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Theorem log2ublem2 20789
Description: Lemma for log2ub 20791. (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 11314 . . . 4  |-  (  T. 
->  ( 0 ... K
)  e.  Fin )
3 elfznn0 11085 . . . . . 6  |-  ( n  e.  ( 0 ... K )  ->  n  e.  NN0 )
43adantl 454 . . . . 5  |-  ( (  T.  /\  n  e.  ( 0 ... K
) )  ->  n  e.  NN0 )
5 2re 10071 . . . . . 6  |-  2  e.  RR
6 3nn 10136 . . . . . . . 8  |-  3  e.  NN
7 2nn0 10240 . . . . . . . . . 10  |-  2  e.  NN0
8 nn0mulcl 10258 . . . . . . . . . 10  |-  ( ( 2  e.  NN0  /\  n  e.  NN0 )  -> 
( 2  x.  n
)  e.  NN0 )
97, 8mpan 653 . . . . . . . . 9  |-  ( n  e.  NN0  ->  ( 2  x.  n )  e. 
NN0 )
10 nn0p1nn 10261 . . . . . . . . 9  |-  ( ( 2  x.  n )  e.  NN0  ->  ( ( 2  x.  n )  +  1 )  e.  NN )
119, 10syl 16 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( ( 2  x.  n )  +  1 )  e.  NN )
12 nnmulcl 10025 . . . . . . . 8  |-  ( ( 3  e.  NN  /\  ( ( 2  x.  n )  +  1 )  e.  NN )  ->  ( 3  x.  ( ( 2  x.  n )  +  1 ) )  e.  NN )
136, 11, 12sylancr 646 . . . . . . 7  |-  ( n  e.  NN0  ->  ( 3  x.  ( ( 2  x.  n )  +  1 ) )  e.  NN )
14 9nn 10142 . . . . . . . 8  |-  9  e.  NN
15 nnexpcl 11396 . . . . . . . 8  |-  ( ( 9  e.  NN  /\  n  e.  NN0 )  -> 
( 9 ^ n
)  e.  NN )
1614, 15mpan 653 . . . . . . 7  |-  ( n  e.  NN0  ->  ( 9 ^ n )  e.  NN )
1713, 16nnmulcld 10049 . . . . . 6  |-  ( n  e.  NN0  ->  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) )  e.  NN )
18 nndivre 10037 . . . . . 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 646 . . . . 5  |-  ( n  e.  NN0  ->  ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n
) ) )  e.  RR )
204, 19syl 16 . . . 4  |-  ( (  T.  /\  n  e.  ( 0 ... K
) )  ->  (
2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) )  e.  RR )
212, 20fsumrecl 12530 . . 3  |-  (  T. 
->  sum_ n  e.  ( 0 ... K ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) )  e.  RR )
2221trud 1333 . 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 10260 . . . . 5  |-  ( 2  x.  N )  e. 
NN0
25 nn0p1nn 10261 . . . . 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 10026 . . 3  |-  ( 3  x.  ( ( 2  x.  N )  +  1 ) )  e.  NN
28 nnexpcl 11396 . . . 4  |-  ( ( 9  e.  NN  /\  N  e.  NN0 )  -> 
( 9 ^ N
)  e.  NN )
2914, 23, 28mp2an 655 . . 3  |-  ( 9 ^ N )  e.  NN
3027, 29nnmulcli 10026 . 2  |-  ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N ) )  e.  NN
31 log2ublem2.2 . . 3  |-  B  e. 
NN0
327, 31nn0mulcli 10260 . 2  |-  ( 2  x.  B )  e. 
NN0
33 log2ublem2.3 . . 3  |-  F  e. 
NN0
347, 33nn0mulcli 10260 . 2  |-  ( 2  x.  F )  e. 
NN0
35 nn0uz 10522 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
3623, 35eleqtri 2510 . . . . . 6  |-  N  e.  ( ZZ>= `  0 )
3736a1i 11 . . . . 5  |-  (  T. 
->  N  e.  ( ZZ>=
`  0 ) )
38 elfznn0 11085 . . . . . . 7  |-  ( n  e.  ( 0 ... N )  ->  n  e.  NN0 )
3938adantl 454 . . . . . 6  |-  ( (  T.  /\  n  e.  ( 0 ... N
) )  ->  n  e.  NN0 )
4019recnd 9116 . . . . . 6  |-  ( n  e.  NN0  ->  ( 2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n
) ) )  e.  CC )
4139, 40syl 16 . . . . 5  |-  ( (  T.  /\  n  e.  ( 0 ... N
) )  ->  (
2  /  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) ) )  e.  CC )
42 oveq2 6091 . . . . . . . . 9  |-  ( n  =  N  ->  (
2  x.  n )  =  ( 2  x.  N ) )
4342oveq1d 6098 . . . . . . . 8  |-  ( n  =  N  ->  (
( 2  x.  n
)  +  1 )  =  ( ( 2  x.  N )  +  1 ) )
4443oveq2d 6099 . . . . . . 7  |-  ( n  =  N  ->  (
3  x.  ( ( 2  x.  n )  +  1 ) )  =  ( 3  x.  ( ( 2  x.  N )  +  1 ) ) )
45 oveq2 6091 . . . . . . 7  |-  ( n  =  N  ->  (
9 ^ n )  =  ( 9 ^ N ) )
4644, 45oveq12d 6101 . . . . . 6  |-  ( n  =  N  ->  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) )  =  ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N
) ) )
4746oveq2d 6099 . . . . 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 12539 . . . 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 1333 . . 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 6094 . . . . 5  |-  ( 0 ... ( N  - 
1 ) )  =  ( 0 ... K
)
5251sumeq1i 12494 . . . 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 6093 . . 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 2458 . 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 10072 . . . 4  |-  2  e.  CC
5631nn0cni 10235 . . . 4  |-  B  e.  CC
5733nn0cni 10235 . . . 4  |-  F  e.  CC
5855, 56, 57adddii 9102 . . 3  |-  ( 2  x.  ( B  +  F ) )  =  ( ( 2  x.  B )  +  ( 2  x.  F ) )
59 log2ublem2.6 . . . 4  |-  ( B  +  F )  =  G
6059oveq2i 6094 . . 3  |-  ( 2  x.  ( B  +  F ) )  =  ( 2  x.  G
)
6158, 60eqtr3i 2460 . 2  |-  ( ( 2  x.  B )  +  ( 2  x.  F ) )  =  ( 2  x.  G
)
62 7nn 10140 . . . . . . . . 9  |-  7  e.  NN
6362nnnn0i 10231 . . . . . . . 8  |-  7  e.  NN0
64 nnexpcl 11396 . . . . . . . 8  |-  ( ( 3  e.  NN  /\  7  e.  NN0 )  -> 
( 3 ^ 7 )  e.  NN )
656, 63, 64mp2an 655 . . . . . . 7  |-  ( 3 ^ 7 )  e.  NN
66 5nn 10138 . . . . . . . 8  |-  5  e.  NN
6766, 62nnmulcli 10026 . . . . . . 7  |-  ( 5  x.  7 )  e.  NN
6865, 67nnmulcli 10026 . . . . . 6  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  NN
6968nnrei 10011 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR
7069, 5remulcli 9106 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )  e.  RR
7170leidi 9563 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )  <_ 
( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )
726nnnn0i 10231 . . . . . . . . . . . 12  |-  3  e.  NN0
73 nnexpcl 11396 . . . . . . . . . . . 12  |-  ( ( 9  e.  NN  /\  3  e.  NN0 )  -> 
( 9 ^ 3 )  e.  NN )
7414, 72, 73mp2an 655 . . . . . . . . . . 11  |-  ( 9 ^ 3 )  e.  NN
7574nncni 10012 . . . . . . . . . 10  |-  ( 9 ^ 3 )  e.  CC
7667nncni 10012 . . . . . . . . . 10  |-  ( 5  x.  7 )  e.  CC
7775, 76mulcomi 9098 . . . . . . . . 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 10235 . . . . . . . . . . . . . 14  |-  M  e.  CC
8123nn0cni 10235 . . . . . . . . . . . . . 14  |-  N  e.  CC
8280, 81addcomi 9259 . . . . . . . . . . . . 13  |-  ( M  +  N )  =  ( N  +  M
)
8378, 82eqtr3i 2460 . . . . . . . . . . . 12  |-  3  =  ( N  +  M )
8483oveq2i 6094 . . . . . . . . . . 11  |-  ( 9 ^ 3 )  =  ( 9 ^ ( N  +  M )
)
8514nncni 10012 . . . . . . . . . . . 12  |-  9  e.  CC
86 expadd 11424 . . . . . . . . . . . 12  |-  ( ( 9  e.  CC  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
9 ^ ( N  +  M ) )  =  ( ( 9 ^ N )  x.  ( 9 ^ M
) ) )
8785, 23, 79, 86mp3an 1280 . . . . . . . . . . 11  |-  ( 9 ^ ( N  +  M ) )  =  ( ( 9 ^ N )  x.  (
9 ^ M ) )
8884, 87eqtri 2458 . . . . . . . . . 10  |-  ( 9 ^ 3 )  =  ( ( 9 ^ N )  x.  (
9 ^ M ) )
8988oveq2i 6094 . . . . . . . . 9  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 3 ) )  =  ( ( 5  x.  7 )  x.  (
( 9 ^ N
)  x.  ( 9 ^ M ) ) )
9029nncni 10012 . . . . . . . . . 10  |-  ( 9 ^ N )  e.  CC
91 nnexpcl 11396 . . . . . . . . . . . 12  |-  ( ( 9  e.  NN  /\  M  e.  NN0 )  -> 
( 9 ^ M
)  e.  NN )
9214, 79, 91mp2an 655 . . . . . . . . . . 11  |-  ( 9 ^ M )  e.  NN
9392nncni 10012 . . . . . . . . . 10  |-  ( 9 ^ M )  e.  CC
9476, 90, 93mul12i 9263 . . . . . . . . 9  |-  ( ( 5  x.  7 )  x.  ( ( 9 ^ N )  x.  ( 9 ^ M
) ) )  =  ( ( 9 ^ N )  x.  (
( 5  x.  7 )  x.  ( 9 ^ M ) ) )
9577, 89, 943eqtri 2462 . . . . . . . 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 6094 . . . . . . . 8  |-  ( ( 9 ^ N )  x.  ( ( 5  x.  7 )  x.  ( 9 ^ M
) ) )  =  ( ( 9 ^ N )  x.  (
( ( 2  x.  N )  +  1 )  x.  F ) )
9895, 97eqtri 2458 . . . . . . 7  |-  ( ( 9 ^ 3 )  x.  ( 5  x.  7 ) )  =  ( ( 9 ^ N )  x.  (
( ( 2  x.  N )  +  1 )  x.  F ) )
9998oveq2i 6094 . . . . . 6  |-  ( 3  x.  ( ( 9 ^ 3 )  x.  ( 5  x.  7 ) ) )  =  ( 3  x.  (
( 9 ^ N
)  x.  ( ( ( 2  x.  N
)  +  1 )  x.  F ) ) )
100 df-7 10065 . . . . . . . . . 10  |-  7  =  ( 6  +  1 )
101100oveq2i 6094 . . . . . . . . 9  |-  ( 3 ^ 7 )  =  ( 3 ^ (
6  +  1 ) )
102 3cn 10074 . . . . . . . . . . 11  |-  3  e.  CC
103 6nn0 10244 . . . . . . . . . . 11  |-  6  e.  NN0
104 expp1 11390 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  6  e.  NN0 )  -> 
( 3 ^ (
6  +  1 ) )  =  ( ( 3 ^ 6 )  x.  3 ) )
105102, 103, 104mp2an 655 . . . . . . . . . 10  |-  ( 3 ^ ( 6  +  1 ) )  =  ( ( 3 ^ 6 )  x.  3 )
106 expmul 11427 . . . . . . . . . . . . 13  |-  ( ( 3  e.  CC  /\  2  e.  NN0  /\  3  e.  NN0 )  ->  (
3 ^ ( 2  x.  3 ) )  =  ( ( 3 ^ 2 ) ^
3 ) )
107102, 7, 72, 106mp3an 1280 . . . . . . . . . . . 12  |-  ( 3 ^ ( 2  x.  3 ) )  =  ( ( 3 ^ 2 ) ^ 3 )
10855, 102mulcomi 9098 . . . . . . . . . . . . . 14  |-  ( 2  x.  3 )  =  ( 3  x.  2 )
109 3t2e6 10130 . . . . . . . . . . . . . 14  |-  ( 3  x.  2 )  =  6
110108, 109eqtri 2458 . . . . . . . . . . . . 13  |-  ( 2  x.  3 )  =  6
111110oveq2i 6094 . . . . . . . . . . . 12  |-  ( 3 ^ ( 2  x.  3 ) )  =  ( 3 ^ 6 )
112 sq3 11480 . . . . . . . . . . . . 13  |-  ( 3 ^ 2 )  =  9
113112oveq1i 6093 . . . . . . . . . . . 12  |-  ( ( 3 ^ 2 ) ^ 3 )  =  ( 9 ^ 3 )
114107, 111, 1133eqtr3i 2466 . . . . . . . . . . 11  |-  ( 3 ^ 6 )  =  ( 9 ^ 3 )
115114oveq1i 6093 . . . . . . . . . 10  |-  ( ( 3 ^ 6 )  x.  3 )  =  ( ( 9 ^ 3 )  x.  3 )
116105, 115eqtri 2458 . . . . . . . . 9  |-  ( 3 ^ ( 6  +  1 ) )  =  ( ( 9 ^ 3 )  x.  3 )
11775, 102mulcomi 9098 . . . . . . . . 9  |-  ( ( 9 ^ 3 )  x.  3 )  =  ( 3  x.  (
9 ^ 3 ) )
118101, 116, 1173eqtri 2462 . . . . . . . 8  |-  ( 3 ^ 7 )  =  ( 3  x.  (
9 ^ 3 ) )
119118oveq1i 6093 . . . . . . 7  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  =  ( ( 3  x.  ( 9 ^ 3 ) )  x.  (
5  x.  7 ) )
120102, 75, 76mulassi 9101 . . . . . . 7  |-  ( ( 3  x.  ( 9 ^ 3 ) )  x.  ( 5  x.  7 ) )  =  ( 3  x.  (
( 9 ^ 3 )  x.  ( 5  x.  7 ) ) )
121119, 120eqtri 2458 . . . . . 6  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  =  ( 3  x.  (
( 9 ^ 3 )  x.  ( 5  x.  7 ) ) )
12226nncni 10012 . . . . . . . . 9  |-  ( ( 2  x.  N )  +  1 )  e.  CC
123102, 122, 90mul32i 9264 . . . . . . . 8  |-  ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N ) )  =  ( ( 3  x.  ( 9 ^ N
) )  x.  (
( 2  x.  N
)  +  1 ) )
124123oveq1i 6093 . . . . . . 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 9097 . . . . . . . 8  |-  ( 3  x.  ( 9 ^ N ) )  e.  CC
126125, 122, 57mulassi 9101 . . . . . . 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 9097 . . . . . . . 8  |-  ( ( ( 2  x.  N
)  +  1 )  x.  F )  e.  CC
128102, 90, 127mulassi 9101 . . . . . . 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 2462 . . . . . 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 2468 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  =  ( ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N
) )  x.  F
)
131130oveq2i 6094 . . . 4  |-  ( 2  x.  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  =  ( 2  x.  (
( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  (
9 ^ N ) )  x.  F ) )
13265nncni 10012 . . . . . 6  |-  ( 3 ^ 7 )  e.  CC
133132, 76mulcli 9097 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  CC
134133, 55mulcomi 9098 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )  =  ( 2  x.  (
( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )
13530nncni 10012 . . . . 5  |-  ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N ) )  e.  CC
136135, 55, 57mul12i 9263 . . . 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 2468 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )  =  ( ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N
) )  x.  (
2  x.  F ) )
13871, 137breqtri 4237 . 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 20788 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 360    T. wtru 1326    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997    <_ cle 9123    - cmin 9293    / cdiv 9679   NNcn 10002   2c2 10051   3c3 10052   5c5 10054   6c6 10055   7c7 10056   9c9 10058   NN0cn0 10223   ZZ>=cuz 10490   ...cfz 11045   ^cexp 11384   sum_csu 12481
This theorem is referenced by:  log2ublem3  20790
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 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fzo 11138  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482
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