 15014 . . . . . . . . . . . 12  |-  ( 3 ^ 3 )  = ; 2
7
23717, 98deccl 11054 . . . . . . . . . . . . 13  |- ; 2 7  e.  NN0
238 eqid 2420 . . . . . . . . . . . . 13  |- ; 2 7  = ; 2 7
23962, 137deccl 11054 . . . . . . . . . . . . 13  |- ; 1 8  e.  NN0
240 eqid 2420 . . . . . . . . . . . . . 14  |- ; 1 8  = ; 1 8
241 2t2e4 10748 . . . . . . . . . . . . . . . 16  |-  ( 2  x.  2 )  =  4
242241, 171oveq12i 6308 . . . . . . . . . . . . . . 15  |-  ( ( 2  x.  2 )  +  ( 1  +  2 ) )  =  ( 4  +  3 )
243 4p3e7 10734 . . . . . . . . . . . . . . 15  |-  ( 4  +  3 )  =  7
244242, 243eqtri 2449 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  2 )  +  ( 1  +  2 ) )  =  7
245 7t2e14 11122 . . . . . . . . . . . . . . 15  |-  ( 7  x.  2 )  = ; 1
4
246 1p1e2 10712 . . . . . . . . . . . . . . 15  |-  ( 1  +  1 )  =  2
247 8cn 10684 . . . . . . . . . . . . . . . 16  |-  8  e.  CC
248 8p4e12 11097 . . . . . . . . . . . . . . . 16  |-  ( 8  +  4 )  = ; 1
2
249247, 79, 248addcomli 9814 . . . . . . . . . . . . . . 15  |-  ( 4  +  8 )  = ; 1
2
25062, 10, 137, 245, 246, 17, 249decaddci 11085 . . . . . . . . . . . . . 14  |-  ( ( 7  x.  2 )  +  8 )  = ; 2
2
25117, 98, 62, 137, 238, 240, 17, 17, 17, 244, 250decmac 11079 . . . . . . . . . . . . 13  |-  ( (; 2
7  x.  2 )  + ; 1 8 )  = ; 7
2
25275, 80, 245mulcomli 9639 . . . . . . . . . . . . . . 15  |-  ( 2  x.  7 )  = ; 1
4
253 4p4e8 10735 . . . . . . . . . . . . . . 15  |-  ( 4  +  4 )  =  8
25462, 10, 10, 252, 253decaddi 11084 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  7 )  +  4 )  = ; 1
8
25598, 17, 98, 238, 150, 10, 254, 194decmul1c 11087 . . . . . . . . . . . . 13  |-  (; 2 7  x.  7 )  = ;; 1 8 9
256237, 17, 98, 238, 150, 239, 251, 255decmul2c 11088 . . . . . . . . . . . 12  |-  (; 2 7  x. ; 2 7 )  = ;; 7 2 9
25754, 54, 235, 236, 256numexp2x 15003 . . . . . . . . . . 11  |-  ( 3 ^ 6 )  = ;; 7 2 9
258 eqid 2420 . . . . . . . . . . . 12  |- ; 7 2  = ; 7 2
259176oveq1i 6306 . . . . . . . . . . . . 13  |-  ( ( 7  x.  3 )  +  0 )  =  (; 2 1  +  0 )
260231nn0cni 10870 . . . . . . . . . . . . . 14  |- ; 2 1  e.  CC
261260addid1i 9809 . . . . . . . . . . . . 13  |-  (; 2 1  +  0 )  = ; 2 1
262259, 261eqtri 2449 . . . . . . . . . . . 12  |-  ( ( 7  x.  3 )  +  0 )  = ; 2
1
263235oveq1i 6306 . . . . . . . . . . . . 13  |-  ( ( 2  x.  3 )  +  2 )  =  ( 6  +  2 )
264 6p2e8 10740 . . . . . . . . . . . . 13  |-  ( 6  +  2 )  =  8
265263, 264eqtri 2449 . . . . . . . . . . . 12  |-  ( ( 2  x.  3 )  +  2 )  =  8
26698, 17, 57, 17, 258, 199, 54, 262, 265decma 11078 . . . . . . . . . . 11  |-  ( (; 7
2  x.  3 )  +  2 )  = ;; 2 1 8
267 9t3e27 11136 . . . . . . . . . . 11  |-  ( 9  x.  3 )  = ; 2
7
26854, 233, 150, 257, 98, 17, 266, 267decmul1c 11087 . . . . . . . . . 10  |-  ( ( 3 ^ 6 )  x.  3 )  = ;;; 2 1 8 7
26954, 60, 64, 268numexpp1 15002 . . . . . . . . 9  |-  ( 3 ^ 7 )  = ;;; 2 1 8 7
27062, 98deccl 11054 . . . . . . . . . 10  |- ; 1 7  e.  NN0
271270, 98deccl 11054 . . . . . . . . 9  |- ;; 1 7 7  e.  NN0
272 eqid 2420 . . . . . . . . . 10  |- ;; 2 1 8  = ;; 2 1 8
273 eqid 2420 . . . . . . . . . 10  |- ;; 1 7 7  = ;; 1 7 7
27417, 57deccl 11054 . . . . . . . . . . 11  |- ; 2 0  e.  NN0
275274, 54deccl 11054 . . . . . . . . . 10  |- ;; 2 0 3  e.  NN0
27617, 17deccl 11054 . . . . . . . . . . 11  |- ; 2 2  e.  NN0
277 eqid 2420 . . . . . . . . . . 11  |- ; 2 1  = ; 2 1
278 eqid 2420 . . . . . . . . . . . 12  |- ; 1 7  = ; 1 7
279 eqid 2420 . . . . . . . . . . . 12  |- ;; 2 0 3  = ;; 2 0 3
280 eqid 2420 . . . . . . . . . . . . . 14  |- ; 2 0  = ; 2 0
28180addid2i 9810 . . . . . . . . . . . . . 14  |-  ( 0  +  2 )  =  2
2824addid1i 9809 . . . . . . . . . . . . . 14  |-  ( 1  +  0 )  =  1
28357, 62, 17, 57, 201, 280, 281, 282decadd 11081 . . . . . . . . . . . . 13  |-  ( 1  + ; 2 0 )  = ; 2
1
28417, 62, 246, 283decsuc 11063 . . . . . . . . . . . 12  |-  ( ( 1  + ; 2 0 )  +  1 )  = ; 2 2
285 7p3e10 10744 . . . . . . . . . . . 12  |-  ( 7  +  3 )  =  10
28662, 98, 274, 54, 278, 279, 284, 285decaddc2 11083 . . . . . . . . . . 11  |-  (; 1 7  + ;; 2 0 3 )  = ;; 2 2 0
287 eqid 2420 . . . . . . . . . . . 12  |- ;; 2 5 3  = ;; 2 5 3
288 eqid 2420 . . . . . . . . . . . . 13  |- ; 2 2  = ; 2 2
289 eqid 2420 . . . . . . . . . . . . 13  |- ; 2 5  = ; 2 5
290 2p2e4 10716 . . . . . . . . . . . . 13  |-  ( 2  +  2 )  =  4
29176, 80, 220addcomli 9814 . . . . . . . . . . . . 13  |-  ( 2  +  5 )  =  7
29217, 17, 17, 55, 288, 289, 290, 291decadd 11081 . . . . . . . . . . . 12  |-  (; 2 2  + ; 2 5 )  = ; 4
7
29355dec0h 11056 . . . . . . . . . . . . . 14  |-  5  = ; 0 5
294195, 293eqtri 2449 . . . . . . . . . . . . 13  |-  ( 4  +  1 )  = ; 0
5
295241, 200oveq12i 6308 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
296295, 195eqtri 2449 . . . . . . . . . . . . 13  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  5
297 5t2e10 10753 . . . . . . . . . . . . . . 15  |-  ( 5  x.  2 )  =  10
298 dec10 11070 . . . . . . . . . . . . . . 15  |-  10  = ; 1 0
299297, 298eqtri 2449 . . . . . . . . . . . . . 14  |-  ( 5  x.  2 )  = ; 1
0
30076addid2i 9810 . . . . . . . . . . . . . 14  |-  ( 0  +  5 )  =  5
30162, 57, 55, 299, 300decaddi 11084 . . . . . . . . . . . . 13  |-  ( ( 5  x.  2 )  +  5 )  = ; 1
5
30217, 55, 57, 55, 289, 294, 17, 55, 62, 296, 301decmac 11079 . . . . . . . . . . . 12  |-  ( (; 2
5  x.  2 )  +  ( 4  +  1 ) )  = ; 5
5
303234oveq1i 6306 . . . . . . . . . . . . 13  |-  ( ( 3  x.  2 )  +  7 )  =  ( 6  +  7 )
304 7p6e13 11094 . . . . . . . . . . . . . 14  |-  ( 7  +  6 )  = ; 1
3
30575, 178, 304addcomli 9814 . . . . . . . . . . . . 13  |-  ( 6  +  7 )  = ; 1
3
306303, 305eqtri 2449 . . . . . . . . . . . 12  |-  ( ( 3  x.  2 )  +  7 )  = ; 1
3
307229, 54, 10, 98, 287, 292, 17, 54, 62, 302, 306decmac 11079 . . . . . . . . . . 11  |-  ( (;; 2 5 3  x.  2 )  +  (; 2
2  + ; 2 5 ) )  = ;; 5 5 3
308230nn0cni 10870 . . . . . . . . . . . . . 14  |- ;; 2 5 3  e.  CC
309308mulid1i 9634 . . . . . . . . . . . . 13  |-  (;; 2 5 3  x.  1 )  = ;; 2 5 3
310309oveq1i 6306 . . . . . . . . . . . 12  |-  ( (;; 2 5 3  x.  1 )  +  0 )  =  (;; 2 5 3  +  0 )
311308addid1i 9809 . . . . . . . . . . . 12  |-  (;; 2 5 3  +  0 )  = ;; 2 5 3
312310, 311eqtri 2449 . . . . . . . . . . 11  |-  ( (;; 2 5 3  x.  1 )  +  0 )  = ;; 2 5 3
31317, 62, 276, 57, 277, 286, 230, 54, 229, 307, 312decma2c 11080 . . . . . . . . . 10  |-  ( (;; 2 5 3  x. ; 2
1 )  +  (; 1
7  + ;; 2 0 3 ) )  = ;;; 5 5 3 3
31498dec0h 11056 . . . . . . . . . . 11  |-  7  = ; 0 7
31579addid2i 9810 . . . . . . . . . . . . . 14  |-  ( 0  +  4 )  =  4
316315oveq2i 6307 . . . . . . . . . . . . 13  |-  ( ( 2  x.  8 )  +  ( 0  +  4 ) )  =  ( ( 2  x.  8 )  +  4 )
317 8t2e16 11128 . . . . . . . . . . . . . . 15  |-  ( 8  x.  2 )  = ; 1
6
318247, 80, 317mulcomli 9639 . . . . . . . . . . . . . 14  |-  ( 2  x.  8 )  = ; 1
6
319 6p4e10 10742 . . . . . . . . . . . . . 14  |-  ( 6  +  4 )  =  10
32062, 60, 10, 318, 246, 319decaddci2 11086 . . . . . . . . . . . . 13  |-  ( ( 2  x.  8 )  +  4 )  = ; 2
0
321316, 320eqtri 2449 . . . . . . . . . . . 12  |-  ( ( 2  x.  8 )  +  ( 0  +  4 ) )  = ; 2
0
322 8t5e40 11131 . . . . . . . . . . . . . 14  |-  ( 8  x.  5 )  = ; 4
0
323247, 76, 322mulcomli 9639 . . . . . . . . . . . . 13  |-  ( 5  x.  8 )  = ; 4
0
32410, 57, 54, 323, 184decaddi 11084 . . . . . . . . . . . 12  |-  ( ( 5  x.  8 )  +  3 )  = ; 4
3
32517, 55, 57, 54, 289, 186, 137, 54, 10, 321, 324decmac 11079 . . . . . . . . . . 11  |-  ( (; 2
5  x.  8 )  +  ( 0  +  3 ) )  = ;; 2 0 3
326 8t3e24 11129 . . . . . . . . . . . . 13  |-  ( 8  x.  3 )  = ; 2
4
327247, 3, 326mulcomli 9639 . . . . . . . . . . . 12  |-  ( 3  x.  8 )  = ; 2
4
328 2p1e3 10722 . . . . . . . . . . . 12  |-  ( 2  +  1 )  =  3
329 7p4e11 11092 . . . . . . . . . . . . 13  |-  ( 7  +  4 )  = ; 1
1
33075, 79, 329addcomli 9814 . . . . . . . . . . . 12  |-  ( 4  +  7 )  = ; 1
1
33117, 10, 98, 327, 328, 62, 330decaddci 11085 . . . . . . . . . . 11  |-  ( ( 3  x.  8 )  +  7 )  = ; 3
1
332229, 54, 57, 98, 287, 314, 137, 62, 54, 325, 331decmac 11079 . . . . . . . . . 10  |-  ( (;; 2 5 3  x.  8 )  +  7 )  = ;;; 2 0 3 1
333231, 137, 270, 98, 272, 273, 230, 62, 275, 313, 332decma2c 11080 . . . . . . . . 9  |-  ( (;; 2 5 3  x. ;; 2 1 8 )  + ;; 1 7 7 )  = ;;;; 5 5 3 3 1
334184oveq2i 6307 . . . . . . . . . . . 12  |-  ( ( 2  x.  7 )  +  ( 0  +  3 ) )  =  ( ( 2  x.  7 )  +  3 )
33562, 10, 54, 252, 243decaddi 11084 . . . . . . . . . . . 12  |-  ( ( 2  x.  7 )  +  3 )  = ; 1
7
336334, 335eqtri 2449 . . . . . . . . . . 11  |-  ( ( 2  x.  7 )  +  ( 0  +  3 ) )  = ; 1
7
33754, 55, 17, 78, 220decaddi 11084 . . . . . . . . . . 11  |-  ( ( 5  x.  7 )  +  2 )  = ; 3
7
33817, 55, 57, 17, 289, 199, 98, 98, 54, 336, 337decmac 11079 . . . . . . . . . 10  |-  ( (; 2
5  x.  7 )  +  2 )  = ;; 1 7 7
33998, 229, 54, 287, 62, 17, 338, 177decmul1c 11087 . . . . . . . . 9  |-  (;; 2 5 3  x.  7 )  = ;;; 1 7 7 1
340230, 232, 98, 269, 62, 271, 333, 339decmul2c 11088 . . . . . . . 8  |-  (;; 2 5 3  x.  (
3 ^ 7 ) )  = ;;;;; 5 5 3 3 1 1
341 eqid 2420 . . . . . . . . . . 11  |- ;;;; 5 5 3 3 1  = ;;;; 5 5 3 3 1
342 eqid 2420 . . . . . . . . . . . . . 14  |- ;;; 5 5 3 3  = ;;; 5 5 3 3
343 eqid 2420 . . . . . . . . . . . . . . 15  |- ;; 5 5 3  = ;; 5 5 3
344 eqid 2420 . . . . . . . . . . . . . . . 16  |- ; 5 5  = ; 5 5
345281, 199eqtri 2449 . . . . . . . . . . . . . . . 16  |-  ( 0  +  2 )  = ; 0
2
346184oveq2i 6307 . . . . . . . . . . . . . . . . 17  |-  ( ( 5  x.  7 )  +  ( 0  +  3 ) )  =  ( ( 5  x.  7 )  +  3 )
347346, 174eqtri 2449 . . . . . . . . . . . . . . . 16  |-  ( ( 5  x.  7 )  +  ( 0  +  3 ) )  = ; 3
8
34855, 55, 57, 17, 344, 345, 98, 98, 54, 347, 337decmac 11079 . . . . . . . . . . . . . . 15  |-  ( (; 5
5  x.  7 )  +  ( 0  +  2 ) )  = ;; 3 8 7
34917, 62, 17, 177, 171decaddi 11084 . . . . . . . . . . . . . . 15  |-  ( ( 3  x.  7 )  +  2 )  = ; 2
3
350225, 54, 57, 17, 343, 199, 98, 54, 17, 348, 349decmac 11079 . . . . . . . . . . . . . 14  |-  ( (;; 5 5 3  x.  7 )  +  2 )  = ;;; 3 8 7 3
35198, 226, 54, 342, 62, 17, 350, 177decmul1c 11087 . . . . . . . . . . . . 13  |-  (;;; 5 5 3 3  x.  7 )  = ;;;; 3 8 7 3 1
352351oveq1i 6306 . . . . . . . . . . . 12  |-  ( (;;; 5 5 3 3  x.  7 )  +  0 )  =  (;;;; 3 8 7 3 1  +  0 )
353141nn0cni 10870 . . . . . . . . . . . . 13  |- ;;;; 3 8 7 3 1  e.  CC
354353addid1i 9809 . . . . . . . . . . . 12  |-  (;;;; 3 8 7 3 1  +  0 )  = ;;;; 3 8 7 3 1
355352, 354eqtri 2449 . . . . . . . . . . 11  |-  ( (;;; 5 5 3 3  x.  7 )  +  0 )  = ;;;; 3 8 7 3 1
35675mulid2i 9635 . . . . . . . . . . . 12  |-  ( 1  x.  7 )  =  7
357356, 314eqtri 2449 . . . . . . . . . . 11  |-  ( 1  x.  7 )  = ; 0
7
35898, 227, 62, 341, 98, 57, 355, 357decmul1c 11087 . . . . . . . . . 10  |-  (;;;; 5 5 3 3 1  x.  7 )  = ;;;;; 3 8 7 3 1 7
359358oveq1i 6306 . . . . . . . . 9  |-  ( (;;;; 5 5 3 3 1  x.  7 )  +  0 )  =  (;;;;; 3 8 7 3 1 7  +  0 )
360143nn0cni 10870 . . . . . . . . . 10  |- ;;;;; 3 8 7 3 1 7  e.  CC
361360addid1i 9809 . . . . . . . . 9  |-  (;;;;; 3 8 7 3 1 7  +  0 )  = ;;;;; 3 8 7 3 1 7
362359, 361eqtri 2449 . . . . . . . 8  |-  ( (;;;; 5 5 3 3 1  x.  7 )  +  0 )  = ;;;;; 3 8 7 3 1 7
36398, 228, 62, 340, 98, 57, 362, 357decmul1c 11087 . . . . . . 7  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  = ;;;;;; 3 8 7 3 1 7 7
364147, 224, 3633brtr4i 4445 . . . . . 6  |-  (;;;; 5 3 0 5 7  x. ; 7 3 )  < 
( (;; 2 5 3  x.  (
3 ^ 7 ) )  x.  7 )
36598, 54deccl 11054 . . . . . . . . 9  |- ; 7 3  e.  NN0
366128, 365nn0mulcli 10897 . . . . . . . 8  |-  (;;;; 5 3 0 5 7  x. ; 7 3 )  e. 
NN0
367366nn0rei 10869 . . . . . . 7  |-  (;;;; 5 3 0 5 7  x. ; 7 3 )  e.  RR
36854, 98nn0expcli 12284 . . . . . . . . . 10  |-  ( 3 ^ 7 )  e. 
NN0
369230, 368nn0mulcli 10897 . . . . . . . . 9  |-  (;; 2 5 3  x.  (
3 ^ 7 ) )  e.  NN0
370369, 98nn0mulcli 10897 . . . . . . . 8  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  e.  NN0
371370nn0rei 10869 . . . . . . 7  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  e.  RR
37267nnrei 10607 . . . . . . 7  |-  5  e.  RR
37367nngt0i 10632 . . . . . . 7  |-  0  <  5
374367, 371, 372, 373ltmul1ii 10524 . . . . . 6  |-  ( (;;;; 5 3 0 5 7  x. ; 7 3 )  < 
( (;; 2 5 3  x.  (
3 ^ 7 ) )  x.  7 )  <-> 
( (;;;; 5 3 0 5 7  x. ; 7 3 )  x.  5 )  <  (
( (;; 2 5 3  x.  (
3 ^ 7 ) )  x.  7 )  x.  5 ) )
375364, 374mpbi 211 . . . . 5  |-  ( (;;;; 5 3 0 5 7  x. ; 7 3 )  x.  5 )  <  (
( (;; 2 5 3  x.  (
3 ^ 7 ) )  x.  7 )  x.  5 )
376128nn0cni 10870 . . . . . . 7  |- ;;;; 5 3 0 5 7  e.  CC
377365nn0cni 10870 . . . . . . 7  |- ; 7 3  e.  CC
378376, 377, 76mulassi 9641 . . . . . 6  |-  ( (;;;; 5 3 0 5 7  x. ; 7 3 )  x.  5 )  =  (;;;; 5 3 0 5 7  x.  (; 7 3  x.  5 ) )
37954, 55, 158, 77decsuc 11063 . . . . . . . 8  |-  ( ( 7  x.  5 )  +  1 )  = ; 3
6
38076, 3, 206mulcomli 9639 . . . . . . . 8  |-  ( 3  x.  5 )  = ; 1
5
38155, 98, 54, 148, 55, 62, 379, 380decmul1c 11087 . . . . . . 7  |-  (; 7 3  x.  5 )  = ;; 3 6 5
382381oveq2i 6307 . . . . . 6  |-  (;;;; 5 3 0 5 7  x.  (; 7 3  x.  5 ) )  =  (;;;; 5 3 0 5 7  x. ;; 3 6 5 )
383378, 382eqtri 2449 . . . . 5  |-  ( (;;;; 5 3 0 5 7  x. ; 7 3 )  x.  5 )  =  (;;;; 5 3 0 5 7  x. ;; 3 6 5 )
384308, 101mulcli 9637 . . . . . . 7  |-  (;; 2 5 3  x.  (
3 ^ 7 ) )  e.  CC
385384, 75, 76mulassi 9641 . . . . . 6  |-  ( ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  x.  5 )  =  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  ( 7  x.  5 ) )
38675, 76mulcomi 9638 . . . . . . . 8  |-  ( 7  x.  5 )  =  ( 5  x.  7 )
387386oveq2i 6307 . . . . . . 7  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  (
7  x.  5 ) )  =  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  (
5  x.  7 ) )
388308, 101, 102mulassi 9641 . . . . . . 7  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  (
5  x.  7 ) )  =  (;; 2 5 3  x.  (
( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )
389387, 388eqtri 2449 . . . . . 6  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  (
7  x.  5 ) )  =  (;; 2 5 3  x.  (
( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )
390385, 389eqtri 2449 . . . . 5  |-  ( ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  x.  5 )  =  (;; 2 5 3  x.  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) ) )
391375, 383, 3903brtr3i 4444 . . . 4  |-  (;;;; 5 3 0 5 7  x. ;; 3 6 5 )  <  (;; 2 5 3  x.  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) ) )
39254, 60deccl 11054 . . . . . . . 8  |- ; 3 6  e.  NN0
393392, 67decnncl 11053 . . . . . . 7  |- ;; 3 6 5  e.  NN
394393nnrei 10607 . . . . . 6  |- ;; 3 6 5  e.  RR
395393nngt0i 10632 . . . . . 6  |-  0  < ;; 3 6 5
396394, 395pm3.2i 456 . . . . 5  |-  (;; 3 6 5  e.  RR  /\  0  < ;; 3 6 5 )
397230nn0rei 10869 . . . . 5  |- ;; 2 5 3  e.  RR
398 lt2mul2div 10472 . . . . 5  |-  ( ( (;;;; 5 3 0 5 7  e.  RR  /\  (;; 3 6 5  e.  RR  /\  0  < ;; 3 6 5 ) )  /\  (;; 2 5 3  e.  RR  /\  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR  /\  0  <  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) ) ) )  ->  (
(;;;; 5 3 0 5 7  x. ;; 3 6 5 )  <  (;; 2 5 3  x.  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) ) )  <->  (;;;; 5 3 0 5 7  /  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  < 
(;; 2 5 3  / ;; 3 6 5 ) ) )
399129, 396, 397, 133, 398mp4an 677 . . . 4  |-  ( (;;;; 5 3 0 5 7  x. ;; 3 6 5 )  < 
(;; 2 5 3  x.  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  <-> 
(;;;; 5 3 0 5 7  /  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  < 
(;; 2 5 3  / ;; 3 6 5 ) )
400391, 399mpbi 211 . . 3  |-  (;;;; 5 3 0 5 7  /  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  < 
(;; 2 5 3  / ;; 3 6 5 )
401 nndivre 10634 . . . . 5  |-  ( (;;;; 5 3 0 5 7  e.  RR  /\  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) )  e.  NN )  ->  (;;;; 5 3 0 5 7  /  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  e.  RR )
402129, 130, 401mp2an 676 . . . 4  |-  (;;;; 5 3 0 5 7  /  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  e.  RR
403 nndivre 10634 . . . . 5  |-  ( (;; 2 5 3  e.  RR  /\ ;; 3 6 5  e.  NN )  ->  (;; 2 5 3  / ;; 3 6 5 )  e.  RR )
404397, 393, 403mp2an 676 . . . 4  |-  (;; 2 5 3  / ;; 3 6 5 )  e.  RR
405127, 402, 404lelttri 9750 . . 3  |-  ( ( ( sum_ n  e.  ( 0 ... 3 ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) )  +  ( 3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) )  <_  (;;;; 5 3 0 5 7  /  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  /\  (;;;; 5 3 0 5 7  /  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  <  (;; 2 5 3  / ;; 3 6 5 ) )  ->  ( sum_ n  e.  ( 0 ... 3
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  +  ( 3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) )  <  (;; 2 5 3  / ;; 3 6 5 ) )
406136, 400, 405mp2an 676 . 2  |-  ( sum_ n  e.  ( 0 ... 3 ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n
) ) )  +  ( 3  /  (
( 4  x.  (
( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) )  <  (;; 2 5 3  / ;; 3 6 5 )
40732, 127, 404lelttri 9750 . 2  |-  ( ( ( log `  2
)  <_  ( sum_ n  e.  ( 0 ... 3 ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n
) ) )  +  ( 3  /  (
( 4  x.  (
( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) ) )  /\  ( sum_ n  e.  ( 0 ... 3 ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) )  +  ( 3  / 
( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  (
9 ^ 4 ) ) ) )  < 
(;; 2 5 3  / ;; 3 6 5 ) )  ->  ( log `  2
)  <  (;; 2 5 3  / ;; 3 6 5 ) )
40852, 406, 407mp2an 676 1  |-  ( log `  2 )  < 
(;; 2 5 3  / ;; 3 6 5 )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   T. wtru 1438    e. wcel 1867   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   CCcc 9526   RRcr 9527   0cc0 9528   1c1 9529    + caddc 9531    x. cmul 9533    < clt 9664    <_ cle 9665    - cmin 9849    / cdiv 10258   NNcn 10598   2c2 10648   3c3 10649   4c4 10650   5c5 10651   6c6 10652   7c7 10653   8c8 10654   9c9 10655   10c10 10656   NN0cn0 10858  ;cdc 11040   RR+crp 11291   [,]cicc 11627   ...cfz 11771   ^cexp 12258   sum_csu 13719   logclog 23366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606  ax-addf 9607  ax-mulf 9608
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6917  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8016  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-q 11254  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-ioo 11628  df-ioc 11629  df-ico 11630  df-icc 11631  df-fz 11772  df-fzo 11903  df-fl 12014  df-mod 12083  df-seq 12200  df-exp 12259  df-fac 12446  df-bc 12474  df-hash 12502  df-shft 13098  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-limsup 13493  df-clim 13519  df-rlim 13520  df-sum 13720  df-ef 14088  df-sin 14090  df-cos 14091  df-tan 14092  df-pi 14093  df-dvds 14273  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-ress 15080  df-plusg 15155  df-mulr 15156  df-starv 15157  df-sca 15158  df-vsca 15159  df-ip 15160  df-tset 15161  df-ple 15162  df-ds 15164  df-unif 15165  df-hom 15166  df-cco 15167  df-rest 15273  df-topn 15274  df-0g 15292  df-gsum 15293  df-topgen 15294  df-pt 15295  df-prds 15298  df-xrs 15352  df-qtop 15357  df-imas 15358  df-xps 15360  df-mre 15436  df-mrc 15437  df-acs 15439  df-mgm 16432  df-sgrp 16471  df-mnd 16481  df-submnd 16527  df-mulg 16620  df-cntz 16915  df-cmn 17360  df-psmet 18890  df-xmet 18891  df-met 18892  df-bl 18893  df-mopn 18894  df-fbas 18895  df-fg 18896  df-cnfld 18899  df-top 19845  df-bases 19846  df-topon 19847  df-topsp 19848  df-cld 19958  df-ntr 19959  df-cls 19960  df-nei 20038  df-lp 20076  df-perf 20077  df-cn 20167  df-cnp 20168  df-haus 20255  df-cmp 20326  df-tx 20501  df-hmeo 20694  df-fil 20785  df-fm 20877  df-flim 20878  df-flf 20879  df-xms 21259  df-ms 21260  df-tms 21261  df-cncf 21799  df-limc 22695  df-dv 22696  df-ulm 23194  df-log 23368  df-atan 23655
This theorem is referenced by:  log2le1  23738  birthday  23742
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