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Theorem log2ub 22229
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 10370 . . . . . . . . . . 11  |-  4  =  ( 3  +  1 )
21oveq1i 6090 . . . . . . . . . 10  |-  ( 4  -  1 )  =  ( ( 3  +  1 )  -  1 )
3 3cn 10384 . . . . . . . . . . 11  |-  3  e.  CC
4 ax-1cn 9328 . . . . . . . . . . 11  |-  1  e.  CC
5 pncan 9604 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  1  e.  CC )  ->  ( ( 3  +  1 )  -  1 )  =  3 )
63, 4, 5mp2an 665 . . . . . . . . . 10  |-  ( ( 3  +  1 )  -  1 )  =  3
72, 6eqtri 2453 . . . . . . . . 9  |-  ( 4  -  1 )  =  3
87oveq2i 6091 . . . . . . . 8  |-  ( 0 ... ( 4  -  1 ) )  =  ( 0 ... 3
)
98sumeq1i 13159 . . . . . . 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
) ) )
109oveq2i 6091 . . . . . 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 ) ) ) )
11 4nn0 10586 . . . . . . 7  |-  4  e.  NN0
12 log2tlbnd 22225 . . . . . . 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 ) ) ) ) )
1311, 12ax-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 ) ) ) )
1410, 13eqeltrri 2504 . . . . 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 ) ) ) )
15 0re 9374 . . . . . 6  |-  0  e.  RR
16 3re 10383 . . . . . . 7  |-  3  e.  RR
17 4nn 10469 . . . . . . . . 9  |-  4  e.  NN
18 2nn0 10584 . . . . . . . . . 10  |-  2  e.  NN0
19 1nn 10321 . . . . . . . . . 10  |-  1  e.  NN
2018, 11, 19numnncl 10751 . . . . . . . . 9  |-  ( ( 2  x.  4 )  +  1 )  e.  NN
2117, 20nnmulcli 10334 . . . . . . . 8  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  e.  NN
22 9nn 10474 . . . . . . . . 9  |-  9  e.  NN
23 nnexpcl 11862 . . . . . . . . 9  |-  ( ( 9  e.  NN  /\  4  e.  NN0 )  -> 
( 9 ^ 4 )  e.  NN )
2422, 11, 23mp2an 665 . . . . . . . 8  |-  ( 9 ^ 4 )  e.  NN
2521, 24nnmulcli 10334 . . . . . . 7  |-  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) )  e.  NN
26 nndivre 10345 . . . . . . 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 )
2716, 25, 26mp2an 665 . . . . . 6  |-  ( 3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) )  e.  RR
2815, 27elicc2i 11349 . . . . 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 ) ) ) ) )
2914, 28mpbi 208 . . . 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 ) ) ) )
3029simp3i 992 . . 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 ) ) )
31 2rp 10984 . . . . 5  |-  2  e.  RR+
32 relogcl 21912 . . . . 5  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
3331, 32ax-mp 5 . . . 4  |-  ( log `  2 )  e.  RR
34 fzfid 11779 . . . . . 6  |-  ( T. 
->  ( 0 ... 3
)  e.  Fin )
35 2re 10379 . . . . . . 7  |-  2  e.  RR
36 3nn 10468 . . . . . . . . 9  |-  3  e.  NN
37 elfznn0 11468 . . . . . . . . . . . 12  |-  ( n  e.  ( 0 ... 3 )  ->  n  e.  NN0 )
3837adantl 463 . . . . . . . . . . 11  |-  ( ( T.  /\  n  e.  ( 0 ... 3
) )  ->  n  e.  NN0 )
39 nn0mulcl 10604 . . . . . . . . . . 11  |-  ( ( 2  e.  NN0  /\  n  e.  NN0 )  -> 
( 2  x.  n
)  e.  NN0 )
4018, 38, 39sylancr 656 . . . . . . . . . 10  |-  ( ( T.  /\  n  e.  ( 0 ... 3
) )  ->  (
2  x.  n )  e.  NN0 )
41 nn0p1nn 10607 . . . . . . . . . 10  |-  ( ( 2  x.  n )  e.  NN0  ->  ( ( 2  x.  n )  +  1 )  e.  NN )
4240, 41syl 16 . . . . . . . . 9  |-  ( ( T.  /\  n  e.  ( 0 ... 3
) )  ->  (
( 2  x.  n
)  +  1 )  e.  NN )
43 nnmulcl 10333 . . . . . . . . 9  |-  ( ( 3  e.  NN  /\  ( ( 2  x.  n )  +  1 )  e.  NN )  ->  ( 3  x.  ( ( 2  x.  n )  +  1 ) )  e.  NN )
4436, 42, 43sylancr 656 . . . . . . . 8  |-  ( ( T.  /\  n  e.  ( 0 ... 3
) )  ->  (
3  x.  ( ( 2  x.  n )  +  1 ) )  e.  NN )
45 nnexpcl 11862 . . . . . . . . 9  |-  ( ( 9  e.  NN  /\  n  e.  NN0 )  -> 
( 9 ^ n
)  e.  NN )
4622, 38, 45sylancr 656 . . . . . . . 8  |-  ( ( T.  /\  n  e.  ( 0 ... 3
) )  ->  (
9 ^ n )  e.  NN )
4744, 46nnmulcld 10357 . . . . . . 7  |-  ( ( T.  /\  n  e.  ( 0 ... 3
) )  ->  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) )  e.  NN )
48 nndivre 10345 . . . . . . 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 )
4935, 47, 48sylancr 656 . . . . . 6  |-  ( ( T.  /\  n  e.  ( 0 ... 3
) )  ->  (
2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) )  e.  RR )
5034, 49fsumrecl 13195 . . . . 5  |-  ( T. 
->  sum_ n  e.  ( 0 ... 3 ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) )  e.  RR )
5150trud 1371 . . . 4  |-  sum_ n  e.  ( 0 ... 3
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  e.  RR
5233, 51, 27lesubadd2i 9888 . . 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 ) ) ) ) )
5330, 52mpbi 208 . 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 ) ) ) )
54 log2ublem3 22228 . . . . 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
55 3nn0 10585 . . . . 5  |-  3  e.  NN0
56 5nn0 10587 . . . . . . . . 9  |-  5  e.  NN0
5756, 55deccl 10757 . . . . . . . 8  |- ; 5 3  e.  NN0
58 0nn0 10582 . . . . . . . 8  |-  0  e.  NN0
5957, 58deccl 10757 . . . . . . 7  |- ;; 5 3 0  e.  NN0
6059, 56deccl 10757 . . . . . 6  |- ;;; 5 3 0 5  e.  NN0
61 6nn0 10588 . . . . . 6  |-  6  e.  NN0
6260, 61deccl 10757 . . . . 5  |- ;;;; 5 3 0 5 6  e.  NN0
63 1nn0 10583 . . . . 5  |-  1  e.  NN0
64 eqid 2433 . . . . 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 ) ) ) )
65 6p1e7 10438 . . . . . 6  |-  ( 6  +  1 )  =  7
66 eqid 2433 . . . . . 6  |- ;;;; 5 3 0 5 6  = ;;;; 5 3 0 5 6
6760, 61, 65, 66decsuc 10766 . . . . 5  |-  (;;;; 5 3 0 5 6  +  1 )  = ;;;; 5 3 0 5 7
68 5nn 10470 . . . . . . . . . 10  |-  5  e.  NN
69 7nn 10472 . . . . . . . . . 10  |-  7  e.  NN
7068, 69nnmulcli 10334 . . . . . . . . 9  |-  ( 5  x.  7 )  e.  NN
7170nnrei 10319 . . . . . . . 8  |-  ( 5  x.  7 )  e.  RR
7221nnrei 10319 . . . . . . . 8  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  e.  RR
73 6nn 10471 . . . . . . . . . 10  |-  6  e.  NN
74 5lt6 10486 . . . . . . . . . 10  |-  5  <  6
7555, 56, 73, 74declt 10764 . . . . . . . . 9  |- ; 3 5  < ; 3 6
76 7cn 10393 . . . . . . . . . 10  |-  7  e.  CC
77 5cn 10389 . . . . . . . . . 10  |-  5  e.  CC
78 7t5e35 10828 . . . . . . . . . 10  |-  ( 7  x.  5 )  = ; 3
5
7976, 77, 78mulcomli 9381 . . . . . . . . 9  |-  ( 5  x.  7 )  = ; 3
5
80 4cn 10387 . . . . . . . . . . . . . 14  |-  4  e.  CC
81 2cn 10380 . . . . . . . . . . . . . 14  |-  2  e.  CC
82 4t2e8 10463 . . . . . . . . . . . . . 14  |-  ( 4  x.  2 )  =  8
8380, 81, 82mulcomli 9381 . . . . . . . . . . . . 13  |-  ( 2  x.  4 )  =  8
8483oveq1i 6090 . . . . . . . . . . . 12  |-  ( ( 2  x.  4 )  +  1 )  =  ( 8  +  1 )
85 8p1e9 10440 . . . . . . . . . . . 12  |-  ( 8  +  1 )  =  9
8684, 85eqtri 2453 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  +  1 )  =  9
8786oveq2i 6091 . . . . . . . . . 10  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  =  ( 4  x.  9 )
88 9cn 10397 . . . . . . . . . . 11  |-  9  e.  CC
89 9t4e36 10840 . . . . . . . . . . 11  |-  ( 9  x.  4 )  = ; 3
6
9088, 80, 89mulcomli 9381 . . . . . . . . . 10  |-  ( 4  x.  9 )  = ; 3
6
9187, 90eqtri 2453 . . . . . . . . 9  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  = ; 3
6
9275, 79, 913brtr4i 4308 . . . . . . . 8  |-  ( 5  x.  7 )  < 
( 4  x.  (
( 2  x.  4 )  +  1 ) )
9371, 72, 92ltleii 9485 . . . . . . 7  |-  ( 5  x.  7 )  <_ 
( 4  x.  (
( 2  x.  4 )  +  1 ) )
9424nngt0i 10343 . . . . . . . 8  |-  0  <  ( 9 ^ 4 )
9524nnrei 10319 . . . . . . . . 9  |-  ( 9 ^ 4 )  e.  RR
9671, 72, 95lemul2i 10244 . . . . . . . 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 ) ) ) ) )
9794, 96ax-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 ) ) ) )
9893, 97mpbi 208 . . . . . 6  |-  ( ( 9 ^ 4 )  x.  ( 5  x.  7 ) )  <_ 
( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) )
99 7nn0 10589 . . . . . . . . . 10  |-  7  e.  NN0
100 nnexpcl 11862 . . . . . . . . . 10  |-  ( ( 3  e.  NN  /\  7  e.  NN0 )  -> 
( 3 ^ 7 )  e.  NN )
10136, 99, 100mp2an 665 . . . . . . . . 9  |-  ( 3 ^ 7 )  e.  NN
102101nncni 10320 . . . . . . . 8  |-  ( 3 ^ 7 )  e.  CC
10370nncni 10320 . . . . . . . 8  |-  ( 5  x.  7 )  e.  CC
104102, 103, 3mul32i 9553 . . . . . . 7  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  3 )  =  ( ( ( 3 ^ 7 )  x.  3 )  x.  (
5  x.  7 ) )
10580, 81mulcomi 9380 . . . . . . . . . . . 12  |-  ( 4  x.  2 )  =  ( 2  x.  4 )
106 df-8 10374 . . . . . . . . . . . 12  |-  8  =  ( 7  +  1 )
10782, 105, 1063eqtr3i 2461 . . . . . . . . . . 11  |-  ( 2  x.  4 )  =  ( 7  +  1 )
108107oveq2i 6091 . . . . . . . . . 10  |-  ( 3 ^ ( 2  x.  4 ) )  =  ( 3 ^ (
7  +  1 ) )
109 expmul 11893 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  2  e.  NN0  /\  4  e.  NN0 )  ->  (
3 ^ ( 2  x.  4 ) )  =  ( ( 3 ^ 2 ) ^
4 ) )
1103, 18, 11, 109mp3an 1307 . . . . . . . . . 10  |-  ( 3 ^ ( 2  x.  4 ) )  =  ( ( 3 ^ 2 ) ^ 4 )
111108, 110eqtr3i 2455 . . . . . . . . 9  |-  ( 3 ^ ( 7  +  1 ) )  =  ( ( 3 ^ 2 ) ^ 4 )
112 expp1 11856 . . . . . . . . . 10  |-  ( ( 3  e.  CC  /\  7  e.  NN0 )  -> 
( 3 ^ (
7  +  1 ) )  =  ( ( 3 ^ 7 )  x.  3 ) )
1133, 99, 112mp2an 665 . . . . . . . . 9  |-  ( 3 ^ ( 7  +  1 ) )  =  ( ( 3 ^ 7 )  x.  3 )
114 sq3 11947 . . . . . . . . . 10  |-  ( 3 ^ 2 )  =  9
115114oveq1i 6090 . . . . . . . . 9  |-  ( ( 3 ^ 2 ) ^ 4 )  =  ( 9 ^ 4 )
116111, 113, 1153eqtr3i 2461 . . . . . . . 8  |-  ( ( 3 ^ 7 )  x.  3 )  =  ( 9 ^ 4 )
117116oveq1i 6090 . . . . . . 7  |-  ( ( ( 3 ^ 7 )  x.  3 )  x.  ( 5  x.  7 ) )  =  ( ( 9 ^ 4 )  x.  (
5  x.  7 ) )
118104, 117eqtri 2453 . . . . . 6  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  3 )  =  ( ( 9 ^ 4 )  x.  (
5  x.  7 ) )
11921nncni 10320 . . . . . . . . 9  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  e.  CC
12024nncni 10320 . . . . . . . . 9  |-  ( 9 ^ 4 )  e.  CC
121119, 120mulcomi 9380 . . . . . . . 8  |-  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) )  =  ( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) )
122121oveq1i 6090 . . . . . . 7  |-  ( ( ( 4  x.  (
( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) )  x.  1 )  =  ( ( ( 9 ^ 4 )  x.  ( 4  x.  (
( 2  x.  4 )  +  1 ) ) )  x.  1 )
123120, 119mulcli 9379 . . . . . . . 8  |-  ( ( 9 ^ 4 )  x.  ( 4  x.  ( ( 2  x.  4 )  +  1 ) ) )  e.  CC
124123mulid1i 9376 . . . . . . 7  |-  ( ( ( 9 ^ 4 )  x.  ( 4  x.  ( ( 2  x.  4 )  +  1 ) ) )  x.  1 )  =  ( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) )
125122, 124eqtri 2453 . . . . . 6  |-  ( ( ( 4  x.  (
( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) )  x.  1 )  =  ( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) )
12698, 118, 1253brtr4i 4308 . . . . 5  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  3 )  <_ 
( ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) )  x.  1 )
12754, 51, 55, 25, 62, 63, 64, 67, 126log2ublem1 22226 . . . 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
12851, 27readdcli 9387 . . . . 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
12960, 99deccl 10757 . . . . . 6  |- ;;;; 5 3 0 5 7  e.  NN0
130129nn0rei 10578 . . . . 5  |- ;;;; 5 3 0 5 7  e.  RR
131101, 70nnmulcli 10334 . . . . . . 7  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  NN
132131nnrei 10319 . . . . . 6  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR
133131nngt0i 10343 . . . . . 6  |-  0  <  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) )
134132, 133pm3.2i 452 . . . . 5  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR  /\  0  <  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) ) )
135 lemuldiv2 10200 . . . . 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 ) ) ) ) )
136128, 130, 134, 135mp3an 1307 . . . 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 ) ) ) )
137127, 136mpbi 208 . . 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 ) ) )
138 8nn0 10590 . . . . . . . . . . . . 13  |-  8  e.  NN0
13955, 138deccl 10757 . . . . . . . . . . . 12  |- ; 3 8  e.  NN0
140139, 99deccl 10757 . . . . . . . . . . 11  |- ;; 3 8 7  e.  NN0
141140, 55deccl 10757 . . . . . . . . . 10  |- ;;; 3 8 7 3  e.  NN0
142141, 63deccl 10757 . . . . . . . . 9  |- ;;;; 3 8 7 3 1  e.  NN0
143142, 61deccl 10757 . . . . . . . 8  |- ;;;;; 3 8 7 3 1 6  e.  NN0
144142, 99deccl 10757 . . . . . . . 8  |- ;;;;; 3 8 7 3 1 7  e.  NN0
145 1lt10 10520 . . . . . . . 8  |-  1  <  10
146 6lt7 10491 . . . . . . . . 9  |-  6  <  7
147142, 61, 69, 146declt 10764 . . . . . . . 8  |- ;;;;; 3 8 7 3 1 6  < ;;;;; 3 8 7 3 1 7
148143, 144, 63, 99, 145, 147decltc 10765 . . . . . . 7  |- ;;;;;; 3 8 7 3 1 6 1  < ;;;;;; 3 8 7 3 1 7 7
149 eqid 2433 . . . . . . . 8  |- ; 7 3  = ; 7 3
15063, 56deccl 10757 . . . . . . . . . . 11  |- ; 1 5  e.  NN0
151 9nn0 10591 . . . . . . . . . . 11  |-  9  e.  NN0
152150, 151deccl 10757 . . . . . . . . . 10  |- ;; 1 5 9  e.  NN0
153152, 63deccl 10757 . . . . . . . . 9  |- ;;; 1 5 9 1  e.  NN0
154153, 99deccl 10757 . . . . . . . 8  |- ;;;; 1 5 9 1 7  e.  NN0
155 eqid 2433 . . . . . . . . 9  |- ;;;; 5 3 0 5 7  = ;;;; 5 3 0 5 7
156 eqid 2433 . . . . . . . . 9  |- ;;;; 1 5 9 1 7  = ;;;; 1 5 9 1 7
157 eqid 2433 . . . . . . . . . 10  |- ;;; 5 3 0 5  = ;;; 5 3 0 5
158 eqid 2433 . . . . . . . . . . 11  |- ;;; 1 5 9 1  = ;;; 1 5 9 1
159 5p1e6 10437 . . . . . . . . . . . 12  |-  ( 5  +  1 )  =  6
16077, 4, 159addcomli 9549 . . . . . . . . . . 11  |-  ( 1  +  5 )  =  6
161152, 63, 56, 158, 160decaddi 10787 . . . . . . . . . 10  |-  (;;; 1 5 9 1  +  5 )  = ;;; 1 5 9 6
16263, 61deccl 10757 . . . . . . . . . . 11  |- ; 1 6  e.  NN0
163 eqid 2433 . . . . . . . . . . 11  |- ;; 5 3 0  = ;; 5 3 0
164 eqid 2433 . . . . . . . . . . . 12  |- ;; 1 5 9  = ;; 1 5 9
165 eqid 2433 . . . . . . . . . . . . 13  |- ; 1 5  = ; 1 5
16663, 56, 159, 165decsuc 10766 . . . . . . . . . . . 12  |-  (; 1 5  +  1 )  = ; 1 6
167 9p4e13 10807 . . . . . . . . . . . 12  |-  ( 9  +  4 )  = ; 1
3
168150, 151, 11, 164, 166, 55, 167decaddci 10788 . . . . . . . . . . 11  |-  (;; 1 5 9  +  4 )  = ;; 1 6 3
169 eqid 2433 . . . . . . . . . . . 12  |- ; 5 3  = ; 5 3
170162nn0cni 10579 . . . . . . . . . . . . 13  |- ; 1 6  e.  CC
171170addid1i 9544 . . . . . . . . . . . 12  |-  (; 1 6  +  0 )  = ; 1 6
172 1p2e3 10434 . . . . . . . . . . . . . 14  |-  ( 1  +  2 )  =  3
173172oveq2i 6091 . . . . . . . . . . . . 13  |-  ( ( 5  x.  7 )  +  ( 1  +  2 ) )  =  ( ( 5  x.  7 )  +  3 )
174 5p3e8 10448 . . . . . . . . . . . . . 14  |-  ( 5  +  3 )  =  8
17555, 56, 55, 79, 174decaddi 10787 . . . . . . . . . . . . 13  |-  ( ( 5  x.  7 )  +  3 )  = ; 3
8
176173, 175eqtri 2453 . . . . . . . . . . . 12  |-  ( ( 5  x.  7 )  +  ( 1  +  2 ) )  = ; 3
8
177 7t3e21 10826 . . . . . . . . . . . . . 14  |-  ( 7  x.  3 )  = ; 2
1
17876, 3, 177mulcomli 9381 . . . . . . . . . . . . 13  |-  ( 3  x.  7 )  = ; 2
1
179 6cn 10391 . . . . . . . . . . . . . 14  |-  6  e.  CC
180179, 4, 65addcomli 9549 . . . . . . . . . . . . 13  |-  ( 1  +  6 )  =  7
18118, 63, 61, 178, 180decaddi 10787 . . . . . . . . . . . 12  |-  ( ( 3  x.  7 )  +  6 )  = ; 2
7
18256, 55, 63, 61, 169, 171, 99, 99, 18, 176, 181decmac 10782 . . . . . . . . . . 11  |-  ( (; 5
3  x.  7 )  +  (; 1 6  +  0 ) )  = ;; 3 8 7
18376mul02i 9546 . . . . . . . . . . . . 13  |-  ( 0  x.  7 )  =  0
184183oveq1i 6090 . . . . . . . . . . . 12  |-  ( ( 0  x.  7 )  +  3 )  =  ( 0  +  3 )
1853addid2i 9545 . . . . . . . . . . . . 13  |-  ( 0  +  3 )  =  3
18655dec0h 10759 . . . . . . . . . . . . 13  |-  3  = ; 0 3
187185, 186eqtri 2453 . . . . . . . . . . . 12  |-  ( 0  +  3 )  = ; 0
3
188184, 187eqtri 2453 . . . . . . . . . . 11  |-  ( ( 0  x.  7 )  +  3 )  = ; 0
3
18957, 58, 162, 55, 163, 168, 99, 55, 58, 182, 188decmac 10782 . . . . . . . . . 10  |-  ( (;; 5 3 0  x.  7 )  +  (;; 1 5 9  +  4 ) )  = ;;; 3 8 7 3
190 3p1e4 10435 . . . . . . . . . . 11  |-  ( 3  +  1 )  =  4
191 6p5e11 10793 . . . . . . . . . . . 12  |-  ( 6  +  5 )  = ; 1
1
192179, 77, 191addcomli 9549 . . . . . . . . . . 11  |-  ( 5  +  6 )  = ; 1
1
19355, 56, 61, 79, 190, 63, 192decaddci 10788 . . . . . . . . . 10  |-  ( ( 5  x.  7 )  +  6 )  = ; 4
1
19459, 56, 152, 61, 157, 161, 99, 63, 11, 189, 193decmac 10782 . . . . . . . . 9  |-  ( (;;; 5 3 0 5  x.  7 )  +  (;;; 1 5 9 1  +  5 ) )  = ;;;; 3 8 7 3 1
195 7t7e49 10830 . . . . . . . . . 10  |-  ( 7  x.  7 )  = ; 4
9
196 4p1e5 10436 . . . . . . . . . 10  |-  ( 4  +  1 )  =  5
197 9p7e16 10810 . . . . . . . . . 10  |-  ( 9  +  7 )  = ; 1
6
19811, 151, 99, 195, 196, 61, 197decaddci 10788 . . . . . . . . 9  |-  ( ( 7  x.  7 )  +  7 )  = ; 5
6
19960, 99, 153, 99, 155, 156, 99, 61, 56, 194, 198decmac 10782 . . . . . . . 8  |-  ( (;;;; 5 3 0 5 7  x.  7 )  + ;;;; 1 5 9 1 7 )  = ;;;;; 3 8 7 3 1 6
20018dec0h 10759 . . . . . . . . . 10  |-  2  = ; 0 2
2014addid2i 9545 . . . . . . . . . . . 12  |-  ( 0  +  1 )  =  1
20263dec0h 10759 . . . . . . . . . . . 12  |-  1  = ; 0 1
203201, 202eqtri 2453 . . . . . . . . . . 11  |-  ( 0  +  1 )  = ; 0
1
204 00id 9532 . . . . . . . . . . . . 13  |-  ( 0  +  0 )  =  0
20558dec0h 10759 . . . . . . . . . . . . 13  |-  0  = ; 0 0
206204, 205eqtri 2453 . . . . . . . . . . . 12  |-  ( 0  +  0 )  = ; 0
0
207 5t3e15 10817 . . . . . . . . . . . . . 14  |-  ( 5  x.  3 )  = ; 1
5
208207oveq1i 6090 . . . . . . . . . . . . 13  |-  ( ( 5  x.  3 )  +  0 )  =  (; 1 5  +  0 )
209150nn0cni 10579 . . . . . . . . . . . . . 14  |- ; 1 5  e.  CC
210209addid1i 9544 . . . . . . . . . . . . 13  |-  (; 1 5  +  0 )  = ; 1 5
211208, 210eqtri 2453 . . . . . . . . . . . 12  |-  ( ( 5  x.  3 )  +  0 )  = ; 1
5
212 3t3e9 10462 . . . . . . . . . . . . . 14  |-  ( 3  x.  3 )  =  9
213212oveq1i 6090 . . . . . . . . . . . . 13  |-  ( ( 3  x.  3 )  +  0 )  =  ( 9  +  0 )
21488addid1i 9544 . . . . . . . . . . . . 13  |-  ( 9  +  0 )  =  9
215213, 214eqtri 2453 . . . . . . . . . . . 12  |-  ( ( 3  x.  3 )  +  0 )  =  9
21656, 55, 58, 58, 169, 206, 55, 211, 215decma 10781 . . . . . . . . . . 11  |-  ( (; 5
3  x.  3 )  +  ( 0  +  0 ) )  = ;; 1 5 9
2173mul02i 9546 . . . . . . . . . . . . 13  |-  ( 0  x.  3 )  =  0
218217oveq1i 6090 . . . . . . . . . . . 12  |-  ( ( 0  x.  3 )  +  1 )  =  ( 0  +  1 )
219218, 203eqtri 2453 . . . . . . . . . . 11  |-  ( ( 0  x.  3 )  +  1 )  = ; 0
1
22057, 58, 58, 63, 163, 203, 55, 63, 58, 216, 219decmac 10782 . . . . . . . . . 10  |-  ( (;; 5 3 0  x.  3 )  +  ( 0  +  1 ) )  = ;;; 1 5 9 1
221 5p2e7 10447 . . . . . . . . . . 11  |-  ( 5  +  2 )  =  7
22263, 56, 18, 207, 221decaddi 10787 . . . . . . . . . 10  |-  ( ( 5  x.  3 )  +  2 )  = ; 1
7
22359, 56, 58, 18, 157, 200, 55, 99, 63, 220, 222decmac 10782 . . . . . . . . 9  |-  ( (;;; 5 3 0 5  x.  3 )  +  2 )  = ;;;; 1 5 9 1 7
22455, 60, 99, 155, 63, 18, 223, 177decmul1c 10790 . . . . . . . 8  |-  (;;;; 5 3 0 5 7  x.  3 )  = ;;;;; 1 5 9 1 7 1
225129, 99, 55, 149, 63, 154, 199, 224decmul2c 10791 . . . . . . 7  |-  (;;;; 5 3 0 5 7  x. ; 7 3 )  = ;;;;;; 3 8 7 3 1 6 1
22656, 56deccl 10757 . . . . . . . . . . 11  |- ; 5 5  e.  NN0
227226, 55deccl 10757 . . . . . . . . . 10  |- ;; 5 5 3  e.  NN0
228227, 55deccl 10757 . . . . . . . . 9  |- ;;; 5 5 3 3  e.  NN0
229228, 63deccl 10757 . . . . . . . 8  |- ;;;; 5 5 3 3 1  e.  NN0
23018, 56deccl 10757 . . . . . . . . . 10  |- ; 2 5  e.  NN0
231230, 55deccl 10757 . . . . . . . . 9  |- ;; 2 5 3  e.  NN0
23218, 63deccl 10757 . . . . . . . . . 10  |- ; 2 1  e.  NN0
233232, 138deccl 10757 . . . . . . . . 9  |- ;; 2 1 8  e.  NN0
23499, 18deccl 10757 . . . . . . . . . . 11  |- ; 7 2  e.  NN0
235 3t2e6 10461 . . . . . . . . . . . . 13  |-  ( 3  x.  2 )  =  6
2363, 81, 235mulcomli 9381 . . . . . . . . . . . 12  |-  ( 2  x.  3 )  =  6
237 3exp3 14101 . . . . . . . . . . . 12  |-  ( 3 ^ 3 )  = ; 2
7
23818, 99deccl 10757 . . . . . . . . . . . . 13  |- ; 2 7  e.  NN0
239 eqid 2433 . . . . . . . . . . . . 13  |- ; 2 7  = ; 2 7
24063, 138deccl 10757 . . . . . . . . . . . . 13  |- ; 1 8  e.  NN0
241 eqid 2433 . . . . . . . . . . . . . 14  |- ; 1 8  = ; 1 8
242 2t2e4 10459 . . . . . . . . . . . . . . . 16  |-  ( 2  x.  2 )  =  4
243242, 172oveq12i 6092 . . . . . . . . . . . . . . 15  |-  ( ( 2  x.  2 )  +  ( 1  +  2 ) )  =  ( 4  +  3 )
244 4p3e7 10445 . . . . . . . . . . . . . . 15  |-  ( 4  +  3 )  =  7
245243, 244eqtri 2453 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  2 )  +  ( 1  +  2 ) )  =  7
246 7t2e14 10825 . . . . . . . . . . . . . . 15  |-  ( 7  x.  2 )  = ; 1
4
247 1p1e2 10423 . . . . . . . . . . . . . . 15  |-  ( 1  +  1 )  =  2
248 8cn 10395 . . . . . . . . . . . . . . . 16  |-  8  e.  CC
249 8p4e12 10800 . . . . . . . . . . . . . . . 16  |-  ( 8  +  4 )  = ; 1
2
250248, 80, 249addcomli 9549 . . . . . . . . . . . . . . 15  |-  ( 4  +  8 )  = ; 1
2
25163, 11, 138, 246, 247, 18, 250decaddci 10788 . . . . . . . . . . . . . 14  |-  ( ( 7  x.  2 )  +  8 )  = ; 2
2
25218, 99, 63, 138, 239, 241, 18, 18, 18, 245, 251decmac 10782 . . . . . . . . . . . . 13  |-  ( (; 2
7  x.  2 )  + ; 1 8 )  = ; 7
2
25376, 81, 246mulcomli 9381 . . . . . . . . . . . . . . 15  |-  ( 2  x.  7 )  = ; 1
4
254 4p4e8 10446 . . . . . . . . . . . . . . 15  |-  ( 4  +  4 )  =  8
25563, 11, 11, 253, 254decaddi 10787 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  7 )  +  4 )  = ; 1
8
25699, 18, 99, 239, 151, 11, 255, 195decmul1c 10790 . . . . . . . . . . . . 13  |-  (; 2 7  x.  7 )  = ;; 1 8 9
257238, 18, 99, 239, 151, 240, 252, 256decmul2c 10791 . . . . . . . . . . . 12  |-  (; 2 7  x. ; 2 7 )  = ;; 7 2 9
25855, 55, 236, 237, 257numexp2x 14091 . . . . . . . . . . 11  |-  ( 3 ^ 6 )  = ;; 7 2 9
259 eqid 2433 . . . . . . . . . . . 12  |- ; 7 2  = ; 7 2
260177oveq1i 6090 . . . . . . . . . . . . 13  |-  ( ( 7  x.  3 )  +  0 )  =  (; 2 1  +  0 )
261232nn0cni 10579 . . . . . . . . . . . . . 14  |- ; 2 1  e.  CC
262261addid1i 9544 . . . . . . . . . . . . 13  |-  (; 2 1  +  0 )  = ; 2 1
263260, 262eqtri 2453 . . . . . . . . . . . 12  |-  ( ( 7  x.  3 )  +  0 )  = ; 2
1
264236oveq1i 6090 . . . . . . . . . . . . 13  |-  ( ( 2  x.  3 )  +  2 )  =  ( 6  +  2 )
265 6p2e8 10451 . . . . . . . . . . . . 13  |-  ( 6  +  2 )  =  8
266264, 265eqtri 2453 . . . . . . . . . . . 12  |-  ( ( 2  x.  3 )  +  2 )  =  8
26799, 18, 58, 18, 259, 200, 55, 263, 266decma 10781 . . . . . . . . . . 11  |-  ( (; 7
2  x.  3 )  +  2 )  = ;; 2 1 8
268 9t3e27 10839 . . . . . . . . . . 11  |-  ( 9  x.  3 )  = ; 2
7
26955, 234, 151, 258, 99, 18, 267, 268decmul1c 10790 . . . . . . . . . 10  |-  ( ( 3 ^ 6 )  x.  3 )  = ;;; 2 1 8 7
27055, 61, 65, 269numexpp1 14090 . . . . . . . . 9  |-  ( 3 ^ 7 )  = ;;; 2 1 8 7
27163, 99deccl 10757 . . . . . . . . . 10  |- ; 1 7  e.  NN0
272271, 99deccl 10757 . . . . . . . . 9  |- ;; 1 7 7  e.  NN0
273 eqid 2433 . . . . . . . . . 10  |- ;; 2 1 8  = ;; 2 1 8
274 eqid 2433 . . . . . . . . . 10  |- ;; 1 7 7  = ;; 1 7 7
27518, 58deccl 10757 . . . . . . . . . . 11  |- ; 2 0  e.  NN0
276275, 55deccl 10757 . . . . . . . . . 10  |- ;; 2 0 3  e.  NN0
27718, 18deccl 10757 . . . . . . . . . . 11  |- ; 2 2  e.  NN0
278 eqid 2433 . . . . . . . . . . 11  |- ; 2 1  = ; 2 1
279 eqid 2433 . . . . . . . . . . . 12  |- ; 1 7  = ; 1 7
280 eqid 2433 . . . . . . . . . . . 12  |- ;; 2 0 3  = ;; 2 0 3
281 eqid 2433 . . . . . . . . . . . . . 14  |- ; 2 0  = ; 2 0
28281addid2i 9545 . . . . . . . . . . . . . 14  |-  ( 0  +  2 )  =  2
2834addid1i 9544 . . . . . . . . . . . . . 14  |-  ( 1  +  0 )  =  1
28458, 63, 18, 58, 202, 281, 282, 283decadd 10784 . . . . . . . . . . . . 13  |-  ( 1  + ; 2 0 )  = ; 2
1
28518, 63, 247, 284decsuc 10766 . . . . . . . . . . . 12  |-  ( ( 1  + ; 2 0 )  +  1 )  = ; 2 2
286 7p3e10 10455 . . . . . . . . . . . 12  |-  ( 7  +  3 )  =  10
28763, 99, 275, 55, 279, 280, 285, 286decaddc2 10786 . . . . . . . . . . 11  |-  (; 1 7  + ;; 2 0 3 )  = ;; 2 2 0
288 eqid 2433 . . . . . . . . . . . 12  |- ;; 2 5 3  = ;; 2 5 3
289 eqid 2433 . . . . . . . . . . . . 13  |- ; 2 2  = ; 2 2
290 eqid 2433 . . . . . . . . . . . . 13  |- ; 2 5  = ; 2 5
291 2p2e4 10427 . . . . . . . . . . . . 13  |-  ( 2  +  2 )  =  4
29277, 81, 221addcomli 9549 . . . . . . . . . . . . 13  |-  ( 2  +  5 )  =  7
29318, 18, 18, 56, 289, 290, 291, 292decadd 10784 . . . . . . . . . . . 12  |-  (; 2 2  + ; 2 5 )  = ; 4
7
29456dec0h 10759 . . . . . . . . . . . . . 14  |-  5  = ; 0 5
295196, 294eqtri 2453 . . . . . . . . . . . . 13  |-  ( 4  +  1 )  = ; 0
5
296242, 201oveq12i 6092 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
297296, 196eqtri 2453 . . . . . . . . . . . . 13  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  5
298 5t2e10 10464 . . . . . . . . . . . . . . 15  |-  ( 5  x.  2 )  =  10
299 dec10 10773 . . . . . . . . . . . . . . 15  |-  10  = ; 1 0
300298, 299eqtri 2453 . . . . . . . . . . . . . 14  |-  ( 5  x.  2 )  = ; 1
0
30177addid2i 9545 . . . . . . . . . . . . . 14  |-  ( 0  +  5 )  =  5
30263, 58, 56, 300, 301decaddi 10787 . . . . . . . . . . . . 13  |-  ( ( 5  x.  2 )  +  5 )  = ; 1
5
30318, 56, 58, 56, 290, 295, 18, 56, 63, 297, 302decmac 10782 . . . . . . . . . . . 12  |-  ( (; 2
5  x.  2 )  +  ( 4  +  1 ) )  = ; 5
5
304235oveq1i 6090 . . . . . . . . . . . . 13  |-  ( ( 3  x.  2 )  +  7 )  =  ( 6  +  7 )
305 7p6e13 10797 . . . . . . . . . . . . . 14  |-  ( 7  +  6 )  = ; 1
3
30676, 179, 305addcomli 9549 . . . . . . . . . . . . 13  |-  ( 6  +  7 )  = ; 1
3
307304, 306eqtri 2453 . . . . . . . . . . . 12  |-  ( ( 3  x.  2 )  +  7 )  = ; 1
3
308230, 55, 11, 99, 288, 293, 18, 55, 63, 303, 307decmac 10782 . . . . . . . . . . 11  |-  ( (;; 2 5 3  x.  2 )  +  (; 2
2  + ; 2 5 ) )  = ;; 5 5 3
309231nn0cni 10579 . . . . . . . . . . . . . 14  |- ;; 2 5 3  e.  CC
310309mulid1i 9376 . . . . . . . . . . . . 13  |-  (;; 2 5 3  x.  1 )  = ;; 2 5 3
311310oveq1i 6090 . . . . . . . . . . . 12  |-  ( (;; 2 5 3  x.  1 )  +  0 )  =  (;; 2 5 3  +  0 )
312309addid1i 9544 . . . . . . . . . . . 12  |-  (;; 2 5 3  +  0 )  = ;; 2 5 3
313311, 312eqtri 2453 . . . . . . . . . . 11  |-  ( (;; 2 5 3  x.  1 )  +  0 )  = ;; 2 5 3
31418, 63, 277, 58, 278, 287, 231, 55, 230, 308, 313decma2c 10783 . . . . . . . . . 10  |-  ( (;; 2 5 3  x. ; 2
1 )  +  (; 1
7  + ;; 2 0 3 ) )  = ;;; 5 5 3 3
31599dec0h 10759 . . . . . . . . . . 11  |-  7  = ; 0 7
31680addid2i 9545 . . . . . . . . . . . . . 14  |-  ( 0  +  4 )  =  4
317316oveq2i 6091 . . . . . . . . . . . . 13  |-  ( ( 2  x.  8 )  +  ( 0  +  4 ) )  =  ( ( 2  x.  8 )  +  4 )
318 8t2e16 10831 . . . . . . . . . . . . . . 15  |-  ( 8  x.  2 )  = ; 1
6
319248, 81, 318mulcomli 9381 . . . . . . . . . . . . . 14  |-  ( 2  x.  8 )  = ; 1
6
320 6p4e10 10453 . . . . . . . . . . . . . 14  |-  ( 6  +  4 )  =  10
32163, 61, 11, 319, 247, 320decaddci2 10789 . . . . . . . . . . . . 13  |-  ( ( 2  x.  8 )  +  4 )  = ; 2
0
322317, 321eqtri 2453 . . . . . . . . . . . 12  |-  ( ( 2  x.  8 )  +  ( 0  +  4 ) )  = ; 2
0
323 8t5e40 10834 . . . . . . . . . . . . . 14  |-  ( 8  x.  5 )  = ; 4
0
324248, 77, 323mulcomli 9381 . . . . . . . . . . . . 13  |-  ( 5  x.  8 )  = ; 4
0
32511, 58, 55, 324, 185decaddi 10787 . . . . . . . . . . . 12  |-  ( ( 5  x.  8 )  +  3 )  = ; 4
3
32618, 56, 58, 55, 290, 187, 138, 55, 11, 322, 325decmac 10782 . . . . . . . . . . 11  |-  ( (; 2
5  x.  8 )  +  ( 0  +  3 ) )  = ;; 2 0 3
327 8t3e24 10832 . . . . . . . . . . . . 13  |-  ( 8  x.  3 )  = ; 2
4
328248, 3, 327mulcomli 9381 . . . . . . . . . . . 12  |-  ( 3  x.  8 )  = ; 2
4
329 2p1e3 10433 . . . . . . . . . . . 12  |-  ( 2  +  1 )  =  3
330 7p4e11 10795 . . . . . . . . . . . . 13  |-  ( 7  +  4 )  = ; 1
1
33176, 80, 330addcomli 9549 . . . . . . . . . . . 12  |-  ( 4  +  7 )  = ; 1
1
33218, 11, 99, 328, 329, 63, 331decaddci 10788 . . . . . . . . . . 11  |-  ( ( 3  x.  8 )  +  7 )  = ; 3
1
333230, 55, 58, 99, 288, 315, 138, 63, 55, 326, 332decmac 10782 . . . . . . . . . 10  |-  ( (;; 2 5 3  x.  8 )  +  7 )  = ;;; 2 0 3 1
334232, 138, 271, 99, 273, 274, 231, 63, 276, 314, 333decma2c 10783 . . . . . . . . 9  |-  ( (;; 2 5 3  x. ;; 2 1 8 )  + ;; 1 7 7 )  = ;;;; 5 5 3 3 1
335185oveq2i 6091 . . . . . . . . . . . 12  |-  ( ( 2  x.  7 )  +  ( 0  +  3 ) )  =  ( ( 2  x.  7 )  +  3 )
33663, 11, 55, 253, 244decaddi 10787 . . . . . . . . . . . 12  |-  ( ( 2  x.  7 )  +  3 )  = ; 1
7
337335, 336eqtri 2453 . . . . . . . . . . 11  |-  ( ( 2  x.  7 )  +  ( 0  +  3 ) )  = ; 1
7
33855, 56, 18, 79, 221decaddi 10787 . . . . . . . . . . 11  |-  ( ( 5  x.  7 )  +  2 )  = ; 3
7
33918, 56, 58, 18, 290, 200, 99, 99, 55, 337, 338decmac 10782 . . . . . . . . . 10  |-  ( (; 2
5  x.  7 )  +  2 )  = ;; 1 7 7
34099, 230, 55, 288, 63, 18, 339, 178decmul1c 10790 . . . . . . . . 9  |-  (;; 2 5 3  x.  7 )  = ;;; 1 7 7 1
341231, 233, 99, 270, 63, 272, 334, 340decmul2c 10791 . . . . . . . 8  |-  (;; 2 5 3  x.  (
3 ^ 7 ) )  = ;;;;; 5 5 3 3 1 1
342 eqid 2433 . . . . . . . . . . 11  |- ;;;; 5 5 3 3 1  = ;;;; 5 5 3 3 1
343 eqid 2433 . . . . . . . . . . . . . 14  |- ;;; 5 5 3 3  = ;;; 5 5 3 3
344 eqid 2433 . . . . . . . . . . . . . . 15  |- ;; 5 5 3  = ;; 5 5 3
345 eqid 2433 . . . . . . . . . . . . . . . 16  |- ; 5 5  = ; 5 5
346282, 200eqtri 2453 . . . . . . . . . . . . . . . 16  |-  ( 0  +  2 )  = ; 0
2
347185oveq2i 6091 . . . . . . . . . . . . . . . . 17  |-  ( ( 5  x.  7 )  +  ( 0  +  3 ) )  =  ( ( 5  x.  7 )  +  3 )
348347, 175eqtri 2453 . . . . . . . . . . . . . . . 16  |-  ( ( 5  x.  7 )  +  ( 0  +  3 ) )  = ; 3
8
34956, 56, 58, 18, 345, 346, 99, 99, 55, 348, 338decmac 10782 . . . . . . . . . . . . . . 15  |-  ( (; 5
5  x.  7 )  +  ( 0  +  2 ) )  = ;; 3 8 7
35018, 63, 18, 178, 172decaddi 10787 . . . . . . . . . . . . . . 15  |-  ( ( 3  x.  7 )  +  2 )  = ; 2
3
351226, 55, 58, 18, 344, 200, 99, 55, 18, 349, 350decmac 10782 . . . . . . . . . . . . . 14  |-  ( (;; 5 5 3  x.  7 )  +  2 )  = ;;; 3 8 7 3
35299, 227, 55, 343, 63, 18, 351, 178decmul1c 10790 . . . . . . . . . . . . 13  |-  (;;; 5 5 3 3  x.  7 )  = ;;;; 3 8 7 3 1
353352oveq1i 6090 . . . . . . . . . . . 12  |-  ( (;;; 5 5 3 3  x.  7 )  +  0 )  =  (;;;; 3 8 7 3 1  +  0 )
354142nn0cni 10579 . . . . . . . . . . . . 13  |- ;;;; 3 8 7 3 1  e.  CC
355354addid1i 9544 . . . . . . . . . . . 12  |-  (;;;; 3 8 7 3 1  +  0 )  = ;;;; 3 8 7 3 1
356353, 355eqtri 2453 . . . . . . . . . . 11  |-  ( (;;; 5 5 3 3  x.  7 )  +  0 )  = ;;;; 3 8 7 3 1
35776mulid2i 9377 . . . . . . . . . . . 12  |-  ( 1  x.  7 )  =  7
358357, 315eqtri 2453 . . . . . . . . . . 11  |-  ( 1  x.  7 )  = ; 0
7
35999, 228, 63, 342, 99, 58, 356, 358decmul1c 10790 . . . . . . . . . 10  |-  (;;;; 5 5 3 3 1  x.  7 )  = ;;;;; 3 8 7 3 1 7
360359oveq1i 6090 . . . . . . . . 9  |-  ( (;;;; 5 5 3 3 1  x.  7 )  +  0 )  =  (;;;;; 3 8 7 3 1 7  +  0 )
361144nn0cni 10579 . . . . . . . . . 10  |- ;;;;; 3 8 7 3 1 7  e.  CC
362361addid1i 9544 . . . . . . . . 9  |-  (;;;;; 3 8 7 3 1 7  +  0 )  = ;;;;; 3 8 7 3 1 7
363360, 362eqtri 2453 . . . . . . . 8  |-  ( (;;;; 5 5 3 3 1  x.  7 )  +  0 )  = ;;;;; 3 8 7 3 1 7
36499, 229, 63, 341, 99, 58, 363, 358decmul1c 10790 . . . . . . 7  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  = ;;;;;; 3 8 7 3 1 7 7
365148, 225, 3643brtr4i 4308 . . . . . 6  |-  (;;;; 5 3 0 5 7  x. ; 7 3 )  < 
( (;; 2 5 3  x.  (
3 ^ 7 ) )  x.  7 )
36699, 55deccl 10757 . . . . . . . . 9  |- ; 7 3  e.  NN0
367129, 366nn0mulcli 10606 . . . . . . . 8  |-  (;;;; 5 3 0 5 7  x. ; 7 3 )  e. 
NN0
368367nn0rei 10578 . . . . . . 7  |-  (;;;; 5 3 0 5 7  x. ; 7 3 )  e.  RR
36955, 99nn0expcli 11875 . . . . . . . . . 10  |-  ( 3 ^ 7 )  e. 
NN0
370231, 369nn0mulcli 10606 . . . . . . . . 9  |-  (;; 2 5 3  x.  (
3 ^ 7 ) )  e.  NN0
371370, 99nn0mulcli 10606 . . . . . . . 8  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  e.  NN0
372371nn0rei 10578 . . . . . . 7  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  e.  RR
37368nnrei 10319 . . . . . . 7  |-  5  e.  RR
37468nngt0i 10343 . . . . . . 7  |-  0  <  5
375368, 372, 373, 374ltmul1ii 10249 . . . . . 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 ) )
376365, 375mpbi 208 . . . . 5  |-  ( (;;;; 5 3 0 5 7  x. ; 7 3 )  x.  5 )  <  (
( (;; 2 5 3  x.  (
3 ^ 7 ) )  x.  7 )  x.  5 )
377129nn0cni 10579 . . . . . . 7  |- ;;;; 5 3 0 5 7  e.  CC
378366nn0cni 10579 . . . . . . 7  |- ; 7 3  e.  CC
379377, 378, 77mulassi 9383 . . . . . 6  |-  ( (;;;; 5 3 0 5 7  x. ; 7 3 )  x.  5 )  =  (;;;; 5 3 0 5 7  x.  (; 7 3  x.  5 ) )
38055, 56, 159, 78decsuc 10766 . . . . . . . 8  |-  ( ( 7  x.  5 )  +  1 )  = ; 3
6
38177, 3, 207mulcomli 9381 . . . . . . . 8  |-  ( 3  x.  5 )  = ; 1
5
38256, 99, 55, 149, 56, 63, 380, 381decmul1c 10790 . . . . . . 7  |-  (; 7 3  x.  5 )  = ;; 3 6 5
383382oveq2i 6091 . . . . . 6  |-  (;;;; 5 3 0 5 7  x.  (; 7 3  x.  5 ) )  =  (;;;; 5 3 0 5 7  x. ;; 3 6 5 )
384379, 383eqtri 2453 . . . . 5  |-  ( (;;;; 5 3 0 5 7  x. ; 7 3 )  x.  5 )  =  (;;;; 5 3 0 5 7  x. ;; 3 6 5 )
385309, 102mulcli 9379 . . . . . . 7  |-  (;; 2 5 3  x.  (
3 ^ 7 ) )  e.  CC
386385, 76, 77mulassi 9383 . . . . . 6  |-  ( ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  x.  5 )  =  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  ( 7  x.  5 ) )
38776, 77mulcomi 9380 . . . . . . . 8  |-  ( 7  x.  5 )  =  ( 5  x.  7 )
388387oveq2i 6091 . . . . . . 7  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  (
7  x.  5 ) )  =  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  (
5  x.  7 ) )
389309, 102, 103mulassi 9383 . . . . . . 7  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  (
5  x.  7 ) )  =  (;; 2 5 3  x.  (
( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )
390388, 389eqtri 2453 . . . . . 6  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  (
7  x.  5 ) )  =  (;; 2 5 3  x.  (
( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )
391386, 390eqtri 2453 . . . . 5  |-  ( ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  x.  5 )  =  (;; 2 5 3  x.  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) ) )
392376, 384, 3913brtr3i 4307 . . . 4  |-  (;;;; 5 3 0 5 7  x. ;; 3 6 5 )  <  (;; 2 5 3  x.  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) ) )
39355, 61deccl 10757 . . . . . . . 8  |- ; 3 6  e.  NN0
394393, 68decnncl 10756 . . . . . . 7  |- ;; 3 6 5  e.  NN
395394nnrei 10319 . . . . . 6  |- ;; 3 6 5  e.  RR
396394nngt0i 10343 . . . . . 6  |-  0  < ;; 3 6 5
397395, 396pm3.2i 452 . . . . 5  |-  (;; 3 6 5  e.  RR  /\  0  < ;; 3 6 5 )
398231nn0rei 10578 . . . . 5  |- ;; 2 5 3  e.  RR
399 lt2mul2div 10196 . . . . 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 ) ) )
400130, 397, 398, 134, 399mp4an 666 . . . 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 ) )
401392, 400mpbi 208 . . 3  |-  (;;;; 5 3 0 5 7  /  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  < 
(;; 2 5 3  / ;; 3 6 5 )
402 nndivre 10345 . . . . 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 )
403130, 131, 402mp2an 665 . . . 4  |-  (;;;; 5 3 0 5 7  /  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  e.  RR
404 nndivre 10345 . . . . 5  |-  ( (;; 2 5 3  e.  RR  /\ ;; 3 6 5  e.  NN )  ->  (;; 2 5 3  / ;; 3 6 5 )  e.  RR )
405398, 394, 404mp2an 665 . . . 4  |-  (;; 2 5 3  / ;; 3 6 5 )  e.  RR
406128, 403, 405lelttri 9489 . . 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 ) )
407137, 401, 406mp2an 665 . 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 )
40833, 128, 405lelttri 9489 . 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 ) )
40953, 407, 408mp2an 665 1  |-  ( log `  2 )  < 
(;; 2 5 3  / ;; 3 6 5 )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 958    = wceq 1362   T. wtru 1363    e. wcel 1755   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   CCcc 9268   RRcr 9269   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275    < clt 9406    <_ cle 9407    - cmin 9583    / cdiv 9981   NNcn 10310   2c2 10359   3c3 10360   4c4 10361   5c5 10362   6c6 10363   7c7 10364   8c8 10365   9c9 10366   10c10 10367   NN0cn0 10567  ;cdc 10743   RR+crp 10979   [,]cicc 11291   ...cfz 11424   ^cexp 11849   sum_csu 13147   logclog 21891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-shft 12540  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-limsup 12933  df-clim 12950  df-rlim 12951  df-sum 13148  df-ef 13336  df-sin 13338  df-cos 13339  df-tan 13340  df-pi 13341  df-dvds 13519  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-cmp 18832  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-ulm 21727  df-log 21893  df-atan 22147
This theorem is referenced by:  birthday  22233  log2le1  26320
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