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Theorem log2ublem2 20243
Description: Lemma for log2ub 20245. (Contributed by Mario Carneiro, 17-Apr-2015.)
Hypotheses
Ref Expression
log2ublem2.1  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  sum_ n  e.  ( 0 ... K ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) ) )  <_  (
2  x.  B )
log2ublem2.2  |-  B  e. 
NN0
log2ublem2.3  |-  F  e. 
NN0
log2ublem2.4  |-  N  e. 
NN0
log2ublem2.5  |-  ( N  -  1 )  =  K
log2ublem2.6  |-  ( B  +  F )  =  G
log2ublem2.7  |-  M  e. 
NN0
log2ublem2.8  |-  ( M  +  N )  =  3
log2ublem2.9  |-  ( ( 5  x.  7 )  x.  ( 9 ^ M ) )  =  ( ( ( 2  x.  N )  +  1 )  x.  F
)
Assertion
Ref Expression
log2ublem2  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  sum_ n  e.  ( 0 ... N ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) ) )  <_  (
2  x.  G )
Distinct variable groups:    n, K    n, N
Allowed substitution hints:    B( n)    F( n)    G( n)    M( n)

Proof of Theorem log2ublem2
StepHypRef Expression
1 log2ublem2.1 . 2  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  sum_ n  e.  ( 0 ... K ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) ) )  <_  (
2  x.  B )
2 fzfid 11035 . . . 4  |-  (  T. 
->  ( 0 ... K
)  e.  Fin )
3 elfznn0 10822 . . . . . 6  |-  ( n  e.  ( 0 ... K )  ->  n  e.  NN0 )
43adantl 452 . . . . 5  |-  ( (  T.  /\  n  e.  ( 0 ... K
) )  ->  n  e.  NN0 )
5 2re 9815 . . . . . 6  |-  2  e.  RR
6 3nn 9878 . . . . . . . 8  |-  3  e.  NN
7 2nn0 9982 . . . . . . . . . 10  |-  2  e.  NN0
8 nn0mulcl 10000 . . . . . . . . . 10  |-  ( ( 2  e.  NN0  /\  n  e.  NN0 )  -> 
( 2  x.  n
)  e.  NN0 )
97, 8mpan 651 . . . . . . . . 9  |-  ( n  e.  NN0  ->  ( 2  x.  n )  e. 
NN0 )
10 nn0p1nn 10003 . . . . . . . . 9  |-  ( ( 2  x.  n )  e.  NN0  ->  ( ( 2  x.  n )  +  1 )  e.  NN )
119, 10syl 15 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( ( 2  x.  n )  +  1 )  e.  NN )
12 nnmulcl 9769 . . . . . . . 8  |-  ( ( 3  e.  NN  /\  ( ( 2  x.  n )  +  1 )  e.  NN )  ->  ( 3  x.  ( ( 2  x.  n )  +  1 ) )  e.  NN )
136, 11, 12sylancr 644 . . . . . . 7  |-  ( n  e.  NN0  ->  ( 3  x.  ( ( 2  x.  n )  +  1 ) )  e.  NN )
14 9nn 9884 . . . . . . . 8  |-  9  e.  NN
15 nnexpcl 11116 . . . . . . . 8  |-  ( ( 9  e.  NN  /\  n  e.  NN0 )  -> 
( 9 ^ n
)  e.  NN )
1614, 15mpan 651 . . . . . . 7  |-  ( n  e.  NN0  ->  ( 9 ^ n )  e.  NN )
1713, 16nnmulcld 9793 . . . . . 6  |-  ( n  e.  NN0  ->  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) )  e.  NN )
18 nndivre 9781 . . . . . 6  |-  ( ( 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 )
195, 17, 18sylancr 644 . . . . 5  |-  ( n  e.  NN0  ->  ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n
) ) )  e.  RR )
204, 19syl 15 . . . 4  |-  ( (  T.  /\  n  e.  ( 0 ... K
) )  ->  (
2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) )  e.  RR )
212, 20fsumrecl 12207 . . 3  |-  (  T. 
->  sum_ n  e.  ( 0 ... K ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) )  e.  RR )
2221trud 1314 . 2  |-  sum_ n  e.  ( 0 ... K
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  e.  RR
23 log2ublem2.4 . . . . . 6  |-  N  e. 
NN0
247, 23nn0mulcli 10002 . . . . 5  |-  ( 2  x.  N )  e. 
NN0
25 nn0p1nn 10003 . . . . 5  |-  ( ( 2  x.  N )  e.  NN0  ->  ( ( 2  x.  N )  +  1 )  e.  NN )
2624, 25ax-mp 8 . . . 4  |-  ( ( 2  x.  N )  +  1 )  e.  NN
276, 26nnmulcli 9770 . . 3  |-  ( 3  x.  ( ( 2  x.  N )  +  1 ) )  e.  NN
28 nnexpcl 11116 . . . 4  |-  ( ( 9  e.  NN  /\  N  e.  NN0 )  -> 
( 9 ^ N
)  e.  NN )
2914, 23, 28mp2an 653 . . 3  |-  ( 9 ^ N )  e.  NN
3027, 29nnmulcli 9770 . 2  |-  ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N ) )  e.  NN
31 log2ublem2.2 . . 3  |-  B  e. 
NN0
327, 31nn0mulcli 10002 . 2  |-  ( 2  x.  B )  e. 
NN0
33 log2ublem2.3 . . 3  |-  F  e. 
NN0
347, 33nn0mulcli 10002 . 2  |-  ( 2  x.  F )  e. 
NN0
35 nn0uz 10262 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
3623, 35eleqtri 2355 . . . . . 6  |-  N  e.  ( ZZ>= `  0 )
3736a1i 10 . . . . 5  |-  (  T. 
->  N  e.  ( ZZ>=
`  0 ) )
38 elfznn0 10822 . . . . . . 7  |-  ( n  e.  ( 0 ... N )  ->  n  e.  NN0 )
3938adantl 452 . . . . . 6  |-  ( (  T.  /\  n  e.  ( 0 ... N
) )  ->  n  e.  NN0 )
4019recnd 8861 . . . . . 6  |-  ( n  e.  NN0  ->  ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n
) ) )  e.  CC )
4139, 40syl 15 . . . . 5  |-  ( (  T.  /\  n  e.  ( 0 ... N
) )  ->  (
2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) )  e.  CC )
42 oveq2 5866 . . . . . . . . 9  |-  ( n  =  N  ->  (
2  x.  n )  =  ( 2  x.  N ) )
4342oveq1d 5873 . . . . . . . 8  |-  ( n  =  N  ->  (
( 2  x.  n
)  +  1 )  =  ( ( 2  x.  N )  +  1 ) )
4443oveq2d 5874 . . . . . . 7  |-  ( n  =  N  ->  (
3  x.  ( ( 2  x.  n )  +  1 ) )  =  ( 3  x.  ( ( 2  x.  N )  +  1 ) ) )
45 oveq2 5866 . . . . . . 7  |-  ( n  =  N  ->  (
9 ^ n )  =  ( 9 ^ N ) )
4644, 45oveq12d 5876 . . . . . 6  |-  ( n  =  N  ->  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) )  =  ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N
) ) )
4746oveq2d 5874 . . . . 5  |-  ( n  =  N  ->  (
2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) )  =  ( 2  / 
( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  (
9 ^ N ) ) ) )
4837, 41, 47fsumm1 12216 . . . 4  |-  (  T. 
->  sum_ n  e.  ( 0 ... N ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) )  =  ( sum_ n  e.  ( 0 ... ( N  -  1 ) ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n
) ) )  +  ( 2  /  (
( 3  x.  (
( 2  x.  N
)  +  1 ) )  x.  ( 9 ^ N ) ) ) ) )
4948trud 1314 . . 3  |-  sum_ n  e.  ( 0 ... N
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  =  (
sum_ n  e.  (
0 ... ( N  - 
1 ) ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) )  +  ( 2  / 
( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  (
9 ^ N ) ) ) )
50 log2ublem2.5 . . . . . 6  |-  ( N  -  1 )  =  K
5150oveq2i 5869 . . . . 5  |-  ( 0 ... ( N  - 
1 ) )  =  ( 0 ... K
)
5251sumeq1i 12171 . . . 4  |-  sum_ n  e.  ( 0 ... ( N  -  1 ) ) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  =  sum_ n  e.  ( 0 ... K ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n
) ) )
5352oveq1i 5868 . . 3  |-  ( sum_ n  e.  ( 0 ... ( N  -  1 ) ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n
) ) )  +  ( 2  /  (
( 3  x.  (
( 2  x.  N
)  +  1 ) )  x.  ( 9 ^ N ) ) ) )  =  (
sum_ n  e.  (
0 ... K ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) )  +  ( 2  / 
( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  (
9 ^ N ) ) ) )
5449, 53eqtri 2303 . 2  |-  sum_ n  e.  ( 0 ... N
) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  =  (
sum_ n  e.  (
0 ... K ) ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) )  +  ( 2  / 
( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  (
9 ^ N ) ) ) )
55 2cn 9816 . . . 4  |-  2  e.  CC
5631nn0cni 9977 . . . 4  |-  B  e.  CC
5733nn0cni 9977 . . . 4  |-  F  e.  CC
5855, 56, 57adddii 8847 . . 3  |-  ( 2  x.  ( B  +  F ) )  =  ( ( 2  x.  B )  +  ( 2  x.  F ) )
59 log2ublem2.6 . . . 4  |-  ( B  +  F )  =  G
6059oveq2i 5869 . . 3  |-  ( 2  x.  ( B  +  F ) )  =  ( 2  x.  G
)
6158, 60eqtr3i 2305 . 2  |-  ( ( 2  x.  B )  +  ( 2  x.  F ) )  =  ( 2  x.  G
)
62 7nn 9882 . . . . . . . . 9  |-  7  e.  NN
6362nnnn0i 9973 . . . . . . . 8  |-  7  e.  NN0
64 nnexpcl 11116 . . . . . . . 8  |-  ( ( 3  e.  NN  /\  7  e.  NN0 )  -> 
( 3 ^ 7 )  e.  NN )
656, 63, 64mp2an 653 . . . . . . 7  |-  ( 3 ^ 7 )  e.  NN
66 5nn 9880 . . . . . . . 8  |-  5  e.  NN
6766, 62nnmulcli 9770 . . . . . . 7  |-  ( 5  x.  7 )  e.  NN
6865, 67nnmulcli 9770 . . . . . 6  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  NN
6968nnrei 9755 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR
7069, 5remulcli 8851 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )  e.  RR
7170leidi 9307 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )  <_ 
( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )
726nnnn0i 9973 . . . . . . . . . . . 12  |-  3  e.  NN0
73 nnexpcl 11116 . . . . . . . . . . . 12  |-  ( ( 9  e.  NN  /\  3  e.  NN0 )  -> 
( 9 ^ 3 )  e.  NN )
7414, 72, 73mp2an 653 . . . . . . . . . . 11  |-  ( 9 ^ 3 )  e.  NN
7574nncni 9756 . . . . . . . . . 10  |-  ( 9 ^ 3 )  e.  CC
7667nncni 9756 . . . . . . . . . 10  |-  ( 5  x.  7 )  e.  CC
7775, 76mulcomi 8843 . . . . . . . . 9  |-  ( ( 9 ^ 3 )  x.  ( 5  x.  7 ) )  =  ( ( 5  x.  7 )  x.  (
9 ^ 3 ) )
78 log2ublem2.8 . . . . . . . . . . . . 13  |-  ( M  +  N )  =  3
79 log2ublem2.7 . . . . . . . . . . . . . . 15  |-  M  e. 
NN0
8079nn0cni 9977 . . . . . . . . . . . . . 14  |-  M  e.  CC
8123nn0cni 9977 . . . . . . . . . . . . . 14  |-  N  e.  CC
8280, 81addcomi 9003 . . . . . . . . . . . . 13  |-  ( M  +  N )  =  ( N  +  M
)
8378, 82eqtr3i 2305 . . . . . . . . . . . 12  |-  3  =  ( N  +  M )
8483oveq2i 5869 . . . . . . . . . . 11  |-  ( 9 ^ 3 )  =  ( 9 ^ ( N  +  M )
)
8514nncni 9756 . . . . . . . . . . . 12  |-  9  e.  CC
86 expadd 11144 . . . . . . . . . . . 12  |-  ( ( 9  e.  CC  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
9 ^ ( N  +  M ) )  =  ( ( 9 ^ N )  x.  ( 9 ^ M
) ) )
8785, 23, 79, 86mp3an 1277 . . . . . . . . . . 11  |-  ( 9 ^ ( N  +  M ) )  =  ( ( 9 ^ N )  x.  (
9 ^ M ) )
8884, 87eqtri 2303 . . . . . . . . . 10  |-  ( 9 ^ 3 )  =  ( ( 9 ^ N )  x.  (
9 ^ M ) )
8988oveq2i 5869 . . . . . . . . 9  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 3 ) )  =  ( ( 5  x.  7 )  x.  (
( 9 ^ N
)  x.  ( 9 ^ M ) ) )
9029nncni 9756 . . . . . . . . . 10  |-  ( 9 ^ N )  e.  CC
91 nnexpcl 11116 . . . . . . . . . . . 12  |-  ( ( 9  e.  NN  /\  M  e.  NN0 )  -> 
( 9 ^ M
)  e.  NN )
9214, 79, 91mp2an 653 . . . . . . . . . . 11  |-  ( 9 ^ M )  e.  NN
9392nncni 9756 . . . . . . . . . 10  |-  ( 9 ^ M )  e.  CC
9476, 90, 93mul12i 9007 . . . . . . . . 9  |-  ( ( 5  x.  7 )  x.  ( ( 9 ^ N )  x.  ( 9 ^ M
) ) )  =  ( ( 9 ^ N )  x.  (
( 5  x.  7 )  x.  ( 9 ^ M ) ) )
9577, 89, 943eqtri 2307 . . . . . . . 8  |-  ( ( 9 ^ 3 )  x.  ( 5  x.  7 ) )  =  ( ( 9 ^ N )  x.  (
( 5  x.  7 )  x.  ( 9 ^ M ) ) )
96 log2ublem2.9 . . . . . . . . 9  |-  ( ( 5  x.  7 )  x.  ( 9 ^ M ) )  =  ( ( ( 2  x.  N )  +  1 )  x.  F
)
9796oveq2i 5869 . . . . . . . 8  |-  ( ( 9 ^ N )  x.  ( ( 5  x.  7 )  x.  ( 9 ^ M
) ) )  =  ( ( 9 ^ N )  x.  (
( ( 2  x.  N )  +  1 )  x.  F ) )
9895, 97eqtri 2303 . . . . . . 7  |-  ( ( 9 ^ 3 )  x.  ( 5  x.  7 ) )  =  ( ( 9 ^ N )  x.  (
( ( 2  x.  N )  +  1 )  x.  F ) )
9998oveq2i 5869 . . . . . 6  |-  ( 3  x.  ( ( 9 ^ 3 )  x.  ( 5  x.  7 ) ) )  =  ( 3  x.  (
( 9 ^ N
)  x.  ( ( ( 2  x.  N
)  +  1 )  x.  F ) ) )
100 df-7 9809 . . . . . . . . . 10  |-  7  =  ( 6  +  1 )
101100oveq2i 5869 . . . . . . . . 9  |-  ( 3 ^ 7 )  =  ( 3 ^ (
6  +  1 ) )
102 3cn 9818 . . . . . . . . . . 11  |-  3  e.  CC
103 6nn0 9986 . . . . . . . . . . 11  |-  6  e.  NN0
104 expp1 11110 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  6  e.  NN0 )  -> 
( 3 ^ (
6  +  1 ) )  =  ( ( 3 ^ 6 )  x.  3 ) )
105102, 103, 104mp2an 653 . . . . . . . . . 10  |-  ( 3 ^ ( 6  +  1 ) )  =  ( ( 3 ^ 6 )  x.  3 )
106 expmul 11147 . . . . . . . . . . . . 13  |-  ( ( 3  e.  CC  /\  2  e.  NN0  /\  3  e.  NN0 )  ->  (
3 ^ ( 2  x.  3 ) )  =  ( ( 3 ^ 2 ) ^
3 ) )
107102, 7, 72, 106mp3an 1277 . . . . . . . . . . . 12  |-  ( 3 ^ ( 2  x.  3 ) )  =  ( ( 3 ^ 2 ) ^ 3 )
10855, 102mulcomi 8843 . . . . . . . . . . . . . 14  |-  ( 2  x.  3 )  =  ( 3  x.  2 )
109 3t2e6 9872 . . . . . . . . . . . . . 14  |-  ( 3  x.  2 )  =  6
110108, 109eqtri 2303 . . . . . . . . . . . . 13  |-  ( 2  x.  3 )  =  6
111110oveq2i 5869 . . . . . . . . . . . 12  |-  ( 3 ^ ( 2  x.  3 ) )  =  ( 3 ^ 6 )
112 sq3 11200 . . . . . . . . . . . . 13  |-  ( 3 ^ 2 )  =  9
113112oveq1i 5868 . . . . . . . . . . . 12  |-  ( ( 3 ^ 2 ) ^ 3 )  =  ( 9 ^ 3 )
114107, 111, 1133eqtr3i 2311 . . . . . . . . . . 11  |-  ( 3 ^ 6 )  =  ( 9 ^ 3 )
115114oveq1i 5868 . . . . . . . . . 10  |-  ( ( 3 ^ 6 )  x.  3 )  =  ( ( 9 ^ 3 )  x.  3 )
116105, 115eqtri 2303 . . . . . . . . 9  |-  ( 3 ^ ( 6  +  1 ) )  =  ( ( 9 ^ 3 )  x.  3 )
11775, 102mulcomi 8843 . . . . . . . . 9  |-  ( ( 9 ^ 3 )  x.  3 )  =  ( 3  x.  (
9 ^ 3 ) )
118101, 116, 1173eqtri 2307 . . . . . . . 8  |-  ( 3 ^ 7 )  =  ( 3  x.  (
9 ^ 3 ) )
119118oveq1i 5868 . . . . . . 7  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  =  ( ( 3  x.  ( 9 ^ 3 ) )  x.  (
5  x.  7 ) )
120102, 75, 76mulassi 8846 . . . . . . 7  |-  ( ( 3  x.  ( 9 ^ 3 ) )  x.  ( 5  x.  7 ) )  =  ( 3  x.  (
( 9 ^ 3 )  x.  ( 5  x.  7 ) ) )
121119, 120eqtri 2303 . . . . . 6  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  =  ( 3  x.  (
( 9 ^ 3 )  x.  ( 5  x.  7 ) ) )
12226nncni 9756 . . . . . . . . 9  |-  ( ( 2  x.  N )  +  1 )  e.  CC
123102, 122, 90mul32i 9008 . . . . . . . 8  |-  ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N ) )  =  ( ( 3  x.  ( 9 ^ N
) )  x.  (
( 2  x.  N
)  +  1 ) )
124123oveq1i 5868 . . . . . . 7  |-  ( ( ( 3  x.  (
( 2  x.  N
)  +  1 ) )  x.  ( 9 ^ N ) )  x.  F )  =  ( ( ( 3  x.  ( 9 ^ N ) )  x.  ( ( 2  x.  N )  +  1 ) )  x.  F
)
125102, 90mulcli 8842 . . . . . . . 8  |-  ( 3  x.  ( 9 ^ N ) )  e.  CC
126125, 122, 57mulassi 8846 . . . . . . 7  |-  ( ( ( 3  x.  (
9 ^ N ) )  x.  ( ( 2  x.  N )  +  1 ) )  x.  F )  =  ( ( 3  x.  ( 9 ^ N
) )  x.  (
( ( 2  x.  N )  +  1 )  x.  F ) )
127122, 57mulcli 8842 . . . . . . . 8  |-  ( ( ( 2  x.  N
)  +  1 )  x.  F )  e.  CC
128102, 90, 127mulassi 8846 . . . . . . 7  |-  ( ( 3  x.  ( 9 ^ N ) )  x.  ( ( ( 2  x.  N )  +  1 )  x.  F ) )  =  ( 3  x.  (
( 9 ^ N
)  x.  ( ( ( 2  x.  N
)  +  1 )  x.  F ) ) )
129124, 126, 1283eqtri 2307 . . . . . 6  |-  ( ( ( 3  x.  (
( 2  x.  N
)  +  1 ) )  x.  ( 9 ^ N ) )  x.  F )  =  ( 3  x.  (
( 9 ^ N
)  x.  ( ( ( 2  x.  N
)  +  1 )  x.  F ) ) )
13099, 121, 1293eqtr4i 2313 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  =  ( ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N
) )  x.  F
)
131130oveq2i 5869 . . . 4  |-  ( 2  x.  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  =  ( 2  x.  (
( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  (
9 ^ N ) )  x.  F ) )
13265nncni 9756 . . . . . 6  |-  ( 3 ^ 7 )  e.  CC
133132, 76mulcli 8842 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  CC
134133, 55mulcomi 8843 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )  =  ( 2  x.  (
( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )
13530nncni 9756 . . . . 5  |-  ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N ) )  e.  CC
136135, 55, 57mul12i 9007 . . . 4  |-  ( ( ( 3  x.  (
( 2  x.  N
)  +  1 ) )  x.  ( 9 ^ N ) )  x.  ( 2  x.  F ) )  =  ( 2  x.  (
( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  (
9 ^ N ) )  x.  F ) )
137131, 134, 1363eqtr4i 2313 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )  =  ( ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N
) )  x.  (
2  x.  F ) )
13871, 137breqtri 4046 . 2  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )  <_ 
( ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N
) )  x.  (
2  x.  F ) )
1391, 22, 7, 30, 32, 34, 54, 61, 138log2ublem1 20242 1  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  sum_ n  e.  ( 0 ... N ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) ) )  <_  (
2  x.  G )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    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    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   3c3 9796   5c5 9798   6c6 9799   7c7 9800   9c9 9802   NN0cn0 9965   ZZ>=cuz 10230   ...cfz 10782   ^cexp 11104   sum_csu 12158
This theorem is referenced by:  log2ublem3  20244
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
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-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-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  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-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159
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