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Theorem log2ub 20245
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 9806 . . . . . . . . . . 11  |-  4  =  ( 3  +  1 )
21oveq1i 5868 . . . . . . . . . 10  |-  ( 4  -  1 )  =  ( ( 3  +  1 )  -  1 )
3 3cn 9818 . . . . . . . . . . 11  |-  3  e.  CC
4 ax-1cn 8795 . . . . . . . . . . 11  |-  1  e.  CC
5 pncan 9057 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  1  e.  CC )  ->  ( ( 3  +  1 )  -  1 )  =  3 )
63, 4, 5mp2an 653 . . . . . . . . . 10  |-  ( ( 3  +  1 )  -  1 )  =  3
72, 6eqtri 2303 . . . . . . . . 9  |-  ( 4  -  1 )  =  3
87oveq2i 5869 . . . . . . . 8  |-  ( 0 ... ( 4  -  1 ) )  =  ( 0 ... 3
)
98sumeq1i 12171 . . . . . . 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 5869 . . . . . 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 9984 . . . . . . 7  |-  4  e.  NN0
12 log2tlbnd 20241 . . . . . . 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 2354 . . . . 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 8838 . . . . . 6  |-  0  e.  RR
16 3re 9817 . . . . . . 7  |-  3  e.  RR
17 4nn 9879 . . . . . . . . 9  |-  4  e.  NN
18 2nn0 9982 . . . . . . . . . 10  |-  2  e.  NN0
19 1nn 9757 . . . . . . . . . 10  |-  1  e.  NN
2018, 11, 19numnncl 10132 . . . . . . . . 9  |-  ( ( 2  x.  4 )  +  1 )  e.  NN
2117, 20nnmulcli 9770 . . . . . . . 8  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  e.  NN
22 9nn 9884 . . . . . . . . 9  |-  9  e.  NN
23 nnexpcl 11116 . . . . . . . . 9  |-  ( ( 9  e.  NN  /\  4  e.  NN0 )  -> 
( 9 ^ 4 )  e.  NN )
2422, 11, 23mp2an 653 . . . . . . . 8  |-  ( 9 ^ 4 )  e.  NN
2521, 24nnmulcli 9770 . . . . . . 7  |-  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) )  e.  NN
26 nndivre 9781 . . . . . . 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 653 . . . . . 6  |-  ( 3  /  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) ) )  e.  RR
2815, 27elicc2i 10716 . . . . 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 199 . . . 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 966 . . 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 10359 . . . . 5  |-  2  e.  RR+
32 relogcl 19932 . . . . 5  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
3331, 32ax-mp 8 . . . 4  |-  ( log `  2 )  e.  RR
34 fzfid 11035 . . . . . 6  |-  (  T. 
->  ( 0 ... 3
)  e.  Fin )
35 2re 9815 . . . . . . 7  |-  2  e.  RR
36 3nn 9878 . . . . . . . . 9  |-  3  e.  NN
37 elfznn0 10822 . . . . . . . . . . . 12  |-  ( n  e.  ( 0 ... 3 )  ->  n  e.  NN0 )
3837adantl 452 . . . . . . . . . . 11  |-  ( (  T.  /\  n  e.  ( 0 ... 3
) )  ->  n  e.  NN0 )
39 nn0mulcl 10000 . . . . . . . . . . 11  |-  ( ( 2  e.  NN0  /\  n  e.  NN0 )  -> 
( 2  x.  n
)  e.  NN0 )
4018, 38, 39sylancr 644 . . . . . . . . . 10  |-  ( (  T.  /\  n  e.  ( 0 ... 3
) )  ->  (
2  x.  n )  e.  NN0 )
41 nn0p1nn 10003 . . . . . . . . . 10  |-  ( ( 2  x.  n )  e.  NN0  ->  ( ( 2  x.  n )  +  1 )  e.  NN )
4240, 41syl 15 . . . . . . . . 9  |-  ( (  T.  /\  n  e.  ( 0 ... 3
) )  ->  (
( 2  x.  n
)  +  1 )  e.  NN )
43 nnmulcl 9769 . . . . . . . . 9  |-  ( ( 3  e.  NN  /\  ( ( 2  x.  n )  +  1 )  e.  NN )  ->  ( 3  x.  ( ( 2  x.  n )  +  1 ) )  e.  NN )
4436, 42, 43sylancr 644 . . . . . . . 8  |-  ( (  T.  /\  n  e.  ( 0 ... 3
) )  ->  (
3  x.  ( ( 2  x.  n )  +  1 ) )  e.  NN )
45 nnexpcl 11116 . . . . . . . . 9  |-  ( ( 9  e.  NN  /\  n  e.  NN0 )  -> 
( 9 ^ n
)  e.  NN )
4622, 38, 45sylancr 644 . . . . . . . 8  |-  ( (  T.  /\  n  e.  ( 0 ... 3
) )  ->  (
9 ^ n )  e.  NN )
4744, 46nnmulcld 9793 . . . . . . 7  |-  ( (  T.  /\  n  e.  ( 0 ... 3
) )  ->  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) )  e.  NN )
48 nndivre 9781 . . . . . . 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 644 . . . . . 6  |-  ( (  T.  /\  n  e.  ( 0 ... 3
) )  ->  (
2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) )  e.  RR )
5034, 49fsumrecl 12207 . . . . 5  |-  (  T. 
->  sum_ n  e.  ( 0 ... 3 ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) )  e.  RR )
5150trud 1314 . . . 4  |-  sum_ n  e.  ( 0 ... 3
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  e.  RR
5233, 51, 27lesubadd2i 9333 . . 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 199 . 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 20244 . . . . 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 9983 . . . . 5  |-  3  e.  NN0
56 5nn0 9985 . . . . . . . . 9  |-  5  e.  NN0
5756, 55deccl 10138 . . . . . . . 8  |- ; 5 3  e.  NN0
58 0nn0 9980 . . . . . . . 8  |-  0  e.  NN0
5957, 58deccl 10138 . . . . . . 7  |- ;; 5 3 0  e.  NN0
6059, 56deccl 10138 . . . . . 6  |- ;;; 5 3 0 5  e.  NN0
61 6nn0 9986 . . . . . 6  |-  6  e.  NN0
6260, 61deccl 10138 . . . . 5  |- ;;;; 5 3 0 5 6  e.  NN0
63 1nn0 9981 . . . . 5  |-  1  e.  NN0
64 eqid 2283 . . . . 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 9851 . . . . . 6  |-  ( 6  +  1 )  =  7
66 eqid 2283 . . . . . 6  |- ;;;; 5 3 0 5 6  = ;;;; 5 3 0 5 6
6760, 61, 65, 66decsuc 10147 . . . . 5  |-  (;;;; 5 3 0 5 6  +  1 )  = ;;;; 5 3 0 5 7
68 5nn 9880 . . . . . . . . . 10  |-  5  e.  NN
69 7nn 9882 . . . . . . . . . 10  |-  7  e.  NN
7068, 69nnmulcli 9770 . . . . . . . . 9  |-  ( 5  x.  7 )  e.  NN
7170nnrei 9755 . . . . . . . 8  |-  ( 5  x.  7 )  e.  RR
7221nnrei 9755 . . . . . . . 8  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  e.  RR
73 6nn 9881 . . . . . . . . . 10  |-  6  e.  NN
74 5lt6 9896 . . . . . . . . . 10  |-  5  <  6
7555, 56, 73, 74declt 10145 . . . . . . . . 9  |- ; 3 5  < ; 3 6
7669nncni 9756 . . . . . . . . . 10  |-  7  e.  CC
7768nncni 9756 . . . . . . . . . 10  |-  5  e.  CC
78 7t5e35 10209 . . . . . . . . . 10  |-  ( 7  x.  5 )  = ; 3
5
7976, 77, 78mulcomli 8844 . . . . . . . . 9  |-  ( 5  x.  7 )  = ; 3
5
80 4cn 9820 . . . . . . . . . . . . . 14  |-  4  e.  CC
81 2cn 9816 . . . . . . . . . . . . . 14  |-  2  e.  CC
82 4t2e8 9874 . . . . . . . . . . . . . 14  |-  ( 4  x.  2 )  =  8
8380, 81, 82mulcomli 8844 . . . . . . . . . . . . 13  |-  ( 2  x.  4 )  =  8
8483oveq1i 5868 . . . . . . . . . . . 12  |-  ( ( 2  x.  4 )  +  1 )  =  ( 8  +  1 )
85 8p1e9 9853 . . . . . . . . . . . 12  |-  ( 8  +  1 )  =  9
8684, 85eqtri 2303 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  +  1 )  =  9
8786oveq2i 5869 . . . . . . . . . 10  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  =  ( 4  x.  9 )
8822nncni 9756 . . . . . . . . . . 11  |-  9  e.  CC
89 9t4e36 10221 . . . . . . . . . . 11  |-  ( 9  x.  4 )  = ; 3
6
9088, 80, 89mulcomli 8844 . . . . . . . . . 10  |-  ( 4  x.  9 )  = ; 3
6
9187, 90eqtri 2303 . . . . . . . . 9  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  = ; 3
6
9275, 79, 913brtr4i 4051 . . . . . . . 8  |-  ( 5  x.  7 )  < 
( 4  x.  (
( 2  x.  4 )  +  1 ) )
9371, 72, 92ltleii 8941 . . . . . . 7  |-  ( 5  x.  7 )  <_ 
( 4  x.  (
( 2  x.  4 )  +  1 ) )
9424nngt0i 9779 . . . . . . . 8  |-  0  <  ( 9 ^ 4 )
9524nnrei 9755 . . . . . . . . 9  |-  ( 9 ^ 4 )  e.  RR
9671, 72, 95lemul2i 9680 . . . . . . . 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 199 . . . . . 6  |-  ( ( 9 ^ 4 )  x.  ( 5  x.  7 ) )  <_ 
( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) )
99 7nn0 9987 . . . . . . . . . 10  |-  7  e.  NN0
100 nnexpcl 11116 . . . . . . . . . 10  |-  ( ( 3  e.  NN  /\  7  e.  NN0 )  -> 
( 3 ^ 7 )  e.  NN )
10136, 99, 100mp2an 653 . . . . . . . . 9  |-  ( 3 ^ 7 )  e.  NN
102101nncni 9756 . . . . . . . 8  |-  ( 3 ^ 7 )  e.  CC
10370nncni 9756 . . . . . . . 8  |-  ( 5  x.  7 )  e.  CC
104102, 103, 3mul32i 9008 . . . . . . 7  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  3 )  =  ( ( ( 3 ^ 7 )  x.  3 )  x.  (
5  x.  7 ) )
10580, 81mulcomi 8843 . . . . . . . . . . . 12  |-  ( 4  x.  2 )  =  ( 2  x.  4 )
106 df-8 9810 . . . . . . . . . . . 12  |-  8  =  ( 7  +  1 )
10782, 105, 1063eqtr3i 2311 . . . . . . . . . . 11  |-  ( 2  x.  4 )  =  ( 7  +  1 )
108107oveq2i 5869 . . . . . . . . . 10  |-  ( 3 ^ ( 2  x.  4 ) )  =  ( 3 ^ (
7  +  1 ) )
109 expmul 11147 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  2  e.  NN0  /\  4  e.  NN0 )  ->  (
3 ^ ( 2  x.  4 ) )  =  ( ( 3 ^ 2 ) ^
4 ) )
1103, 18, 11, 109mp3an 1277 . . . . . . . . . 10  |-  ( 3 ^ ( 2  x.  4 ) )  =  ( ( 3 ^ 2 ) ^ 4 )
111108, 110eqtr3i 2305 . . . . . . . . 9  |-  ( 3 ^ ( 7  +  1 ) )  =  ( ( 3 ^ 2 ) ^ 4 )
112 expp1 11110 . . . . . . . . . 10  |-  ( ( 3  e.  CC  /\  7  e.  NN0 )  -> 
( 3 ^ (
7  +  1 ) )  =  ( ( 3 ^ 7 )  x.  3 ) )
1133, 99, 112mp2an 653 . . . . . . . . 9  |-  ( 3 ^ ( 7  +  1 ) )  =  ( ( 3 ^ 7 )  x.  3 )
114 sq3 11200 . . . . . . . . . 10  |-  ( 3 ^ 2 )  =  9
115114oveq1i 5868 . . . . . . . . 9  |-  ( ( 3 ^ 2 ) ^ 4 )  =  ( 9 ^ 4 )
116111, 113, 1153eqtr3i 2311 . . . . . . . 8  |-  ( ( 3 ^ 7 )  x.  3 )  =  ( 9 ^ 4 )
117116oveq1i 5868 . . . . . . 7  |-  ( ( ( 3 ^ 7 )  x.  3 )  x.  ( 5  x.  7 ) )  =  ( ( 9 ^ 4 )  x.  (
5  x.  7 ) )
118104, 117eqtri 2303 . . . . . 6  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  3 )  =  ( ( 9 ^ 4 )  x.  (
5  x.  7 ) )
11921nncni 9756 . . . . . . . . 9  |-  ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  e.  CC
12024nncni 9756 . . . . . . . . 9  |-  ( 9 ^ 4 )  e.  CC
121119, 120mulcomi 8843 . . . . . . . 8  |-  ( ( 4  x.  ( ( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) )  =  ( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) )
122121oveq1i 5868 . . . . . . 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 8842 . . . . . . . 8  |-  ( ( 9 ^ 4 )  x.  ( 4  x.  ( ( 2  x.  4 )  +  1 ) ) )  e.  CC
124123mulid1i 8839 . . . . . . 7  |-  ( ( ( 9 ^ 4 )  x.  ( 4  x.  ( ( 2  x.  4 )  +  1 ) ) )  x.  1 )  =  ( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) )
125122, 124eqtri 2303 . . . . . 6  |-  ( ( ( 4  x.  (
( 2  x.  4 )  +  1 ) )  x.  ( 9 ^ 4 ) )  x.  1 )  =  ( ( 9 ^ 4 )  x.  (
4  x.  ( ( 2  x.  4 )  +  1 ) ) )
12698, 118, 1253brtr4i 4051 . . . . 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 20242 . . . 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 8850 . . . . 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 10138 . . . . . 6  |- ;;;; 5 3 0 5 7  e.  NN0
130129nn0rei 9976 . . . . 5  |- ;;;; 5 3 0 5 7  e.  RR
131101, 70nnmulcli 9770 . . . . . . 7  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  NN
132131nnrei 9755 . . . . . 6  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR
133131nngt0i 9779 . . . . . 6  |-  0  <  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) )
134132, 133pm3.2i 441 . . . . 5  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR  /\  0  <  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) ) )
135 lemuldiv2 9636 . . . . 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 1277 . . . 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 199 . . 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 9988 . . . . . . . . . . . . 13  |-  8  e.  NN0
13955, 138deccl 10138 . . . . . . . . . . . 12  |- ; 3 8  e.  NN0
140139, 99deccl 10138 . . . . . . . . . . 11  |- ;; 3 8 7  e.  NN0
141140, 55deccl 10138 . . . . . . . . . 10  |- ;;; 3 8 7 3  e.  NN0
142141, 63deccl 10138 . . . . . . . . 9  |- ;;;; 3 8 7 3 1  e.  NN0
143142, 61deccl 10138 . . . . . . . 8  |- ;;;;; 3 8 7 3 1 6  e.  NN0
144142, 99deccl 10138 . . . . . . . 8  |- ;;;;; 3 8 7 3 1 7  e.  NN0
145 1lt10 9930 . . . . . . . 8  |-  1  <  10
146 6lt7 9901 . . . . . . . . 9  |-  6  <  7
147142, 61, 69, 146declt 10145 . . . . . . . 8  |- ;;;;; 3 8 7 3 1 6  < ;;;;; 3 8 7 3 1 7
148143, 144, 63, 99, 145, 147decltc 10146 . . . . . . 7  |- ;;;;;; 3 8 7 3 1 6 1  < ;;;;;; 3 8 7 3 1 7 7
149 eqid 2283 . . . . . . . 8  |- ; 7 3  = ; 7 3
15063, 56deccl 10138 . . . . . . . . . . 11  |- ; 1 5  e.  NN0
151 9nn0 9989 . . . . . . . . . . 11  |-  9  e.  NN0
152150, 151deccl 10138 . . . . . . . . . 10  |- ;; 1 5 9  e.  NN0
153152, 63deccl 10138 . . . . . . . . 9  |- ;;; 1 5 9 1  e.  NN0
154153, 99deccl 10138 . . . . . . . 8  |- ;;;; 1 5 9 1 7  e.  NN0
155 eqid 2283 . . . . . . . . 9  |- ;;;; 5 3 0 5 7  = ;;;; 5 3 0 5 7
156 eqid 2283 . . . . . . . . 9  |- ;;;; 1 5 9 1 7  = ;;;; 1 5 9 1 7
157 eqid 2283 . . . . . . . . . 10  |- ;;; 5 3 0 5  = ;;; 5 3 0 5
158 eqid 2283 . . . . . . . . . . 11  |- ;;; 1 5 9 1  = ;;; 1 5 9 1
159 5p1e6 9850 . . . . . . . . . . . 12  |-  ( 5  +  1 )  =  6
16077, 4, 159addcomli 9004 . . . . . . . . . . 11  |-  ( 1  +  5 )  =  6
161152, 63, 56, 158, 160decaddi 10168 . . . . . . . . . 10  |-  (;;; 1 5 9 1  +  5 )  = ;;; 1 5 9 6
16263, 61deccl 10138 . . . . . . . . . . 11  |- ; 1 6  e.  NN0
163 eqid 2283 . . . . . . . . . . 11  |- ;; 5 3 0  = ;; 5 3 0
164 eqid 2283 . . . . . . . . . . . 12  |- ;; 1 5 9  = ;; 1 5 9
165 eqid 2283 . . . . . . . . . . . . 13  |- ; 1 5  = ; 1 5
16663, 56, 159, 165decsuc 10147 . . . . . . . . . . . 12  |-  (; 1 5  +  1 )  = ; 1 6
167 9p4e13 10188 . . . . . . . . . . . 12  |-  ( 9  +  4 )  = ; 1
3
168150, 151, 11, 164, 166, 55, 167decaddci 10169 . . . . . . . . . . 11  |-  (;; 1 5 9  +  4 )  = ;; 1 6 3
169 eqid 2283 . . . . . . . . . . . 12  |- ; 5 3  = ; 5 3
170162nn0cni 9977 . . . . . . . . . . . . 13  |- ; 1 6  e.  CC
171170addid1i 8999 . . . . . . . . . . . 12  |-  (; 1 6  +  0 )  = ; 1 6
172 2p1e3 9847 . . . . . . . . . . . . . . 15  |-  ( 2  +  1 )  =  3
17381, 4, 172addcomli 9004 . . . . . . . . . . . . . 14  |-  ( 1  +  2 )  =  3
174173oveq2i 5869 . . . . . . . . . . . . 13  |-  ( ( 5  x.  7 )  +  ( 1  +  2 ) )  =  ( ( 5  x.  7 )  +  3 )
175 5p3e8 9861 . . . . . . . . . . . . . 14  |-  ( 5  +  3 )  =  8
17655, 56, 55, 79, 175decaddi 10168 . . . . . . . . . . . . 13  |-  ( ( 5  x.  7 )  +  3 )  = ; 3
8
177174, 176eqtri 2303 . . . . . . . . . . . 12  |-  ( ( 5  x.  7 )  +  ( 1  +  2 ) )  = ; 3
8
178 7t3e21 10207 . . . . . . . . . . . . . 14  |-  ( 7  x.  3 )  = ; 2
1
17976, 3, 178mulcomli 8844 . . . . . . . . . . . . 13  |-  ( 3  x.  7 )  = ; 2
1
18073nncni 9756 . . . . . . . . . . . . . 14  |-  6  e.  CC
181180, 4, 65addcomli 9004 . . . . . . . . . . . . 13  |-  ( 1  +  6 )  =  7
18218, 63, 61, 179, 181decaddi 10168 . . . . . . . . . . . 12  |-  ( ( 3  x.  7 )  +  6 )  = ; 2
7
18356, 55, 63, 61, 169, 171, 99, 99, 18, 177, 182decmac 10163 . . . . . . . . . . 11  |-  ( (; 5
3  x.  7 )  +  (; 1 6  +  0 ) )  = ;; 3 8 7
18476mul02i 9001 . . . . . . . . . . . . 13  |-  ( 0  x.  7 )  =  0
185184oveq1i 5868 . . . . . . . . . . . 12  |-  ( ( 0  x.  7 )  +  3 )  =  ( 0  +  3 )
1863addid2i 9000 . . . . . . . . . . . . 13  |-  ( 0  +  3 )  =  3
18755dec0h 10140 . . . . . . . . . . . . 13  |-  3  = ; 0 3
188186, 187eqtri 2303 . . . . . . . . . . . 12  |-  ( 0  +  3 )  = ; 0
3
189185, 188eqtri 2303 . . . . . . . . . . 11  |-  ( ( 0  x.  7 )  +  3 )  = ; 0
3
19057, 58, 162, 55, 163, 168, 99, 55, 58, 183, 189decmac 10163 . . . . . . . . . 10  |-  ( (;; 5 3 0  x.  7 )  +  (;; 1 5 9  +  4 ) )  = ;;; 3 8 7 3
191 3p1e4 9848 . . . . . . . . . . 11  |-  ( 3  +  1 )  =  4
192 6p5e11 10174 . . . . . . . . . . . 12  |-  ( 6  +  5 )  = ; 1
1
193180, 77, 192addcomli 9004 . . . . . . . . . . 11  |-  ( 5  +  6 )  = ; 1
1
19455, 56, 61, 79, 191, 63, 193decaddci 10169 . . . . . . . . . 10  |-  ( ( 5  x.  7 )  +  6 )  = ; 4
1
19559, 56, 152, 61, 157, 161, 99, 63, 11, 190, 194decmac 10163 . . . . . . . . 9  |-  ( (;;; 5 3 0 5  x.  7 )  +  (;;; 1 5 9 1  +  5 ) )  = ;;;; 3 8 7 3 1
196 7t7e49 10211 . . . . . . . . . 10  |-  ( 7  x.  7 )  = ; 4
9
197 4p1e5 9849 . . . . . . . . . 10  |-  ( 4  +  1 )  =  5
198 9p7e16 10191 . . . . . . . . . 10  |-  ( 9  +  7 )  = ; 1
6
19911, 151, 99, 196, 197, 61, 198decaddci 10169 . . . . . . . . 9  |-  ( ( 7  x.  7 )  +  7 )  = ; 5
6
20060, 99, 153, 99, 155, 156, 99, 61, 56, 195, 199decmac 10163 . . . . . . . 8  |-  ( (;;;; 5 3 0 5 7  x.  7 )  + ;;;; 1 5 9 1 7 )  = ;;;;; 3 8 7 3 1 6
20118dec0h 10140 . . . . . . . . . 10  |-  2  = ; 0 2
2024addid2i 9000 . . . . . . . . . . . 12  |-  ( 0  +  1 )  =  1
20363dec0h 10140 . . . . . . . . . . . 12  |-  1  = ; 0 1
204202, 203eqtri 2303 . . . . . . . . . . 11  |-  ( 0  +  1 )  = ; 0
1
205 00id 8987 . . . . . . . . . . . . 13  |-  ( 0  +  0 )  =  0
20658dec0h 10140 . . . . . . . . . . . . 13  |-  0  = ; 0 0
207205, 206eqtri 2303 . . . . . . . . . . . 12  |-  ( 0  +  0 )  = ; 0
0
208 5t3e15 10198 . . . . . . . . . . . . . 14  |-  ( 5  x.  3 )  = ; 1
5
209208oveq1i 5868 . . . . . . . . . . . . 13  |-  ( ( 5  x.  3 )  +  0 )  =  (; 1 5  +  0 )
210150nn0cni 9977 . . . . . . . . . . . . . 14  |- ; 1 5  e.  CC
211210addid1i 8999 . . . . . . . . . . . . 13  |-  (; 1 5  +  0 )  = ; 1 5
212209, 211eqtri 2303 . . . . . . . . . . . 12  |-  ( ( 5  x.  3 )  +  0 )  = ; 1
5
213 3t3e9 9873 . . . . . . . . . . . . . 14  |-  ( 3  x.  3 )  =  9
214213oveq1i 5868 . . . . . . . . . . . . 13  |-  ( ( 3  x.  3 )  +  0 )  =  ( 9  +  0 )
21588addid1i 8999 . . . . . . . . . . . . 13  |-  ( 9  +  0 )  =  9
216214, 215eqtri 2303 . . . . . . . . . . . 12  |-  ( ( 3  x.  3 )  +  0 )  =  9
21756, 55, 58, 58, 169, 207, 55, 212, 216decma 10162 . . . . . . . . . . 11  |-  ( (; 5
3  x.  3 )  +  ( 0  +  0 ) )  = ;; 1 5 9
2183mul02i 9001 . . . . . . . . . . . . 13  |-  ( 0  x.  3 )  =  0
219218oveq1i 5868 . . . . . . . . . . . 12  |-  ( ( 0  x.  3 )  +  1 )  =  ( 0  +  1 )
220219, 204eqtri 2303 . . . . . . . . . . 11  |-  ( ( 0  x.  3 )  +  1 )  = ; 0
1
22157, 58, 58, 63, 163, 204, 55, 63, 58, 217, 220decmac 10163 . . . . . . . . . 10  |-  ( (;; 5 3 0  x.  3 )  +  ( 0  +  1 ) )  = ;;; 1 5 9 1
222 5p2e7 9860 . . . . . . . . . . 11  |-  ( 5  +  2 )  =  7
22363, 56, 18, 208, 222decaddi 10168 . . . . . . . . . 10  |-  ( ( 5  x.  3 )  +  2 )  = ; 1
7
22459, 56, 58, 18, 157, 201, 55, 99, 63, 221, 223decmac 10163 . . . . . . . . 9  |-  ( (;;; 5 3 0 5  x.  3 )  +  2 )  = ;;;; 1 5 9 1 7
22555, 60, 99, 155, 63, 18, 224, 178decmul1c 10171 . . . . . . . 8  |-  (;;;; 5 3 0 5 7  x.  3 )  = ;;;;; 1 5 9 1 7 1
226129, 99, 55, 149, 63, 154, 200, 225decmul2c 10172 . . . . . . 7  |-  (;;;; 5 3 0 5 7  x. ; 7 3 )  = ;;;;;; 3 8 7 3 1 6 1
22756, 56deccl 10138 . . . . . . . . . . 11  |- ; 5 5  e.  NN0
228227, 55deccl 10138 . . . . . . . . . 10  |- ;; 5 5 3  e.  NN0
229228, 55deccl 10138 . . . . . . . . 9  |- ;;; 5 5 3 3  e.  NN0
230229, 63deccl 10138 . . . . . . . 8  |- ;;;; 5 5 3 3 1  e.  NN0
23118, 56deccl 10138 . . . . . . . . . 10  |- ; 2 5  e.  NN0
232231, 55deccl 10138 . . . . . . . . 9  |- ;; 2 5 3  e.  NN0
23318, 63deccl 10138 . . . . . . . . . 10  |- ; 2 1  e.  NN0
234233, 138deccl 10138 . . . . . . . . 9  |- ;; 2 1 8  e.  NN0
23599, 18deccl 10138 . . . . . . . . . . 11  |- ; 7 2  e.  NN0
236 3t2e6 9872 . . . . . . . . . . . . 13  |-  ( 3  x.  2 )  =  6
2373, 81, 236mulcomli 8844 . . . . . . . . . . . 12  |-  ( 2  x.  3 )  =  6
238 3exp3 13104 . . . . . . . . . . . 12  |-  ( 3 ^ 3 )  = ; 2
7
23918, 99deccl 10138 . . . . . . . . . . . . 13  |- ; 2 7  e.  NN0
240 eqid 2283 . . . . . . . . . . . . 13  |- ; 2 7  = ; 2 7
24163, 138deccl 10138 . . . . . . . . . . . . 13  |- ; 1 8  e.  NN0
242 eqid 2283 . . . . . . . . . . . . . 14  |- ; 1 8  = ; 1 8
243 2t2e4 9871 . . . . . . . . . . . . . . . 16  |-  ( 2  x.  2 )  =  4
244243, 173oveq12i 5870 . . . . . . . . . . . . . . 15  |-  ( ( 2  x.  2 )  +  ( 1  +  2 ) )  =  ( 4  +  3 )
245 4p3e7 9858 . . . . . . . . . . . . . . 15  |-  ( 4  +  3 )  =  7
246244, 245eqtri 2303 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  2 )  +  ( 1  +  2 ) )  =  7
247 7t2e14 10206 . . . . . . . . . . . . . . 15  |-  ( 7  x.  2 )  = ; 1
4
248 1p1e2 9840 . . . . . . . . . . . . . . 15  |-  ( 1  +  1 )  =  2
249 8nn 9883 . . . . . . . . . . . . . . . . 17  |-  8  e.  NN
250249nncni 9756 . . . . . . . . . . . . . . . 16  |-  8  e.  CC
251 8p4e12 10181 . . . . . . . . . . . . . . . 16  |-  ( 8  +  4 )  = ; 1
2
252250, 80, 251addcomli 9004 . . . . . . . . . . . . . . 15  |-  ( 4  +  8 )  = ; 1
2
25363, 11, 138, 247, 248, 18, 252decaddci 10169 . . . . . . . . . . . . . 14  |-  ( ( 7  x.  2 )  +  8 )  = ; 2
2
25418, 99, 63, 138, 240, 242, 18, 18, 18, 246, 253decmac 10163 . . . . . . . . . . . . 13  |-  ( (; 2
7  x.  2 )  + ; 1 8 )  = ; 7
2
25576, 81, 247mulcomli 8844 . . . . . . . . . . . . . . 15  |-  ( 2  x.  7 )  = ; 1
4
256 4p4e8 9859 . . . . . . . . . . . . . . 15  |-  ( 4  +  4 )  =  8
25763, 11, 11, 255, 256decaddi 10168 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  7 )  +  4 )  = ; 1
8
25899, 18, 99, 240, 151, 11, 257, 196decmul1c 10171 . . . . . . . . . . . . 13  |-  (; 2 7  x.  7 )  = ;; 1 8 9
259239, 18, 99, 240, 151, 241, 254, 258decmul2c 10172 . . . . . . . . . . . 12  |-  (; 2 7  x. ; 2 7 )  = ;; 7 2 9
26055, 55, 237, 238, 259numexp2x 13094 . . . . . . . . . . 11  |-  ( 3 ^ 6 )  = ;; 7 2 9
261 eqid 2283 . . . . . . . . . . . 12  |- ; 7 2  = ; 7 2
262178oveq1i 5868 . . . . . . . . . . . . 13  |-  ( ( 7  x.  3 )  +  0 )  =  (; 2 1  +  0 )
263233nn0cni 9977 . . . . . . . . . . . . . 14  |- ; 2 1  e.  CC
264263addid1i 8999 . . . . . . . . . . . . 13  |-  (; 2 1  +  0 )  = ; 2 1
265262, 264eqtri 2303 . . . . . . . . . . . 12  |-  ( ( 7  x.  3 )  +  0 )  = ; 2
1
266237oveq1i 5868 . . . . . . . . . . . . 13  |-  ( ( 2  x.  3 )  +  2 )  =  ( 6  +  2 )
267 6p2e8 9864 . . . . . . . . . . . . 13  |-  ( 6  +  2 )  =  8
268266, 267eqtri 2303 . . . . . . . . . . . 12  |-  ( ( 2  x.  3 )  +  2 )  =  8
26999, 18, 58, 18, 261, 201, 55, 265, 268decma 10162 . . . . . . . . . . 11  |-  ( (; 7
2  x.  3 )  +  2 )  = ;; 2 1 8
270 9t3e27 10220 . . . . . . . . . . 11  |-  ( 9  x.  3 )  = ; 2
7
27155, 235, 151, 260, 99, 18, 269, 270decmul1c 10171 . . . . . . . . . 10  |-  ( ( 3 ^ 6 )  x.  3 )  = ;;; 2 1 8 7
27255, 61, 65, 271numexpp1 13093 . . . . . . . . 9  |-  ( 3 ^ 7 )  = ;;; 2 1 8 7
27363, 99deccl 10138 . . . . . . . . . 10  |- ; 1 7  e.  NN0
274273, 99deccl 10138 . . . . . . . . 9  |- ;; 1 7 7  e.  NN0
275 eqid 2283 . . . . . . . . . 10  |- ;; 2 1 8  = ;; 2 1 8
276 eqid 2283 . . . . . . . . . 10  |- ;; 1 7 7  = ;; 1 7 7
27718, 58deccl 10138 . . . . . . . . . . 11  |- ; 2 0  e.  NN0
278277, 55deccl 10138 . . . . . . . . . 10  |- ;; 2 0 3  e.  NN0
27918, 18deccl 10138 . . . . . . . . . . 11  |- ; 2 2  e.  NN0
280 eqid 2283 . . . . . . . . . . 11  |- ; 2 1  = ; 2 1
281 eqid 2283 . . . . . . . . . . . 12  |- ; 1 7  = ; 1 7
282 eqid 2283 . . . . . . . . . . . 12  |- ;; 2 0 3  = ;; 2 0 3
283 eqid 2283 . . . . . . . . . . . . . 14  |- ; 2 0  = ; 2 0
28481addid2i 9000 . . . . . . . . . . . . . 14  |-  ( 0  +  2 )  =  2
2854addid1i 8999 . . . . . . . . . . . . . 14  |-  ( 1  +  0 )  =  1
28658, 63, 18, 58, 203, 283, 284, 285decadd 10165 . . . . . . . . . . . . 13  |-  ( 1  + ; 2 0 )  = ; 2
1
28718, 63, 248, 286decsuc 10147 . . . . . . . . . . . 12  |-  ( ( 1  + ; 2 0 )  +  1 )  = ; 2 2
288 7p3e10 9868 . . . . . . . . . . . 12  |-  ( 7  +  3 )  =  10
28963, 99, 277, 55, 281, 282, 287, 288decaddc2 10167 . . . . . . . . . . 11  |-  (; 1 7  + ;; 2 0 3 )  = ;; 2 2 0
290 eqid 2283 . . . . . . . . . . . 12  |- ;; 2 5 3  = ;; 2 5 3
291 eqid 2283 . . . . . . . . . . . . 13  |- ; 2 2  = ; 2 2
292 eqid 2283 . . . . . . . . . . . . 13  |- ; 2 5  = ; 2 5
293 2p2e4 9842 . . . . . . . . . . . . 13  |-  ( 2  +  2 )  =  4
29477, 81, 222addcomli 9004 . . . . . . . . . . . . 13  |-  ( 2  +  5 )  =  7
29518, 18, 18, 56, 291, 292, 293, 294decadd 10165 . . . . . . . . . . . 12  |-  (; 2 2  + ; 2 5 )  = ; 4
7
29656dec0h 10140 . . . . . . . . . . . . . 14  |-  5  = ; 0 5
297197, 296eqtri 2303 . . . . . . . . . . . . 13  |-  ( 4  +  1 )  = ; 0
5
298243, 202oveq12i 5870 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
299298, 197eqtri 2303 . . . . . . . . . . . . 13  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  5
300 5t2e10 9875 . . . . . . . . . . . . . . 15  |-  ( 5  x.  2 )  =  10
301 dec10 10154 . . . . . . . . . . . . . . 15  |-  10  = ; 1 0
302300, 301eqtri 2303 . . . . . . . . . . . . . 14  |-  ( 5  x.  2 )  = ; 1
0
30377addid2i 9000 . . . . . . . . . . . . . 14  |-  ( 0  +  5 )  =  5
30463, 58, 56, 302, 303decaddi 10168 . . . . . . . . . . . . 13  |-  ( ( 5  x.  2 )  +  5 )  = ; 1
5
30518, 56, 58, 56, 292, 297, 18, 56, 63, 299, 304decmac 10163 . . . . . . . . . . . 12  |-  ( (; 2
5  x.  2 )  +  ( 4  +  1 ) )  = ; 5
5
306236oveq1i 5868 . . . . . . . . . . . . 13  |-  ( ( 3  x.  2 )  +  7 )  =  ( 6  +  7 )
307 7p6e13 10178 . . . . . . . . . . . . . 14  |-  ( 7  +  6 )  = ; 1
3
30876, 180, 307addcomli 9004 . . . . . . . . . . . . 13  |-  ( 6  +  7 )  = ; 1
3
309306, 308eqtri 2303 . . . . . . . . . . . 12  |-  ( ( 3  x.  2 )  +  7 )  = ; 1
3
310231, 55, 11, 99, 290, 295, 18, 55, 63, 305, 309decmac 10163 . . . . . . . . . . 11  |-  ( (;; 2 5 3  x.  2 )  +  (; 2
2  + ; 2 5 ) )  = ;; 5 5 3
311232nn0cni 9977 . . . . . . . . . . . . . 14  |- ;; 2 5 3  e.  CC
312311mulid1i 8839 . . . . . . . . . . . . 13  |-  (;; 2 5 3  x.  1 )  = ;; 2 5 3
313312oveq1i 5868 . . . . . . . . . . . 12  |-  ( (;; 2 5 3  x.  1 )  +  0 )  =  (;; 2 5 3  +  0 )
314311addid1i 8999 . . . . . . . . . . . 12  |-  (;; 2 5 3  +  0 )  = ;; 2 5 3
315313, 314eqtri 2303 . . . . . . . . . . 11  |-  ( (;; 2 5 3  x.  1 )  +  0 )  = ;; 2 5 3
31618, 63, 279, 58, 280, 289, 232, 55, 231, 310, 315decma2c 10164 . . . . . . . . . 10  |-  ( (;; 2 5 3  x. ; 2
1 )  +  (; 1
7  + ;; 2 0 3 ) )  = ;;; 5 5 3 3
31799dec0h 10140 . . . . . . . . . . 11  |-  7  = ; 0 7
31880addid2i 9000 . . . . . . . . . . . . . 14  |-  ( 0  +  4 )  =  4
319318oveq2i 5869 . . . . . . . . . . . . 13  |-  ( ( 2  x.  8 )  +  ( 0  +  4 ) )  =  ( ( 2  x.  8 )  +  4 )
320 8t2e16 10212 . . . . . . . . . . . . . . 15  |-  ( 8  x.  2 )  = ; 1
6
321250, 81, 320mulcomli 8844 . . . . . . . . . . . . . 14  |-  ( 2  x.  8 )  = ; 1
6
322 6p4e10 9866 . . . . . . . . . . . . . 14  |-  ( 6  +  4 )  =  10
32363, 61, 11, 321, 248, 322decaddci2 10170 . . . . . . . . . . . . 13  |-  ( ( 2  x.  8 )  +  4 )  = ; 2
0
324319, 323eqtri 2303 . . . . . . . . . . . 12  |-  ( ( 2  x.  8 )  +  ( 0  +  4 ) )  = ; 2
0
325 8t5e40 10215 . . . . . . . . . . . . . 14  |-  ( 8  x.  5 )  = ; 4
0
326250, 77, 325mulcomli 8844 . . . . . . . . . . . . 13  |-  ( 5  x.  8 )  = ; 4
0
32711, 58, 55, 326, 186decaddi 10168 . . . . . . . . . . . 12  |-  ( ( 5  x.  8 )  +  3 )  = ; 4
3
32818, 56, 58, 55, 292, 188, 138, 55, 11, 324, 327decmac 10163 . . . . . . . . . . 11  |-  ( (; 2
5  x.  8 )  +  ( 0  +  3 ) )  = ;; 2 0 3
329 8t3e24 10213 . . . . . . . . . . . . 13  |-  ( 8  x.  3 )  = ; 2
4
330250, 3, 329mulcomli 8844 . . . . . . . . . . . 12  |-  ( 3  x.  8 )  = ; 2
4
331 7p4e11 10176 . . . . . . . . . . . . 13  |-  ( 7  +  4 )  = ; 1
1
33276, 80, 331addcomli 9004 . . . . . . . . . . . 12  |-  ( 4  +  7 )  = ; 1
1
33318, 11, 99, 330, 172, 63, 332decaddci 10169 . . . . . . . . . . 11  |-  ( ( 3  x.  8 )  +  7 )  = ; 3
1
334231, 55, 58, 99, 290, 317, 138, 63, 55, 328, 333decmac 10163 . . . . . . . . . 10  |-  ( (;; 2 5 3  x.  8 )  +  7 )  = ;;; 2 0 3 1
335233, 138, 273, 99, 275, 276, 232, 63, 278, 316, 334decma2c 10164 . . . . . . . . 9  |-  ( (;; 2 5 3  x. ;; 2 1 8 )  + ;; 1 7 7 )  = ;;;; 5 5 3 3 1
336186oveq2i 5869 . . . . . . . . . . . 12  |-  ( ( 2  x.  7 )  +  ( 0  +  3 ) )  =  ( ( 2  x.  7 )  +  3 )
33763, 11, 55, 255, 245decaddi 10168 . . . . . . . . . . . 12  |-  ( ( 2  x.  7 )  +  3 )  = ; 1
7
338336, 337eqtri 2303 . . . . . . . . . . 11  |-  ( ( 2  x.  7 )  +  ( 0  +  3 ) )  = ; 1
7
33955, 56, 18, 79, 222decaddi 10168 . . . . . . . . . . 11  |-  ( ( 5  x.  7 )  +  2 )  = ; 3
7
34018, 56, 58, 18, 292, 201, 99, 99, 55, 338, 339decmac 10163 . . . . . . . . . 10  |-  ( (; 2
5  x.  7 )  +  2 )  = ;; 1 7 7
34199, 231, 55, 290, 63, 18, 340, 179decmul1c 10171 . . . . . . . . 9  |-  (;; 2 5 3  x.  7 )  = ;;; 1 7 7 1
342232, 234, 99, 272, 63, 274, 335, 341decmul2c 10172 . . . . . . . 8  |-  (;; 2 5 3  x.  (
3 ^ 7 ) )  = ;;;;; 5 5 3 3 1 1
343 eqid 2283 . . . . . . . . . . 11  |- ;;;; 5 5 3 3 1  = ;;;; 5 5 3 3 1
344 eqid 2283 . . . . . . . . . . . . . 14  |- ;;; 5 5 3 3  = ;;; 5 5 3 3
345 eqid 2283 . . . . . . . . . . . . . . 15  |- ;; 5 5 3  = ;; 5 5 3
346 eqid 2283 . . . . . . . . . . . . . . . 16  |- ; 5 5  = ; 5 5
347284, 201eqtri 2303 . . . . . . . . . . . . . . . 16  |-  ( 0  +  2 )  = ; 0
2
348186oveq2i 5869 . . . . . . . . . . . . . . . . 17  |-  ( ( 5  x.  7 )  +  ( 0  +  3 ) )  =  ( ( 5  x.  7 )  +  3 )
349348, 176eqtri 2303 . . . . . . . . . . . . . . . 16  |-  ( ( 5  x.  7 )  +  ( 0  +  3 ) )  = ; 3
8
35056, 56, 58, 18, 346, 347, 99, 99, 55, 349, 339decmac 10163 . . . . . . . . . . . . . . 15  |-  ( (; 5
5  x.  7 )  +  ( 0  +  2 ) )  = ;; 3 8 7
35118, 63, 18, 179, 173decaddi 10168 . . . . . . . . . . . . . . 15  |-  ( ( 3  x.  7 )  +  2 )  = ; 2
3
352227, 55, 58, 18, 345, 201, 99, 55, 18, 350, 351decmac 10163 . . . . . . . . . . . . . 14  |-  ( (;; 5 5 3  x.  7 )  +  2 )  = ;;; 3 8 7 3
35399, 228, 55, 344, 63, 18, 352, 179decmul1c 10171 . . . . . . . . . . . . 13  |-  (;;; 5 5 3 3  x.  7 )  = ;;;; 3 8 7 3 1
354353oveq1i 5868 . . . . . . . . . . . 12  |-  ( (;;; 5 5 3 3  x.  7 )  +  0 )  =  (;;;; 3 8 7 3 1  +  0 )
355142nn0cni 9977 . . . . . . . . . . . . 13  |- ;;;; 3 8 7 3 1  e.  CC
356355addid1i 8999 . . . . . . . . . . . 12  |-  (;;;; 3 8 7 3 1  +  0 )  = ;;;; 3 8 7 3 1
357354, 356eqtri 2303 . . . . . . . . . . 11  |-  ( (;;; 5 5 3 3  x.  7 )  +  0 )  = ;;;; 3 8 7 3 1
35876mulid2i 8840 . . . . . . . . . . . 12  |-  ( 1  x.  7 )  =  7
359358, 317eqtri 2303 . . . . . . . . . . 11  |-  ( 1  x.  7 )  = ; 0
7
36099, 229, 63, 343, 99, 58, 357, 359decmul1c 10171 . . . . . . . . . 10  |-  (;;;; 5 5 3 3 1  x.  7 )  = ;;;;; 3 8 7 3 1 7
361360oveq1i 5868 . . . . . . . . 9  |-  ( (;;;; 5 5 3 3 1  x.  7 )  +  0 )  =  (;;;;; 3 8 7 3 1 7  +  0 )
362144nn0cni 9977 . . . . . . . . . 10  |- ;;;;; 3 8 7 3 1 7  e.  CC
363362addid1i 8999 . . . . . . . . 9  |-  (;;;;; 3 8 7 3 1 7  +  0 )  = ;;;;; 3 8 7 3 1 7
364361, 363eqtri 2303 . . . . . . . 8  |-  ( (;;;; 5 5 3 3 1  x.  7 )  +  0 )  = ;;;;; 3 8 7 3 1 7
36599, 230, 63, 342, 99, 58, 364, 359decmul1c 10171 . . . . . . 7  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  = ;;;;;; 3 8 7 3 1 7 7
366148, 226, 3653brtr4i 4051 . . . . . 6  |-  (;;;; 5 3 0 5 7  x. ; 7 3 )  < 
( (;; 2 5 3  x.  (
3 ^ 7 ) )  x.  7 )
36799, 55deccl 10138 . . . . . . . . 9  |- ; 7 3  e.  NN0
368129, 367nn0mulcli 10002 . . . . . . . 8  |-  (;;;; 5 3 0 5 7  x. ; 7 3 )  e. 
NN0
369368nn0rei 9976 . . . . . . 7  |-  (;;;; 5 3 0 5 7  x. ; 7 3 )  e.  RR
37055, 99nn0expcli 11129 . . . . . . . . . 10  |-  ( 3 ^ 7 )  e. 
NN0
371232, 370nn0mulcli 10002 . . . . . . . . 9  |-  (;; 2 5 3  x.  (
3 ^ 7 ) )  e.  NN0
372371, 99nn0mulcli 10002 . . . . . . . 8  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  e.  NN0
373372nn0rei 9976 . . . . . . 7  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  e.  RR
37468nnrei 9755 . . . . . . 7  |-  5  e.  RR
37568nngt0i 9779 . . . . . . 7  |-  0  <  5
376369, 373, 374, 375ltmul1ii 9685 . . . . . 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 199 . . . . 5  |-  ( (;;;; 5 3 0 5 7  x. ; 7 3 )  x.  5 )  <  (
( (;; 2 5 3  x.  (
3 ^ 7 ) )  x.  7 )  x.  5 )
378129nn0cni 9977 . . . . . . 7  |- ;;;; 5 3 0 5 7  e.  CC
379367nn0cni 9977 . . . . . . 7  |- ; 7 3  e.  CC
380378, 379, 77mulassi 8846 . . . . . 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 10147 . . . . . . . 8  |-  ( ( 7  x.  5 )  +  1 )  = ; 3
6
38277, 3, 208mulcomli 8844 . . . . . . . 8  |-  ( 3  x.  5 )  = ; 1
5
38356, 99, 55, 149, 56, 63, 381, 382decmul1c 10171 . . . . . . 7  |-  (; 7 3  x.  5 )  = ;; 3 6 5
384383oveq2i 5869 . . . . . 6  |-  (;;;; 5 3 0 5 7  x.  (; 7 3  x.  5 ) )  =  (;;;; 5 3 0 5 7  x. ;; 3 6 5 )
385380, 384eqtri 2303 . . . . 5  |-  ( (;;;; 5 3 0 5 7  x. ; 7 3 )  x.  5 )  =  (;;;; 5 3 0 5 7  x. ;; 3 6 5 )
386311, 102mulcli 8842 . . . . . . 7  |-  (;; 2 5 3  x.  (
3 ^ 7 ) )  e.  CC
387386, 76, 77mulassi 8846 . . . . . 6  |-  ( ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  x.  5 )  =  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  ( 7  x.  5 ) )
38876, 77mulcomi 8843 . . . . . . . 8  |-  ( 7  x.  5 )  =  ( 5  x.  7 )
389388oveq2i 5869 . . . . . . 7  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  (
7  x.  5 ) )  =  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  (
5  x.  7 ) )
390311, 102, 103mulassi 8846 . . . . . . 7  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  (
5  x.  7 ) )  =  (;; 2 5 3  x.  (
( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )
391389, 390eqtri 2303 . . . . . 6  |-  ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  (
7  x.  5 ) )  =  (;; 2 5 3  x.  (
( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )
392387, 391eqtri 2303 . . . . 5  |-  ( ( (;; 2 5 3  x.  ( 3 ^ 7 ) )  x.  7 )  x.  5 )  =  (;; 2 5 3  x.  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) ) )
393377, 385, 3923brtr3i 4050 . . . 4  |-  (;;;; 5 3 0 5 7  x. ;; 3 6 5 )  <  (;; 2 5 3  x.  ( ( 3 ^ 7 )  x.  (
5  x.  7 ) ) )
39455, 61deccl 10138 . . . . . . . 8  |- ; 3 6  e.  NN0
395394, 68decnncl 10137 . . . . . . 7  |- ;; 3 6 5  e.  NN
396395nnrei 9755 . . . . . 6  |- ;; 3 6 5  e.  RR
397395nngt0i 9779 . . . . . 6  |-  0  < ;; 3 6 5
398396, 397pm3.2i 441 . . . . 5  |-  (;; 3 6 5  e.  RR  /\  0  < ;; 3 6 5 )
399232nn0rei 9976 . . . . 5  |- ;; 2 5 3  e.  RR
400 lt2mul2div 9632 . . . . 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 654 . . . 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 199 . . 3  |-  (;;;; 5 3 0 5 7  /  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  < 
(;; 2 5 3  / ;; 3 6 5 )
403 nndivre 9781 . . . . 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 653 . . . 4  |-  (;;;; 5 3 0 5 7  /  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  e.  RR
405 nndivre 9781 . . . . 5  |-  ( (;; 2 5 3  e.  RR  /\ ;; 3 6 5  e.  NN )  ->  (;; 2 5 3  / ;; 3 6 5 )  e.  RR )
406399, 395, 405mp2an 653 . . . 4  |-  (;; 2 5 3  / ;; 3 6 5 )  e.  RR
407128, 404, 406lelttri 8946 . . 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 653 . 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 8946 . 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 653 1  |-  ( log `  2 )  < 
(;; 2 5 3  / ;; 3 6 5 )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    T. wtru 1307    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   3c3 9796   4c4 9797   5c5 9798   6c6 9799   7c7 9800   8c8 9801   9c9 9802   10c10 9803   NN0cn0 9965  ;cdc 10124   RR+crp 10354   [,]cicc 10659   ...cfz 10782   ^cexp 11104   sum_csu 12158   logclog 19912
This theorem is referenced by:  birthday  20249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-tan 12353  df-pi 12354  df-dvds 12532  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-ulm 19756  df-log 19914  df-atan 20163
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