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Theorem log2ub 20790
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 10061 . . . . . . . . . . 11  |-  4  =  ( 3  +  1 )
21oveq1i 6092 . . . . . . . . . 10  |-  ( 4  -  1 )  =  ( ( 3  +  1 )  -  1 )
3 3cn 10073 . . . . . . . . . . 11  |-  3  e.  CC
4 ax-1cn 9049 . . . . . . . . . . 11  |-  1  e.  CC
5 pncan 9312 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  1  e.  CC )  ->  ( ( 3  +  1 )  -  1 )  =  3 )
63, 4, 5mp2an 655 . . . . . . . . . 10  |-  ( ( 3  +  1 )  -  1 )  =  3
72, 6eqtri 2457 . . . . . . . . 9  |-  ( 4  -  1 )  =  3
87oveq2i 6093 . . . . . . . 8  |-  ( 0 ... ( 4  -  1 ) )  =  ( 0 ... 3
)
98sumeq1i 12493 . . . . . . 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 6093 . . . . . 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 10241 . . . . . . 7  |-  4  e.  NN0
12 log2tlbnd 20786 . . . . . . 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 8 . . . . . 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 2508 . . . . 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 9092 . . . . . 6  |-  0  e.  RR
16 3re 10072 . . . . . . 7  |-  3  e.  RR
17 4nn 10136 . . . . . . . . 9  |-  4  e.  NN
18 2nn0 10239 . . . . . . . . . 10  |-  2  e.  NN0
19 1nn 10012 . . . . . . . . . 10  |-  1  e.  NN
2018, 11, 19numnncl 10391 . . . . . . . . 9  |-  ( ( 2  x.  4 )  +  1 )  e.  NN
2117, 20nnmulcli 10025 . . . . . . . 8  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  e.  NN
22 9nn 10141 . . . . . . . . 9  |-  9  e.  NN
23 nnexpcl 11395 . . . . . . . . 9  |-  ( ( 9  e.  NN  /\  4  e.  NN0 )  -> 
( 9 ^ 4 )  e.  NN )
2422, 11, 23mp2an 655 . . . . . . . 8  |-  ( 9 ^ 4 )  e.  NN
2521, 24nnmulcli 10025 . . . . . . 7  |-  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) )  e.  NN
26 nndivre 10036 . . . . . . 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 655 . . . . . 6  |-  ( 3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) )  e.  RR
2815, 27elicc2i 10977 . . . . 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 201 . . . 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 969 . . 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 10618 . . . . 5  |-  2  e.  RR+
32 relogcl 20474 . . . . 5  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
3331, 32ax-mp 8 . . . 4  |-  ( log `  2 )  e.  RR
34 fzfid 11313 . . . . . 6  |-  (  T. 
->  ( 0 ... 3
)  e.  Fin )
35 2re 10070 . . . . . . 7  |-  2  e.  RR
36 3nn 10135 . . . . . . . . 9  |-  3  e.  NN
37 elfznn0 11084 . . . . . . . . . . . 12  |-  ( n  e.  ( 0 ... 3 )  ->  n  e.  NN0 )
3837adantl 454 . . . . . . . . . . 11  |-  ( (  T.  /\  n  e.  ( 0 ... 3
) )  ->  n  e.  NN0 )
39 nn0mulcl 10257 . . . . . . . . . . 11  |-  ( ( 2  e.  NN0  /\  n  e.  NN0 )  -> 
( 2  x.  n
)  e.  NN0 )
4018, 38, 39sylancr 646 . . . . . . . . . 10  |-  ( (  T.  /\  n  e.  ( 0 ... 3
) )  ->  (
2  x.  n )  e.  NN0 )
41 nn0p1nn 10260 . . . . . . . . . 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 10024 . . . . . . . . 9  |-  ( ( 3  e.  NN  /\  ( ( 2  x.  n )  +  1 )  e.  NN )  ->  ( 3  x.  ( ( 2  x.  n )  +  1 ) )  e.  NN )
4436, 42, 43sylancr 646 . . . . . . . 8  |-  ( (  T.  /\  n  e.  ( 0 ... 3
) )  ->  (
3  x.  ( ( 2  x.  n )  +  1 ) )  e.  NN )
45 nnexpcl 11395 . . . . . . . . 9  |-  ( ( 9  e.  NN  /\  n  e.  NN0 )  -> 
( 9 ^ n
)  e.  NN )
4622, 38, 45sylancr 646 . . . . . . . 8  |-  ( (  T.  /\  n  e.  ( 0 ... 3
) )  ->  (
9 ^ n )  e.  NN )
4744, 46nnmulcld 10048 . . . . . . 7  |-  ( (  T.  /\  n  e.  ( 0 ... 3
) )  ->  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) )  e.  NN )
48 nndivre 10036 . . . . . . 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 646 . . . . . 6  |-  ( (  T.  /\  n  e.  ( 0 ... 3
) )  ->  (
2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) )  e.  RR )
5034, 49fsumrecl 12529 . . . . 5  |-  (  T. 
->  sum_ n  e.  ( 0 ... 3 ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) )  e.  RR )
5150trud 1333 . . . 4  |-  sum_ n  e.  ( 0 ... 3
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  e.  RR
5233, 51, 27lesubadd2i 9588 . . 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 201 . 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 20789 . . . . 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 10240 . . . . 5  |-  3  e.  NN0
56 5nn0 10242 . . . . . . . . 9  |-  5  e.  NN0
5756, 55deccl 10397 . . . . . . . 8  |- ; 5 3  e.  NN0
58 0nn0 10237 . . . . . . . 8  |-  0  e.  NN0
5957, 58deccl 10397 . . . . . . 7  |- ;; 5 3 0  e.  NN0
6059, 56deccl 10397 . . . . . 6  |- ;;; 5 3 0 5  e.  NN0
61 6nn0 10243 . . . . . 6  |-  6  e.  NN0
6260, 61deccl 10397 . . . . 5  |- ;;;; 5 3 0 5 6  e.  NN0
63 1nn0 10238 . . . . 5  |-  1  e.  NN0
64 eqid 2437 . . . . 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 10108 . . . . . 6  |-  ( 6  +  1 )  =  7
66 eqid 2437 . . . . . 6  |- ;;;; 5 3 0 5 6  = ;;;; 5 3 0 5 6
6760, 61, 65, 66decsuc 10406 . . . . 5  |-  (;;;; 5 3 0 5 6  +  1 )  = ;;;; 5 3 0 5 7
68 5nn 10137 . . . . . . . . . 10  |-  5  e.  NN
69 7nn 10139 . . . . . . . . . 10  |-  7  e.  NN
7068, 69nnmulcli 10025 . . . . . . . . 9  |-  ( 5  x.  7 )  e.  NN
7170nnrei 10010 . . . . . . . 8  |-  ( 5  x.  7 )  e.  RR
7221nnrei 10010 . . . . . . . 8  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  e.  RR
73 6nn 10138 . . . . . . . . . 10  |-  6  e.  NN
74 5lt6 10153 . . . . . . . . . 10  |-  5  <  6
7555, 56, 73, 74declt 10404 . . . . . . . . 9  |- ; 3 5  < ; 3 6
7669nncni 10011 . . . . . . . . . 10  |-  7  e.  CC
7768nncni 10011 . . . . . . . . . 10  |-  5  e.  CC
78 7t5e35 10468 . . . . . . . . . 10  |-  ( 7  x.  5 )  = ; 3
5
7976, 77, 78mulcomli 9098 . . . . . . . . 9  |-  ( 5  x.  7 )  = ; 3
5
80 4cn 10075 . . . . . . . . . . . . . 14  |-  4  e.  CC
81 2cn 10071 . . . . . . . . . . . . . 14  |-  2  e.  CC
82 4t2e8 10131 . . . . . . . . . . . . . 14  |-  ( 4  x.  2 )  =  8
8380, 81, 82mulcomli 9098 . . . . . . . . . . . . 13  |-  ( 2  x.  4 )  =  8
8483oveq1i 6092 . . . . . . . . . . . 12  |-  ( ( 2  x.  4 )  +  1 )  =  ( 8  +  1 )
85 8p1e9 10110 . . . . . . . . . . . 12  |-  ( 8  +  1 )  =  9
8684, 85eqtri 2457 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  +  1 )  =  9
8786oveq2i 6093 . . . . . . . . . 10  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  =  ( 4  x.  9 )
8822nncni 10011 . . . . . . . . . . 11  |-  9  e.  CC
89 9t4e36 10480 . . . . . . . . . . 11  |-  ( 9  x.  4 )  = ; 3
6
9088, 80, 89mulcomli 9098 . . . . . . . . . 10  |-  ( 4  x.  9 )  = ; 3
6
9187, 90eqtri 2457 . . . . . . . . 9  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  = ; 3
6
9275, 79, 913brtr4i 4241 . . . . . . . 8  |-  ( 5  x.  7 )  < 
( 4  x.  (
( 2  x.  4 )  +  1 ) )
9371, 72, 92ltleii 9197 . . . . . . 7  |-  ( 5  x.  7 )  <_ 
( 4  x.  (
( 2  x.  4 )  +  1 ) )
9424nngt0i 10034 . . . . . . . 8  |-  0  <  ( 9 ^ 4 )
9524nnrei 10010 . . . . . . . . 9  |-  ( 9 ^ 4 )  e.  RR
9671, 72, 95lemul2i 9935 . . . . . . . 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 8 . . . . . . 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 201 . . . . . 6  |-  ( ( 9 ^ 4 )  x.  ( 5  x.  7 ) )  <_ 
( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) )
99 7nn0 10244 . . . . . . . . . 10  |-  7  e.  NN0
100 nnexpcl 11395 . . . . . . . . . 10  |-  ( ( 3  e.  NN  /\  7  e.  NN0 )  -> 
( 3 ^ 7 )  e.  NN )
10136, 99, 100mp2an 655 . . . . . . . . 9  |-  ( 3 ^ 7 )  e.  NN
102101nncni 10011 . . . . . . . 8  |-  ( 3 ^ 7 )  e.  CC
10370nncni 10011 . . . . . . . 8  |-  ( 5  x.  7 )  e.  CC
104102, 103, 3mul32i 9263 . . . . . . 7  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  3 )  =  ( ( ( 3 ^ 7 )  x.  3 )  x.  (
5  x.  7 ) )
10580, 81mulcomi 9097 . . . . . . . . . . . 12  |-  ( 4  x.  2 )  =  ( 2  x.  4 )
106 df-8 10065 . . . . . . . . . . . 12  |-  8  =  ( 7  +  1 )
10782, 105, 1063eqtr3i 2465 . . . . . . . . . . 11  |-  ( 2  x.  4 )  =  ( 7  +  1 )
108107oveq2i 6093 . . . . . . . . . 10  |-  ( 3 ^ ( 2  x.  4 ) )  =  ( 3 ^ (
7  +  1 ) )
109 expmul 11426 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  2  e.  NN0  /\  4  e.  NN0 )  ->  (
3 ^ ( 2  x.  4 ) )  =  ( ( 3 ^ 2 ) ^
4 ) )
1103, 18, 11, 109mp3an 1280 . . . . . . . . . 10  |-  ( 3 ^ ( 2  x.  4 ) )  =  ( ( 3 ^ 2 ) ^ 4 )
111108, 110eqtr3i 2459 . . . . . . . . 9  |-  ( 3 ^ ( 7  +  1 ) )  =  ( ( 3 ^ 2 ) ^ 4 )
112 expp1 11389 . . . . . . . . . 10  |-  ( ( 3  e.  CC  /\  7  e.  NN0 )  -> 
( 3 ^ (
7  +  1 ) )  =  ( ( 3 ^ 7 )  x.  3 ) )
1133, 99, 112mp2an 655 . . . . . . . . 9  |-  ( 3 ^ ( 7  +  1 ) )  =  ( ( 3 ^ 7 )  x.  3 )
114 sq3 11479 . . . . . . . . . 10  |-  ( 3 ^ 2 )  =  9
115114oveq1i 6092 . . . . . . . . 9  |-  ( ( 3 ^ 2 ) ^ 4 )  =  ( 9 ^ 4 )
116111, 113, 1153eqtr3i 2465 . . . . . . . 8  |-  ( ( 3 ^ 7 )  x.  3 )  =  ( 9 ^ 4 )
117116oveq1i 6092 . . . . . . 7  |-  ( ( ( 3 ^ 7 )  x.  3 )  x.  ( 5  x.  7 ) )  =  ( ( 9 ^ 4 )  x.  (
5  x.  7 ) )
118104, 117eqtri 2457 . . . . . 6  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  3 )  =  ( ( 9 ^ 4 )  x.  (
5  x.  7 ) )
11921nncni 10011 . . . . . . . . 9  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  e.  CC
12024nncni 10011 . . . . . . . . 9  |-  ( 9 ^ 4 )  e.  CC
121119, 120mulcomi 9097 . . . . . . . 8  |-  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) )  =  ( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) )
122121oveq1i 6092 . . . . . . 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 9096 . . . . . . . 8  |-  ( ( 9 ^ 4 )  x.  ( 4  x.  ( ( 2  x.  4 )  +  1 ) ) )  e.  CC
124123mulid1i 9093 . . . . . . 7  |-  ( ( ( 9 ^ 4 )  x.  ( 4  x.  ( ( 2  x.  4 )  +  1 ) ) )  x.  1 )  =  ( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) )
125122, 124eqtri 2457 . . . . . 6  |-  ( ( ( 4  x.  (
( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) )  x.  1 )  =  ( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) )
12698, 118, 1253brtr4i 4241 . . . . 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 20787 . . . 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 9104 . . . . 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 10397 . . . . . 6  |- ;;;; 5 3 0 5 7  e.  NN0
130129nn0rei 10233 . . . . 5  |- ;;;; 5 3 0 5 7  e.  RR
131101, 70nnmulcli 10025 . . . . . . 7  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  NN
132131nnrei 10010 . . . . . 6  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR
133131nngt0i 10034 . . . . . 6  |-  0  <  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) )
134132, 133pm3.2i 443 . . . . 5  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR  /\  0  <  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) ) )
135 lemuldiv2 9891 . . . . 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 1280 . . . 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 201 . . 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 10245 . . . . . . . . . . . . 13  |-  8  e.  NN0
13955, 138deccl 10397 . . . . . . . . . . . 12  |- ; 3 8  e.  NN0
140139, 99deccl 10397 . . . . . . . . . . 11  |- ;; 3 8 7  e.  NN0
141140, 55deccl 10397 . . . . . . . . . 10  |- ;;; 3 8 7 3  e.  NN0
142141, 63deccl 10397 . . . . . . . . 9  |- ;;;; 3 8 7 3 1  e.  NN0
143142, 61deccl 10397 . . . . . . . 8  |- ;;;;; 3 8 7 3 1 6  e.  NN0
144142, 99deccl 10397 . . . . . . . 8  |- ;;;;; 3 8 7 3 1 7  e.  NN0
145 1lt10 10187 . . . . . . . 8  |-  1  <  10
146 6lt7 10158 . . . . . . . . 9  |-  6  <  7
147142, 61, 69, 146declt 10404 . . . . . . . 8  |- ;;;;; 3 8 7 3 1 6  < ;;;;; 3 8 7 3 1 7
148143, 144, 63, 99, 145, 147decltc 10405 . . . . . . 7  |- ;;;;;; 3 8 7 3 1 6 1  < ;;;;;; 3 8 7 3 1 7 7
149 eqid 2437 . . . . . . . 8  |- ; 7 3  = ; 7 3
15063, 56deccl 10397 . . . . . . . . . . 11  |- ; 1 5  e.  NN0
151 9nn0 10246 . . . . . . . . . . 11  |-  9  e.  NN0
152150, 151deccl 10397 . . . . . . . . . 10  |- ;; 1 5 9  e.  NN0
153152, 63deccl 10397 . . . . . . . . 9  |- ;;; 1 5 9 1  e.  NN0
154153, 99deccl 10397 . . . . . . . 8  |- ;;;; 1 5 9 1 7  e.  NN0
155 eqid 2437 . . . . . . . . 9  |- ;;;; 5 3 0 5 7  = ;;;; 5 3 0 5 7
156 eqid 2437 . . . . . . . . 9  |- ;;;; 1 5 9 1 7  = ;;;; 1 5 9 1 7
157 eqid 2437 . . . . . . . . . 10  |- ;;; 5 3 0 5  = ;;; 5 3 0 5
158 eqid 2437 . . . . . . . . . . 11  |- ;;; 1 5 9 1  = ;;; 1 5 9 1
159 5p1e6 10107 . . . . . . . . . . . 12  |-  ( 5  +  1 )  =  6
16077, 4, 159addcomli 9259 . . . . . . . . . . 11  |-  ( 1  +  5 )  =  6
161152, 63, 56, 158, 160decaddi 10427 . . . . . . . . . 10  |-  (;;; 1 5 9 1  +  5 )  = ;;; 1 5 9 6
16263, 61deccl 10397 . . . . . . . . . . 11  |- ; 1 6  e.  NN0
163 eqid 2437 . . . . . . . . . . 11  |- ;; 5 3 0  = ;; 5 3 0
164 eqid 2437 . . . . . . . . . . . 12  |- ;; 1 5 9  = ;; 1 5 9
165 eqid 2437 . . . . . . . . . . . . 13  |- ; 1 5  = ; 1 5
16663, 56, 159, 165decsuc 10406 . . . . . . . . . . . 12  |-  (; 1 5  +  1 )  = ; 1 6
167 9p4e13 10447 . . . . . . . . . . . 12  |-  ( 9  +  4 )  = ; 1
3
168150, 151, 11, 164, 166, 55, 167decaddci 10428 . . . . . . . . . . 11  |-  (;; 1 5 9  +  4 )  = ;; 1 6 3
169 eqid 2437 . . . . . . . . . . . 12  |- ; 5 3  = ; 5 3
170162nn0cni 10234 . . . . . . . . . . . . 13  |- ; 1 6  e.  CC
171170addid1i 9254 . . . . . . . . . . . 12  |-  (; 1 6  +  0 )  = ; 1 6
172 2p1e3 10104 . . . . . . . . . . . . . . 15  |-  ( 2  +  1 )  =  3
17381, 4, 172addcomli 9259 . . . . . . . . . . . . . 14  |-  ( 1  +  2 )  =  3
174173oveq2i 6093 . . . . . . . . . . . . 13  |-  ( ( 5  x.  7 )  +  ( 1  +  2 ) )  =  ( ( 5  x.  7 )  +  3 )
175 5p3e8 10118 . . . . . . . . . . . . . 14  |-  ( 5  +  3 )  =  8
17655, 56, 55, 79, 175decaddi 10427 . . . . . . . . . . . . 13  |-  ( ( 5  x.  7 )  +  3 )  = ; 3
8
177174, 176eqtri 2457 . . . . . . . . . . . 12  |-  ( ( 5  x.  7 )  +  ( 1  +  2 ) )  = ; 3
8
178 7t3e21 10466 . . . . . . . . . . . . . 14  |-  ( 7  x.  3 )  = ; 2
1
17976, 3, 178mulcomli 9098 . . . . . . . . . . . . 13  |-  ( 3  x.  7 )  = ; 2
1
18073nncni 10011 . . . . . . . . . . . . . 14  |-  6  e.  CC
181180, 4, 65addcomli 9259 . . . . . . . . . . . . 13  |-  ( 1  +  6 )  =  7
18218, 63, 61, 179, 181decaddi 10427 . . . . . . . . . . . 12  |-  ( ( 3  x.  7 )  +  6 )  = ; 2
7
18356, 55, 63, 61, 169, 171, 99, 99, 18, 177, 182decmac 10422 . . . . . . . . . . 11  |-  ( (; 5
3  x.  7 )  +  (; 1 6  +  0 ) )  = ;; 3 8 7
18476mul02i 9256 . . . . . . . . . . . . 13  |-  ( 0  x.  7 )  =  0
185184oveq1i 6092 . . . . . . . . . . . 12  |-  ( ( 0  x.  7 )  +  3 )  =  ( 0  +  3 )
1863addid2i 9255 . . . . . . . . . . . . 13  |-  ( 0  +  3 )  =  3
18755dec0h 10399 . . . . . . . . . . . . 13  |-  3  = ; 0 3
188186, 187eqtri 2457 . . . . . . . . . . . 12  |-  ( 0  +  3 )  = ; 0
3
189185, 188eqtri 2457 . . . . . . . . . . 11  |-  ( ( 0  x.  7 )  +  3 )  = ; 0
3
19057, 58, 162, 55, 163, 168, 99, 55, 58, 183, 189decmac 10422 . . . . . . . . . 10  |-  ( (;; 5 3 0  x.  7 )  +  (;; 1 5 9  +  4 ) )  = ;;; 3 8 7 3
191 3p1e4 10105 . . . . . . . . . . 11  |-  ( 3  +  1 )  =  4
192 6p5e11 10433 . . . . . . . . . . . 12  |-  ( 6  +  5 )  = ; 1
1
193180, 77, 192addcomli 9259 . . . . . . . . . . 11  |-  ( 5  +  6 )  = ; 1
1
19455, 56, 61, 79, 191, 63, 193decaddci 10428 . . . . . . . . . 10  |-  ( ( 5  x.  7 )  +  6 )  = ; 4
1
19559, 56, 152, 61, 157, 161, 99, 63, 11, 190, 194decmac 10422 . . . . . . . . 9  |-  ( (;;; 5 3 0 5  x.  7 )  +  (;;; 1 5 9 1  +  5 ) )  = ;;;; 3 8 7 3 1
196 7t7e49 10470 . . . . . . . . . 10  |-  ( 7  x.  7 )  = ; 4
9
197 4p1e5 10106 . . . . . . . . . 10  |-  ( 4  +  1 )  =  5
198 9p7e16 10450 . . . . . . . . . 10  |-  ( 9  +  7 )  = ; 1
6
19911, 151, 99, 196, 197, 61, 198decaddci 10428 . . . . . . . . 9  |-  ( ( 7  x.  7 )  +  7 )  = ; 5
6
20060, 99, 153, 99, 155, 156, 99, 61, 56, 195, 199decmac 10422 . . . . . . . 8  |-  ( (;;;; 5 3 0 5 7  x.  7 )  + ;;;; 1 5 9 1 7 )  = ;;;;; 3 8 7 3 1 6
20118dec0h 10399 . . . . . . . . . 10  |-  2  = ; 0 2
2024addid2i 9255 . . . . . . . . . . . 12  |-  ( 0  +  1 )  =  1
20363dec0h 10399 . . . . . . . . . . . 12  |-  1  = ; 0 1
204202, 203eqtri 2457 . . . . . . . . . . 11  |-  ( 0  +  1 )  = ; 0
1
205 00id 9242 . . . . . . . . . . . . 13  |-  ( 0  +  0 )  =  0
20658dec0h 10399 . . . . . . . . . . . . 13  |-  0  = ; 0 0
207205, 206eqtri 2457 . . . . . . . . . . . 12  |-  ( 0  +  0 )  = ; 0
0
208 5t3e15 10457 . . . . . . . . . . . . . 14  |-  ( 5  x.  3 )  = ; 1
5
209208oveq1i 6092 . . . . . . . . . . . . 13  |-  ( ( 5  x.  3 )  +  0 )  =  (; 1 5  +  0 )
210150nn0cni 10234 . . . . . . . . . . . . . 14  |- ; 1 5  e.  CC
211210addid1i 9254 . . . . . . . . . . . . 13  |-  (; 1 5  +  0 )  = ; 1 5
212209, 211eqtri 2457 . . . . . . . . . . . 12  |-  ( ( 5  x.  3 )  +  0 )  = ; 1
5
213 3t3e9 10130 . . . . . . . . . . . . . 14  |-  ( 3  x.  3 )  =  9
214213oveq1i 6092 . . . . . . . . . . . . 13  |-  ( ( 3  x.  3 )  +  0 )  =  ( 9  +  0 )
21588addid1i 9254 . . . . . . . . . . . . 13  |-  ( 9  +  0 )  =  9
216214, 215eqtri 2457 . . . . . . . . . . . 12  |-  ( ( 3  x.  3 )  +  0 )  =  9
21756, 55, 58, 58, 169, 207, 55, 212, 216decma 10421 . . . . . . . . . . 11  |-  ( (; 5
3  x.  3 )  +  ( 0  +  0 ) )  = ;; 1 5 9
2183mul02i 9256 . . . . . . . . . . . . 13  |-  ( 0  x.  3 )  =  0
219218oveq1i 6092 . . . . . . . . . . . 12  |-  ( ( 0  x.  3 )  +  1 )  =  ( 0  +  1 )
220219, 204eqtri 2457 . . . . . . . . . . 11  |-  ( ( 0  x.  3 )  +  1 )  = ; 0
1
22157, 58, 58, 63, 163, 204, 55, 63, 58, 217, 220decmac 10422 . . . . . . . . . 10  |-  ( (;; 5 3 0  x.  3 )  +  ( 0  +  1 ) )  = ;;; 1 5 9 1
222 5p2e7 10117 . . . . . . . . . . 11  |-  ( 5  +  2 )  =  7
22363, 56, 18, 208, 222decaddi 10427 . . . . . . . . . 10  |-  ( ( 5  x.  3 )  +  2 )  = ; 1
7
22459, 56, 58, 18, 157, 201, 55, 99, 63, 221, 223decmac 10422 . . . . . . . . 9  |-  ( (;;; 5 3 0 5  x.  3 )  +  2 )  = ;;;; 1 5 9 1 7
22555, 60, 99, 155, 63, 18, 224, 178decmul1c 10430 . . . . . . . 8  |-  (;;;; 5 3 0 5 7  x.  3 )  = ;;;;; 1 5 9 1 7 1
226129, 99, 55, 149, 63, 154, 200, 225decmul2c 10431 . . . . . . 7  |-  (;;;; 5 3 0 5 7  x. ; 7 3 )  = ;;;;;; 3 8 7 3 1 6 1
22756, 56deccl 10397 . . . . . . . . . . 11  |- ; 5 5  e.  NN0
228227, 55deccl 10397 . . . . . . . . . 10  |- ;; 5 5 3  e.  NN0
229228, 55deccl 10397 . . . . . . . . 9  |- ;;; 5 5 3 3  e.  NN0
230229, 63deccl 10397 . . . . . . . 8  |- ;;;; 5 5 3 3 1  e.  NN0
23118, 56deccl 10397 . . . . . . . . . 10  |- ; 2 5  e.  NN0
232231, 55deccl 10397 . . . . . . . . 9  |- ;; 2 5 3  e.  NN0
23318, 63deccl 10397 . . . . . . . . . 10  |- ; 2 1  e.  NN0
234233, 138deccl 10397 . . . . . . . . 9  |- ;; 2 1 8  e.  NN0
23599, 18deccl 10397 . . . . . . . . . . 11  |- ; 7 2  e.  NN0
236 3t2e6 10129 . . . . . . . . . . . . 13  |-  ( 3  x.  2 )  =  6
2373, 81, 236mulcomli 9098 . . . . . . . . . . . 12  |-  ( 2  x.  3 )  =  6
238 3exp3 13426 . . . . . . . . . . . 12  |-  ( 3 ^ 3 )  = ; 2
7
23918, 99deccl 10397 . . . . . . . . . . . . 13  |- ; 2 7  e.  NN0
240 eqid 2437 . . . . . . . . . . . . 13  |- ; 2 7  = ; 2 7
24163, 138deccl 10397 . . . . . . . . . . . . 13  |- ; 1 8  e.  NN0
242 eqid 2437 . . . . . . . . . . . . . 14  |- ; 1 8  = ; 1 8
243 2t2e4 10128 . . . . . . . . . . . . . . . 16  |-  ( 2  x.  2 )  =  4
244243, 173oveq12i 6094 . . . . . . . . . . . . . . 15  |-  ( ( 2  x.  2 )  +  ( 1  +  2 ) )  =  ( 4  +  3 )
245 4p3e7 10115 . . . . . . . . . . . . . . 15  |-  ( 4  +  3 )  =  7
246244, 245eqtri 2457 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  2 )  +  ( 1  +  2 ) )  =  7
247 7t2e14 10465 . . . . . . . . . . . . . . 15  |-  ( 7  x.  2 )  = ; 1
4
248 1p1e2 10095 . . . . . . . . . . . . . . 15  |-  ( 1  +  1 )  =  2
249 8nn 10140 . . . . . . . . . . . . . . . . 17  |-  8  e.  NN
250249nncni 10011 . . . . . . . . . . . . . . . 16  |-  8  e.  CC
251 8p4e12 10440 . . . . . . . . . . . . . . . 16  |-  ( 8  +  4 )  = ; 1
2
252250, 80, 251addcomli 9259 . . . . . . . . . . . . . . 15  |-  ( 4  +  8 )  = ; 1
2
25363, 11, 138, 247, 248, 18, 252decaddci 10428 . . . . . . . . . . . . . 14  |-  ( ( 7  x.  2 )  +  8 )  = ; 2
2
25418, 99, 63, 138, 240, 242, 18, 18, 18, 246, 253decmac 10422 . . . . . . . . . . . . 13  |-  ( (; 2
7  x.  2 )  + ; 1 8 )  = ; 7
2
25576, 81, 247mulcomli 9098 . . . . . . . . . . . . . . 15  |-  ( 2  x.  7 )  = ; 1
4
256 4p4e8 10116 . . . . . . . . . . . . . . 15  |-  ( 4  +  4 )  =  8
25763, 11, 11, 255, 256decaddi 10427 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  7 )  +  4 )  = ; 1
8
25899, 18, 99, 240, 151, 11, 257, 196decmul1c 10430 . . . . . . . . . . . . 13  |-  (; 2 7  x.  7 )  = ;; 1 8 9
259239, 18, 99, 240, 151, 241, 254, 258decmul2c 10431 . . . . . . . . . . . 12  |-  (; 2 7  x. ; 2 7 )  = ;; 7 2 9
26055, 55, 237, 238, 259numexp2x 13416 . . . . . . . . . . 11  |-  ( 3 ^ 6 )  = ;; 7 2 9
261 eqid 2437 . . . . . . . . . . . 12  |- ; 7 2  = ; 7 2
262178oveq1i 6092 . . . . . . . . . . . . 13  |-  ( ( 7  x.  3 )  +  0 )  =  (; 2 1  +  0 )
263233nn0cni 10234 . . . . . . . . . . . . . 14  |- ; 2 1  e.  CC
264263addid1i 9254 . . . . . . . . . . . . 13  |-  (; 2 1  +  0 )  = ; 2 1
265262, 264eqtri 2457 . . . . . . . . . . . 12  |-  ( ( 7  x.  3 )  +  0 )  = ; 2
1
266237oveq1i 6092 . . . . . . . . . . . . 13  |-  ( ( 2  x.  3 )  +  2 )  =  ( 6  +  2 )
267 6p2e8 10121 . . . . . . . . . . . . 13  |-  ( 6  +  2 )  =  8
268266, 267eqtri 2457 . . . . . . . . . . . 12  |-  ( ( 2  x.  3 )  +  2 )  =  8
26999, 18, 58, 18, 261, 201, 55, 265, 268decma 10421 . . . . . . . . . . 11  |-  ( (; 7
2  x.  3 )  +  2 )  = ;; 2 1 8
270 9t3e27 10479 . . . . . . . . . . 11  |-  ( 9  x.  3 )  = ; 2
7
27155, 235, 151, 260, 99, 18, 269, 270decmul1c 10430 . . . . . . . . . 10  |-  ( ( 3 ^ 6 )  x.  3 )  = ;;; 2 1 8 7
27255, 61, 65, 271numexpp1 13415 . . . . . . . . 9  |-  ( 3 ^ 7 )  = ;;; 2 1 8 7
27363, 99deccl 10397 . . . . . . . . . 10  |- ; 1 7  e.  NN0
274273, 99deccl 10397 . . . . . . . . 9  |- ;; 1 7 7  e.  NN0
275 eqid 2437 . . . . . . . . . 10  |- ;; 2 1 8  = ;; 2 1 8
276 eqid 2437 . . . . . . . . . 10  |- ;; 1 7 7  = ;; 1 7 7
27718, 58deccl 10397 . . . . . . . . . . 11  |- ; 2 0  e.  NN0
278277, 55deccl 10397 . . . . . . . . . 10  |- ;; 2 0 3  e.  NN0
27918, 18deccl 10397 . . . . . . . . . . 11  |- ; 2 2  e.  NN0
280 eqid 2437 . . . . . . . . . . 11  |- ; 2 1  = ; 2 1
281 eqid 2437 . . . . . . . . . . . 12  |- ; 1 7  = ; 1 7
282 eqid 2437 . . . . . . . . . . . 12  |- ;; 2 0 3  = ;; 2 0 3
283 eqid 2437 . . . . . . . . . . . . . 14  |- ; 2 0  = ; 2 0
28481addid2i 9255 . . . . . . . . . . . . . 14  |-  ( 0  +  2 )  =  2
2854addid1i 9254 . . . . . . . . . . . . . 14  |-  ( 1  +  0 )  =  1
28658, 63, 18, 58, 203, 283, 284, 285decadd 10424 . . . . . . . . . . . . 13  |-  ( 1  + ; 2 0 )  = ; 2
1
28718, 63, 248, 286decsuc 10406 . . . . . . . . . . . 12  |-  ( ( 1  + ; 2 0 )  +  1 )  = ; 2 2
288 7p3e10 10125 . . . . . . . . . . . 12  |-  ( 7  +  3 )  =  10
28963, 99, 277, 55, 281, 282, 287, 288decaddc2 10426 . . . . . . . . . . 11  |-  (; 1 7  + ;; 2 0 3 )  = ;; 2 2 0
290 eqid 2437 . . . . . . . . . . . 12  |- ;; 2 5 3  = ;; 2 5 3
291 eqid 2437 . . . . . . . . . . . . 13  |- ; 2 2  = ; 2 2
292 eqid 2437 . . . . . . . . . . . . 13  |- ; 2 5  = ; 2 5
293 2p2e4 10099 . . . . . . . . . . . . 13  |-  ( 2  +  2 )  =  4
29477, 81, 222addcomli 9259 . . . . . . . . . . . . 13  |-  ( 2  +  5 )  =  7
29518, 18, 18, 56, 291, 292, 293, 294decadd 10424 . . . . . . . . . . . 12  |-  (; 2 2  + ; 2 5 )  = ; 4
7
29656dec0h 10399 . . . . . . . . . . . . . 14  |-  5  = ; 0 5
297197, 296eqtri 2457 . . . . . . . . . . . . 13  |-  ( 4  +  1 )  = ; 0
5
298243, 202oveq12i 6094 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
299298, 197eqtri 2457 . . . . . . . . . . . . 13  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  5
300 5t2e10 10132 . . . . . . . . . . . . . . 15  |-  ( 5  x.  2 )  =  10
301 dec10 10413 . . . . . . . . . . . . . . 15  |-  10  = ; 1 0
302300, 301eqtri 2457 . . . . . . . . . . . . . 14  |-  ( 5  x.  2 )  = ; 1
0
30377addid2i 9255 . . . . . . . . . . . . . 14  |-  ( 0  +  5 )  =  5
30463, 58, 56, 302, 303decaddi 10427 . . . . . . . . . . . . 13  |-  ( ( 5  x.  2 )  +  5 )  = ; 1
5
30518, 56, 58, 56, 292, 297, 18, 56, 63, 299, 304decmac 10422 . . . . . . . . . . . 12  |-  ( (; 2
5  x.  2 )  +  ( 4  +  1 ) )  = ; 5
5
306236oveq1i 6092 . . . . . . . . . . . . 13  |-  ( ( 3  x.  2 )  +  7 )  =  ( 6  +  7 )
307 7p6e13 10437 . . . . . . . . . . . . . 14  |-  ( 7  +  6 )  = ; 1
3
30876, 180, 307addcomli 9259 . . . . . . . . . . . . 13  |-  ( 6  +  7 )  = ; 1
3
309306, 308eqtri 2457 . . . . . . . . . . . 12  |-  ( ( 3  x.  2 )  +  7 )  = ; 1
3
310231, 55, 11, 99, 290, 295, 18, 55, 63, 305, 309decmac 10422 . . . . . . . . . . 11  |-  ( (;; 2 5 3  x.  2 )  +  (; 2
2  + ; 2 5 ) )  = ;; 5 5 3
311232nn0cni 10234 . . . . . . . . . . . . . 14  |- ;; 2 5 3  e.  CC
312311mulid1i 9093 . . . . . . . . . . . . 13  |-  (;; 2 5 3  x.  1 )  = ;; 2 5 3
313312oveq1i 6092 . . . . . . . . . . . 12  |-  ( (;; 2 5 3  x.  1 )  +  0 )  =  (;; 2 5 3  +  0 )
314311addid1i 9254 . . . . . . . . . . . 12  |-  (;; 2 5 3  +  0 )  = ;; 2 5 3
315313, 314eqtri 2457 . . . . . . . . . . 11  |-  ( (;; 2 5 3  x.  1 )  +  0 )  = ;; 2 5 3
31618, 63, 279, 58, 280, 289, 232, 55, 231, 310, 315decma2c 10423 . . . . . . . . . 10  |-  ( (;; 2 5 3  x. ; 2
1 )  +  (; 1
7  + ;; 2 0 3 ) )  = ;;; 5 5 3 3
31799dec0h 10399 . . . . . . . . . . 11  |-  7  = ; 0 7
31880addid2i 9255 . . . . . . . . . . . . . 14  |-  ( 0  +  4 )  =  4
319318oveq2i 6093 . . . . . . . . . . . . 13  |-  ( ( 2  x.  8 )  +  ( 0  +  4 ) )  =  ( ( 2  x.  8 )  +  4 )
320 8t2e16 10471 . . . . . . . . . . . . . . 15  |-  ( 8  x.  2 )  = ; 1
6
321250, 81, 320mulcomli 9098 . . . . . . . . . . . . . 14  |-  ( 2  x.  8 )  = ; 1
6
322 6p4e10 10123 . . . . . . . . . . . . . 14  |-  ( 6  +  4 )  =  10
32363, 61, 11, 321, 248, 322decaddci2 10429 . . . . . . . . . . . . 13  |-  ( ( 2  x.  8 )  +  4 )  = ; 2
0
324319, 323eqtri 2457 . . . . . . . . . . . 12  |-  ( ( 2  x.  8 )  +  ( 0  +  4 ) )  = ; 2
0
325 8t5e40 10474 . . . . . . . . . . . . . 14  |-  ( 8  x.  5 )  = ; 4
0
326250, 77, 325mulcomli 9098 . . . . . . . . . . . . 13  |-  ( 5  x.  8 )  = ; 4
0
32711, 58, 55, 326, 186decaddi 10427 . . . . . . . . . . . 12  |-  ( ( 5  x.  8 )  +  3 )  = ; 4
3
32818, 56, 58, 55, 292, 188, 138, 55, 11, 324, 327decmac 10422 . . . . . . . . . . 11  |-  ( (; 2
5  x.  8 )  +  ( 0  +  3 ) )  = ;; 2 0 3
329 8t3e24 10472 . . . . . . . . . . . . 13  |-  ( 8  x.  3 )  = ; 2
4
330250, 3, 329mulcomli 9098 . . . . . . . . . . . 12  |-  ( 3  x.  8 )  = ; 2
4
331 7p4e11 10435 . . . . . . . . . . . . 13  |-  ( 7  +  4 )  = ; 1
1
33276, 80, 331addcomli 9259 . . . . . . . . . . . 12  |-  ( 4  +  7 )  = ; 1
1
33318, 11, 99, 330, 172, 63, 332decaddci 10428 . . . . . . . . . . 11  |-  ( ( 3  x.  8 )  +  7 )  = ; 3
1
334231, 55, 58, 99, 290, 317, 138, 63, 55, 328, 333decmac 10422 . . . . . . . . . 10  |-  ( (;; 2 5 3  x.  8 )  +  7 )  = ;;; 2 0 3 1
335233, 138, 273, 99, 275, 276, 232, 63, 278, 316, 334decma2c 10423 . . . . . . . . 9  |-  ( (;; 2 5 3  x. ;; 2 1 8 )  + ;; 1 7 7 )  = ;;;; 5 5 3 3 1
336186oveq2i 6093 . . . . . . . . . . . 12  |-  ( ( 2  x.  7 )  +  ( 0  +  3 ) )  =  ( ( 2  x.  7 )  +  3 )
33763, 11, 55, 255, 245decaddi 10427 . . . . . . . . . . . 12  |-  ( ( 2  x.  7 )  +  3 )  = ; 1
7
338336, 337eqtri 2457 . . . . . . . . . . 11  |-  ( ( 2  x.  7 )  +  ( 0  +  3 ) )  = ; 1
7
33955, 56, 18, 79, 222decaddi 10427 . . . . . . . . . . 11  |-  ( ( 5  x.  7 )  +  2 )  = ; 3
7
34018, 56, 58, 18, 292, 201, 99, 99, 55, 338, 339decmac 10422 . . . . . . . . . 10  |-  ( (; 2
5  x.  7 )  +  2 )  = ;; 1 7 7
34199, 231, 55, 290, 63, 18, 340, 179decmul1c 10430 . . . . . . . . 9  |-  (;; 2 5 3  x.  7 )  = ;;; 1 7 7 1
342232, 234, 99, 272, 63, 274, 335, 341decmul2c 10431 . . . . . . . 8  |-  (;; 2 5 3  x.  (
3 ^ 7 ) )  = ;;;;; 5 5 3 3 1 1
343 eqid 2437 . . . . . . . . . . 11  |- ;;;; 5 5 3 3 1  = ;;;; 5 5 3 3 1
344 eqid 2437 . . . . . . . . . . . . . 14  |- ;;; 5 5 3 3  = ;;; 5 5 3 3
345 eqid 2437 . . . . . . . . . . . . . . 15  |- ;; 5 5 3  = ;; 5 5 3
346 eqid 2437 . . . . . . . . . . . . . . . 16  |- ; 5 5  = ; 5 5
347284, 201eqtri 2457 . . . . . . . . . . . . . . . 16  |-  ( 0  +  2 )  = ; 0
2
348186oveq2i 6093 . . . . . . . . . . . . . . . . 17  |-  ( ( 5  x.  7 )  +  ( 0  +  3 ) )  =  ( ( 5  x.  7 )  +  3 )
349348, 176eqtri 2457 . . . . . . . . . . . . . . . 16  |-  ( ( 5  x.  7 )  +  ( 0  +  3 ) )  = ; 3
8
35056, 56, 58, 18, 346, 347, 99, 99, 55, 349, 339decmac 10422 . . . . . . . . . . . . . . 15  |-  ( (; 5
5  x.  7 )  +  ( 0  +  2 ) )  = ;; 3 8 7
35118, 63, 18, 179, 173decaddi 10427 . . . . . . . . . . . . . . 15  |-  ( ( 3  x.  7 )  +  2 )  = ; 2
3
352227, 55, 58, 18, 345, 201, 99, 55, 18, 350, 351decmac 10422 . . . . . . . . . . . . . 14  |-  ( (;; 5 5 3  x.  7 )  +  2 )  = ;;; 3 8 7 3
35399, 228, 55, 344, 63, 18, 352, 179decmul1c 10430 . . . . . . . . . . . . 13  |-  (;;; 5 5 3 3  x.  7 )  = ;;;; 3 8 7 3 1
354353oveq1i 6092 . . . . . . . . . . . 12  |-  ( (;;; 5 5 3 3  x.  7 )  +  0 )  =  (;;;; 3 8 7 3 1  +  0 )
355142nn0cni 10234 . . . . . . . . . . . . 13  |- ;;;; 3 8 7 3 1  e.  CC
356355addid1i 9254 . . . . . . . . . . . 12  |-  (;;;; 3 8 7 3 1  +  0 )  = ;;;; 3 8 7 3 1
357354, 356eqtri 2457 . . . . . . . . . . 11  |-  ( (;;; 5 5 3 3  x.  7 )  +  0 )  = ;;;; 3 8 7 3 1
35876mulid2i 9094 . . . . . . . . . . . 12  |-  ( 1  x.  7 )  =  7
359358, 317eqtri 2457 . . . . . . . . . . 11  |-  ( 1  x.  7 )  = ; 0
7
36099, 229, 63, 343, 99, 58, 357, 359decmul1c 10430 . . . . . . . . . 10  |-  (;;;; 5 5 3 3 1  x.  7 )  = ;;;;; 3 8 7 3 1 7
361360oveq1i 6092 . . . . . . . . 9  |-  ( (;;;; 5 5 3 3 1  x.  7 )  +  0 )  =  (;;;;; 3 8 7 3 1 7  +  0 )
362144nn0cni 10234 . . . . . . . . . 10  |- ;;;;; 3 8 7 3 1 7  e.  CC
363362addid1i 9254 . . . . . . . . 9  |-  (;;;;; 3 8 7 3 1 7  +  0 )  = ;;;;; 3 8 7 3 1 7
364361, 363eqtri 2457 . . . . . . . 8  |-  ( (;;;; 5 5 3 3 1  x.  7 )  +  0 )  = ;;;;; 3 8 7 3 1 7
36599, 230, 63, 342, 99, 58, 364, 359decmul1c 10430 . . . . . . 7  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  = ;;;;;; 3 8 7 3 1 7 7
366148, 226, 3653brtr4i 4241 . . . . . 6  |-  (;;;; 5 3 0 5 7  x. ; 7 3 )  < 
( (;; 2 5 3  x.  (
3 ^ 7 ) )  x.  7 )
36799, 55deccl 10397 . . . . . . . . 9  |- ; 7 3  e.  NN0
368129, 367nn0mulcli 10259 . . . . . . . 8  |-  (;;;; 5 3 0 5 7  x. ; 7 3 )  e. 
NN0
369368nn0rei 10233 . . . . . . 7  |-  (;;;; 5 3 0 5 7  x. ; 7 3 )  e.  RR
37055, 99nn0expcli 11408 . . . . . . . . . 10  |-  ( 3 ^ 7 )  e. 
NN0
371232, 370nn0mulcli 10259 . . . . . . . . 9  |-  (;; 2 5 3  x.  (
3 ^ 7 ) )  e.  NN0
372371, 99nn0mulcli 10259 . . . . . . . 8  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  e.  NN0
373372nn0rei 10233 . . . . . . 7  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  e.  RR
37468nnrei 10010 . . . . . . 7  |-  5  e.  RR
37568nngt0i 10034 . . . . . . 7  |-  0  <  5
376369, 373, 374, 375ltmul1ii 9940 . . . . . 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 ) )
377366, 376mpbi 201 . . . . 5  |-  ( (;;;; 5 3 0 5 7  x. ; 7 3 )  x.  5 )  <  (
( (;; 2 5 3  x.  (
3 ^ 7 ) )  x.  7 )  x.  5 )
378129nn0cni 10234 . . . . . . 7  |- ;;;; 5 3 0 5 7  e.  CC
379367nn0cni 10234 . . . . . . 7  |- ; 7 3  e.  CC
380378, 379, 77mulassi 9100 . . . . . 6  |-  ( (;;;; 5 3 0 5 7  x. ; 7 3 )  x.  5 )  =  (;;;; 5 3 0 5 7  x.  (; 7 3  x.  5 ) )
38155, 56, 159, 78decsuc 10406 . . . . . . . 8  |-  ( ( 7  x.  5 )  +  1 )  = ; 3
6
38277, 3, 208mulcomli 9098 . . . . . . . 8  |-  ( 3  x.  5 )  = ; 1
5
38356, 99, 55, 149, 56, 63, 381, 382decmul1c 10430 . . . . . . 7  |-  (; 7 3  x.  5 )  = ;; 3 6 5
384383oveq2i 6093 . . . . . 6  |-  (;;;; 5 3 0 5 7  x.  (; 7 3  x.  5 ) )  =  (;;;; 5 3 0 5 7  x. ;; 3 6 5 )
385380, 384eqtri 2457 . . . . 5  |-  ( (;;;; 5 3 0 5 7  x. ; 7 3 )  x.  5 )  =  (;;;; 5 3 0 5 7  x. ;; 3 6 5 )
386311, 102mulcli 9096 . . . . . . 7  |-  (;; 2 5 3  x.  (
3 ^ 7 ) )  e.  CC
387386, 76, 77mulassi 9100 . . . . . 6  |-  ( ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  x.  5 )  =  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  ( 7  x.  5 ) )
38876, 77mulcomi 9097 . . . . . . . 8  |-  ( 7  x.  5 )  =  ( 5  x.  7 )
389388oveq2i 6093 . . . . . . 7  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  (
7  x.  5 ) )  =  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  (
5  x.  7 ) )
390311, 102, 103mulassi 9100 . . . . . . 7  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  (
5  x.  7 ) )  =  (;; 2 5 3  x.  (
( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )
391389, 390eqtri 2457 . . . . . 6  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  (
7  x.  5 ) )  =  (;; 2 5 3  x.  (
( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )
392387, 391eqtri 2457 . . . . 5  |-  ( ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  x.  5 )  =  (;; 2 5 3  x.  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) ) )
393377, 385, 3923brtr3i 4240 . . . 4  |-  (;;;; 5 3 0 5 7  x. ;; 3 6 5 )  <  (;; 2 5 3  x.  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) ) )
39455, 61deccl 10397 . . . . . . . 8  |- ; 3 6  e.  NN0
395394, 68decnncl 10396 . . . . . . 7  |- ;; 3 6 5  e.  NN
396395nnrei 10010 . . . . . 6  |- ;; 3 6 5  e.  RR
397395nngt0i 10034 . . . . . 6  |-  0  < ;; 3 6 5
398396, 397pm3.2i 443 . . . . 5  |-  (;; 3 6 5  e.  RR  /\  0  < ;; 3 6 5 )
399232nn0rei 10233 . . . . 5  |- ;; 2 5 3  e.  RR
400 lt2mul2div 9887 . . . . 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 ) ) )
401130, 398, 399, 134, 400mp4an 656 . . . 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 ) )
402393, 401mpbi 201 . . 3  |-  (;;;; 5 3 0 5 7  /  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  < 
(;; 2 5 3  / ;; 3 6 5 )
403 nndivre 10036 . . . . 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 )
404130, 131, 403mp2an 655 . . . 4  |-  (;;;; 5 3 0 5 7  /  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  e.  RR
405 nndivre 10036 . . . . 5  |-  ( (;; 2 5 3  e.  RR  /\ ;; 3 6 5  e.  NN )  ->  (;; 2 5 3  / ;; 3 6 5 )  e.  RR )
406399, 395, 405mp2an 655 . . . 4  |-  (;; 2 5 3  / ;; 3 6 5 )  e.  RR
407128, 404, 406lelttri 9201 . . 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 ) )
408137, 402, 407mp2an 655 . 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 )
40933, 128, 406lelttri 9201 . 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 ) )
41053, 408, 409mp2an 655 1  |-  ( log `  2 )  < 
(;; 2 5 3  / ;; 3 6 5 )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    /\ w3a 937    T. wtru 1326    = wceq 1653    e. wcel 1726   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   CCcc 8989   RRcr 8990   0cc0 8991   1c1 8992    + caddc 8994    x. cmul 8996    < clt 9121    <_ cle 9122    - cmin 9292    / cdiv 9678   NNcn 10001   2c2 10050   3c3 10051   4c4 10052   5c5 10053   6c6 10054   7c7 10055   8c8 10056   9c9 10057   10c10 10058   NN0cn0 10222  ;cdc 10383   RR+crp 10613   [,]cicc 10920   ...cfz 11044   ^cexp 11383   sum_csu 12480   logclog 20453
This theorem is referenced by:  birthday  20794  log2le1  24408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069  ax-addf 9070  ax-mulf 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-2o 6726  df-oadd 6729  df-er 6906  df-map 7021  df-pm 7022  df-ixp 7065  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-fi 7417  df-sup 7447  df-oi 7480  df-card 7827  df-cda 8049  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-uz 10490  df-q 10576  df-rp 10614  df-xneg 10711  df-xadd 10712  df-xmul 10713  df-ioo 10921  df-ioc 10922  df-ico 10923  df-icc 10924  df-fz 11045  df-fzo 11137  df-fl 11203  df-mod 11252  df-seq 11325  df-exp 11384  df-fac 11568  df-bc 11595  df-hash 11620  df-shft 11883  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-limsup 12266  df-clim 12283  df-rlim 12284  df-sum 12481  df-ef 12671  df-sin 12673  df-cos 12674  df-tan 12675  df-pi 12676  df-dvds 12854  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-starv 13545  df-sca 13546  df-vsca 13547  df-tset 13549  df-ple 13550  df-ds 13552  df-unif 13553  df-hom 13554  df-cco 13555  df-rest 13651  df-topn 13652  df-topgen 13668  df-pt 13669  df-prds 13672  df-xrs 13727  df-0g 13728  df-gsum 13729  df-qtop 13734  df-imas 13735  df-xps 13737  df-mre 13812  df-mrc 13813  df-acs 13815  df-mnd 14691  df-submnd 14740  df-mulg 14816  df-cntz 15117  df-cmn 15415  df-psmet 16695  df-xmet 16696  df-met 16697  df-bl 16698  df-mopn 16699  df-fbas 16700  df-fg 16701  df-cnfld 16705  df-top 16964  df-bases 16966  df-topon 16967  df-topsp 16968  df-cld 17084  df-ntr 17085  df-cls 17086  df-nei 17163  df-lp 17201  df-perf 17202  df-cn 17292  df-cnp 17293  df-haus 17380  df-cmp 17451  df-tx 17595  df-hmeo 17788  df-fil 17879  df-fm 17971  df-flim 17972  df-flf 17973  df-xms 18351  df-ms 18352  df-tms 18353  df-cncf 18909  df-limc 19754  df-dv 19755  df-ulm 20294  df-log 20455  df-atan 20708
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