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Theorem log2ublem2 23147
Description: Lemma for log2ub 23149. (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 12059 . . . 4  |-  ( T. 
->  ( 0 ... K
)  e.  Fin )
3 elfznn0 11776 . . . . . 6  |-  ( n  e.  ( 0 ... K )  ->  n  e.  NN0 )
43adantl 466 . . . . 5  |-  ( ( T.  /\  n  e.  ( 0 ... K
) )  ->  n  e.  NN0 )
5 2re 10608 . . . . . 6  |-  2  e.  RR
6 3nn 10697 . . . . . . . 8  |-  3  e.  NN
7 2nn0 10815 . . . . . . . . . 10  |-  2  e.  NN0
8 nn0mulcl 10835 . . . . . . . . . 10  |-  ( ( 2  e.  NN0  /\  n  e.  NN0 )  -> 
( 2  x.  n
)  e.  NN0 )
97, 8mpan 670 . . . . . . . . 9  |-  ( n  e.  NN0  ->  ( 2  x.  n )  e. 
NN0 )
10 nn0p1nn 10838 . . . . . . . . 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 10562 . . . . . . . 8  |-  ( ( 3  e.  NN  /\  ( ( 2  x.  n )  +  1 )  e.  NN )  ->  ( 3  x.  ( ( 2  x.  n )  +  1 ) )  e.  NN )
136, 11, 12sylancr 663 . . . . . . 7  |-  ( n  e.  NN0  ->  ( 3  x.  ( ( 2  x.  n )  +  1 ) )  e.  NN )
14 9nn 10703 . . . . . . . 8  |-  9  e.  NN
15 nnexpcl 12155 . . . . . . . 8  |-  ( ( 9  e.  NN  /\  n  e.  NN0 )  -> 
( 9 ^ n
)  e.  NN )
1614, 15mpan 670 . . . . . . 7  |-  ( n  e.  NN0  ->  ( 9 ^ n )  e.  NN )
1713, 16nnmulcld 10586 . . . . . 6  |-  ( n  e.  NN0  ->  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) )  e.  NN )
18 nndivre 10574 . . . . . 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 663 . . . . 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 13532 . . 3  |-  ( T. 
->  sum_ n  e.  ( 0 ... K ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) )  e.  RR )
2221trud 1390 . 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 10837 . . . . 5  |-  ( 2  x.  N )  e. 
NN0
25 nn0p1nn 10838 . . . . 5  |-  ( ( 2  x.  N )  e.  NN0  ->  ( ( 2  x.  N )  +  1 )  e.  NN )
2624, 25ax-mp 5 . . . 4  |-  ( ( 2  x.  N )  +  1 )  e.  NN
276, 26nnmulcli 10563 . . 3  |-  ( 3  x.  ( ( 2  x.  N )  +  1 ) )  e.  NN
28 nnexpcl 12155 . . . 4  |-  ( ( 9  e.  NN  /\  N  e.  NN0 )  -> 
( 9 ^ N
)  e.  NN )
2914, 23, 28mp2an 672 . . 3  |-  ( 9 ^ N )  e.  NN
3027, 29nnmulcli 10563 . 2  |-  ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N ) )  e.  NN
31 log2ublem2.2 . . 3  |-  B  e. 
NN0
327, 31nn0mulcli 10837 . 2  |-  ( 2  x.  B )  e. 
NN0
33 log2ublem2.3 . . 3  |-  F  e. 
NN0
347, 33nn0mulcli 10837 . 2  |-  ( 2  x.  F )  e. 
NN0
35 nn0uz 11121 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
3623, 35eleqtri 2527 . . . . . 6  |-  N  e.  ( ZZ>= `  0 )
3736a1i 11 . . . . 5  |-  ( T. 
->  N  e.  ( ZZ>=
`  0 ) )
38 elfznn0 11776 . . . . . . 7  |-  ( n  e.  ( 0 ... N )  ->  n  e.  NN0 )
3938adantl 466 . . . . . 6  |-  ( ( T.  /\  n  e.  ( 0 ... N
) )  ->  n  e.  NN0 )
4019recnd 9622 . . . . . 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 6286 . . . . . . . . 9  |-  ( n  =  N  ->  (
2  x.  n )  =  ( 2  x.  N ) )
4342oveq1d 6293 . . . . . . . 8  |-  ( n  =  N  ->  (
( 2  x.  n
)  +  1 )  =  ( ( 2  x.  N )  +  1 ) )
4443oveq2d 6294 . . . . . . 7  |-  ( n  =  N  ->  (
3  x.  ( ( 2  x.  n )  +  1 ) )  =  ( 3  x.  ( ( 2  x.  N )  +  1 ) ) )
45 oveq2 6286 . . . . . . 7  |-  ( n  =  N  ->  (
9 ^ n )  =  ( 9 ^ N ) )
4644, 45oveq12d 6296 . . . . . 6  |-  ( n  =  N  ->  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) )  =  ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N
) ) )
4746oveq2d 6294 . . . . 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 13542 . . . 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 1390 . . 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 6289 . . . . 5  |-  ( 0 ... ( N  - 
1 ) )  =  ( 0 ... K
)
5251sumeq1i 13496 . . . 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 6288 . . 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 2470 . 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 10609 . . . 4  |-  2  e.  CC
5631nn0cni 10810 . . . 4  |-  B  e.  CC
5733nn0cni 10810 . . . 4  |-  F  e.  CC
5855, 56, 57adddii 9606 . . 3  |-  ( 2  x.  ( B  +  F ) )  =  ( ( 2  x.  B )  +  ( 2  x.  F ) )
59 log2ublem2.6 . . . 4  |-  ( B  +  F )  =  G
6059oveq2i 6289 . . 3  |-  ( 2  x.  ( B  +  F ) )  =  ( 2  x.  G
)
6158, 60eqtr3i 2472 . 2  |-  ( ( 2  x.  B )  +  ( 2  x.  F ) )  =  ( 2  x.  G
)
62 7nn 10701 . . . . . . . . 9  |-  7  e.  NN
6362nnnn0i 10806 . . . . . . . 8  |-  7  e.  NN0
64 nnexpcl 12155 . . . . . . . 8  |-  ( ( 3  e.  NN  /\  7  e.  NN0 )  -> 
( 3 ^ 7 )  e.  NN )
656, 63, 64mp2an 672 . . . . . . 7  |-  ( 3 ^ 7 )  e.  NN
66 5nn 10699 . . . . . . . 8  |-  5  e.  NN
6766, 62nnmulcli 10563 . . . . . . 7  |-  ( 5  x.  7 )  e.  NN
6865, 67nnmulcli 10563 . . . . . 6  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  NN
6968nnrei 10548 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR
7069, 5remulcli 9610 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )  e.  RR
7170leidi 10090 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )  <_ 
( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )
726nnnn0i 10806 . . . . . . . . . . . 12  |-  3  e.  NN0
73 nnexpcl 12155 . . . . . . . . . . . 12  |-  ( ( 9  e.  NN  /\  3  e.  NN0 )  -> 
( 9 ^ 3 )  e.  NN )
7414, 72, 73mp2an 672 . . . . . . . . . . 11  |-  ( 9 ^ 3 )  e.  NN
7574nncni 10549 . . . . . . . . . 10  |-  ( 9 ^ 3 )  e.  CC
7667nncni 10549 . . . . . . . . . 10  |-  ( 5  x.  7 )  e.  CC
7775, 76mulcomi 9602 . . . . . . . . 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 10810 . . . . . . . . . . . . . 14  |-  M  e.  CC
8123nn0cni 10810 . . . . . . . . . . . . . 14  |-  N  e.  CC
8280, 81addcomi 9771 . . . . . . . . . . . . 13  |-  ( M  +  N )  =  ( N  +  M
)
8378, 82eqtr3i 2472 . . . . . . . . . . . 12  |-  3  =  ( N  +  M )
8483oveq2i 6289 . . . . . . . . . . 11  |-  ( 9 ^ 3 )  =  ( 9 ^ ( N  +  M )
)
8514nncni 10549 . . . . . . . . . . . 12  |-  9  e.  CC
86 expadd 12184 . . . . . . . . . . . 12  |-  ( ( 9  e.  CC  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
9 ^ ( N  +  M ) )  =  ( ( 9 ^ N )  x.  ( 9 ^ M
) ) )
8785, 23, 79, 86mp3an 1323 . . . . . . . . . . 11  |-  ( 9 ^ ( N  +  M ) )  =  ( ( 9 ^ N )  x.  (
9 ^ M ) )
8884, 87eqtri 2470 . . . . . . . . . 10  |-  ( 9 ^ 3 )  =  ( ( 9 ^ N )  x.  (
9 ^ M ) )
8988oveq2i 6289 . . . . . . . . 9  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 3 ) )  =  ( ( 5  x.  7 )  x.  (
( 9 ^ N
)  x.  ( 9 ^ M ) ) )
9029nncni 10549 . . . . . . . . . 10  |-  ( 9 ^ N )  e.  CC
91 nnexpcl 12155 . . . . . . . . . . . 12  |-  ( ( 9  e.  NN  /\  M  e.  NN0 )  -> 
( 9 ^ M
)  e.  NN )
9214, 79, 91mp2an 672 . . . . . . . . . . 11  |-  ( 9 ^ M )  e.  NN
9392nncni 10549 . . . . . . . . . 10  |-  ( 9 ^ M )  e.  CC
9476, 90, 93mul12i 9775 . . . . . . . . 9  |-  ( ( 5  x.  7 )  x.  ( ( 9 ^ N )  x.  ( 9 ^ M
) ) )  =  ( ( 9 ^ N )  x.  (
( 5  x.  7 )  x.  ( 9 ^ M ) ) )
9577, 89, 943eqtri 2474 . . . . . . . 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 6289 . . . . . . . 8  |-  ( ( 9 ^ N )  x.  ( ( 5  x.  7 )  x.  ( 9 ^ M
) ) )  =  ( ( 9 ^ N )  x.  (
( ( 2  x.  N )  +  1 )  x.  F ) )
9895, 97eqtri 2470 . . . . . . 7  |-  ( ( 9 ^ 3 )  x.  ( 5  x.  7 ) )  =  ( ( 9 ^ N )  x.  (
( ( 2  x.  N )  +  1 )  x.  F ) )
9998oveq2i 6289 . . . . . 6  |-  ( 3  x.  ( ( 9 ^ 3 )  x.  ( 5  x.  7 ) ) )  =  ( 3  x.  (
( 9 ^ N
)  x.  ( ( ( 2  x.  N
)  +  1 )  x.  F ) ) )
100 df-7 10602 . . . . . . . . . 10  |-  7  =  ( 6  +  1 )
101100oveq2i 6289 . . . . . . . . 9  |-  ( 3 ^ 7 )  =  ( 3 ^ (
6  +  1 ) )
102 3cn 10613 . . . . . . . . . . 11  |-  3  e.  CC
103 6nn0 10819 . . . . . . . . . . 11  |-  6  e.  NN0
104 expp1 12149 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  6  e.  NN0 )  -> 
( 3 ^ (
6  +  1 ) )  =  ( ( 3 ^ 6 )  x.  3 ) )
105102, 103, 104mp2an 672 . . . . . . . . . 10  |-  ( 3 ^ ( 6  +  1 ) )  =  ( ( 3 ^ 6 )  x.  3 )
106 expmul 12187 . . . . . . . . . . . . 13  |-  ( ( 3  e.  CC  /\  2  e.  NN0  /\  3  e.  NN0 )  ->  (
3 ^ ( 2  x.  3 ) )  =  ( ( 3 ^ 2 ) ^
3 ) )
107102, 7, 72, 106mp3an 1323 . . . . . . . . . . . 12  |-  ( 3 ^ ( 2  x.  3 ) )  =  ( ( 3 ^ 2 ) ^ 3 )
10855, 102mulcomi 9602 . . . . . . . . . . . . . 14  |-  ( 2  x.  3 )  =  ( 3  x.  2 )
109 3t2e6 10690 . . . . . . . . . . . . . 14  |-  ( 3  x.  2 )  =  6
110108, 109eqtri 2470 . . . . . . . . . . . . 13  |-  ( 2  x.  3 )  =  6
111110oveq2i 6289 . . . . . . . . . . . 12  |-  ( 3 ^ ( 2  x.  3 ) )  =  ( 3 ^ 6 )
112 sq3 12241 . . . . . . . . . . . . 13  |-  ( 3 ^ 2 )  =  9
113112oveq1i 6288 . . . . . . . . . . . 12  |-  ( ( 3 ^ 2 ) ^ 3 )  =  ( 9 ^ 3 )
114107, 111, 1133eqtr3i 2478 . . . . . . . . . . 11  |-  ( 3 ^ 6 )  =  ( 9 ^ 3 )
115114oveq1i 6288 . . . . . . . . . 10  |-  ( ( 3 ^ 6 )  x.  3 )  =  ( ( 9 ^ 3 )  x.  3 )
116105, 115eqtri 2470 . . . . . . . . 9  |-  ( 3 ^ ( 6  +  1 ) )  =  ( ( 9 ^ 3 )  x.  3 )
11775, 102mulcomi 9602 . . . . . . . . 9  |-  ( ( 9 ^ 3 )  x.  3 )  =  ( 3  x.  (
9 ^ 3 ) )
118101, 116, 1173eqtri 2474 . . . . . . . 8  |-  ( 3 ^ 7 )  =  ( 3  x.  (
9 ^ 3 ) )
119118oveq1i 6288 . . . . . . 7  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  =  ( ( 3  x.  ( 9 ^ 3 ) )  x.  (
5  x.  7 ) )
120102, 75, 76mulassi 9605 . . . . . . 7  |-  ( ( 3  x.  ( 9 ^ 3 ) )  x.  ( 5  x.  7 ) )  =  ( 3  x.  (
( 9 ^ 3 )  x.  ( 5  x.  7 ) ) )
121119, 120eqtri 2470 . . . . . 6  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  =  ( 3  x.  (
( 9 ^ 3 )  x.  ( 5  x.  7 ) ) )
12226nncni 10549 . . . . . . . . 9  |-  ( ( 2  x.  N )  +  1 )  e.  CC
123102, 122, 90mul32i 9776 . . . . . . . 8  |-  ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N ) )  =  ( ( 3  x.  ( 9 ^ N
) )  x.  (
( 2  x.  N
)  +  1 ) )
124123oveq1i 6288 . . . . . . 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 9601 . . . . . . . 8  |-  ( 3  x.  ( 9 ^ N ) )  e.  CC
126125, 122, 57mulassi 9605 . . . . . . 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 9601 . . . . . . . 8  |-  ( ( ( 2  x.  N
)  +  1 )  x.  F )  e.  CC
128102, 90, 127mulassi 9605 . . . . . . 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 2474 . . . . . 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 2480 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  =  ( ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N
) )  x.  F
)
131130oveq2i 6289 . . . 4  |-  ( 2  x.  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  =  ( 2  x.  (
( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  (
9 ^ N ) )  x.  F ) )
13265nncni 10549 . . . . . 6  |-  ( 3 ^ 7 )  e.  CC
133132, 76mulcli 9601 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  CC
134133, 55mulcomi 9602 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )  =  ( 2  x.  (
( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )
13530nncni 10549 . . . . 5  |-  ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N ) )  e.  CC
136135, 55, 57mul12i 9775 . . . 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 2480 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )  =  ( ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N
) )  x.  (
2  x.  F ) )
13871, 137breqtri 4457 . 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 23146 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 setvar class
Syntax hints:    /\ wa 369    = wceq 1381   T. wtru 1382    e. wcel 1802   class class class wbr 4434   ` cfv 5575  (class class class)co 6278   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    <_ cle 9629    - cmin 9807    / cdiv 10209   NNcn 10539   2c2 10588   3c3 10589   5c5 10591   6c6 10592   7c7 10593   9c9 10595   NN0cn0 10798   ZZ>=cuz 11087   ...cfz 11678   ^cexp 12142   sum_csu 13484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-int 4269  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-se 4826  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-isom 5584  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-recs 7041  df-rdg 7075  df-1o 7129  df-oadd 7133  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518  df-fin 7519  df-sup 7900  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-div 10210  df-nn 10540  df-2 10597  df-3 10598  df-4 10599  df-5 10600  df-6 10601  df-7 10602  df-8 10603  df-9 10604  df-n0 10799  df-z 10868  df-uz 11088  df-rp 11227  df-fz 11679  df-fzo 11801  df-seq 12084  df-exp 12143  df-hash 12382  df-cj 12908  df-re 12909  df-im 12910  df-sqrt 13044  df-abs 13045  df-clim 13287  df-sum 13485
This theorem is referenced by:  log2ublem3  23148
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