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Theorem log2ublem2 22317
Description: Lemma for log2ub 22319. (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 11787 . . . 4  |-  ( T. 
->  ( 0 ... K
)  e.  Fin )
3 elfznn0 11473 . . . . . 6  |-  ( n  e.  ( 0 ... K )  ->  n  e.  NN0 )
43adantl 466 . . . . 5  |-  ( ( T.  /\  n  e.  ( 0 ... K
) )  ->  n  e.  NN0 )
5 2re 10383 . . . . . 6  |-  2  e.  RR
6 3nn 10472 . . . . . . . 8  |-  3  e.  NN
7 2nn0 10588 . . . . . . . . . 10  |-  2  e.  NN0
8 nn0mulcl 10608 . . . . . . . . . 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 10611 . . . . . . . . 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 10337 . . . . . . . 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 10478 . . . . . . . 8  |-  9  e.  NN
15 nnexpcl 11870 . . . . . . . 8  |-  ( ( 9  e.  NN  /\  n  e.  NN0 )  -> 
( 9 ^ n
)  e.  NN )
1614, 15mpan 670 . . . . . . 7  |-  ( n  e.  NN0  ->  ( 9 ^ n )  e.  NN )
1713, 16nnmulcld 10361 . . . . . 6  |-  ( n  e.  NN0  ->  ( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  ( 9 ^ n ) )  e.  NN )
18 nndivre 10349 . . . . . 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 13203 . . 3  |-  ( T. 
->  sum_ n  e.  ( 0 ... K ) ( 2  /  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) ) )  e.  RR )
2221trud 1378 . 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 10610 . . . . 5  |-  ( 2  x.  N )  e. 
NN0
25 nn0p1nn 10611 . . . . 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 10338 . . 3  |-  ( 3  x.  ( ( 2  x.  N )  +  1 ) )  e.  NN
28 nnexpcl 11870 . . . 4  |-  ( ( 9  e.  NN  /\  N  e.  NN0 )  -> 
( 9 ^ N
)  e.  NN )
2914, 23, 28mp2an 672 . . 3  |-  ( 9 ^ N )  e.  NN
3027, 29nnmulcli 10338 . 2  |-  ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N ) )  e.  NN
31 log2ublem2.2 . . 3  |-  B  e. 
NN0
327, 31nn0mulcli 10610 . 2  |-  ( 2  x.  B )  e. 
NN0
33 log2ublem2.3 . . 3  |-  F  e. 
NN0
347, 33nn0mulcli 10610 . 2  |-  ( 2  x.  F )  e. 
NN0
35 nn0uz 10887 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
3623, 35eleqtri 2510 . . . . . 6  |-  N  e.  ( ZZ>= `  0 )
3736a1i 11 . . . . 5  |-  ( T. 
->  N  e.  ( ZZ>=
`  0 ) )
38 elfznn0 11473 . . . . . . 7  |-  ( n  e.  ( 0 ... N )  ->  n  e.  NN0 )
3938adantl 466 . . . . . 6  |-  ( ( T.  /\  n  e.  ( 0 ... N
) )  ->  n  e.  NN0 )
4019recnd 9404 . . . . . 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 6094 . . . . . . . . 9  |-  ( n  =  N  ->  (
2  x.  n )  =  ( 2  x.  N ) )
4342oveq1d 6101 . . . . . . . 8  |-  ( n  =  N  ->  (
( 2  x.  n
)  +  1 )  =  ( ( 2  x.  N )  +  1 ) )
4443oveq2d 6102 . . . . . . 7  |-  ( n  =  N  ->  (
3  x.  ( ( 2  x.  n )  +  1 ) )  =  ( 3  x.  ( ( 2  x.  N )  +  1 ) ) )
45 oveq2 6094 . . . . . . 7  |-  ( n  =  N  ->  (
9 ^ n )  =  ( 9 ^ N ) )
4644, 45oveq12d 6104 . . . . . 6  |-  ( n  =  N  ->  (
( 3  x.  (
( 2  x.  n
)  +  1 ) )  x.  ( 9 ^ n ) )  =  ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N
) ) )
4746oveq2d 6102 . . . . 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 13212 . . . 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 1378 . . 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 6097 . . . . 5  |-  ( 0 ... ( N  - 
1 ) )  =  ( 0 ... K
)
5251sumeq1i 13167 . . . 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 6096 . . 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 10384 . . . 4  |-  2  e.  CC
5631nn0cni 10583 . . . 4  |-  B  e.  CC
5733nn0cni 10583 . . . 4  |-  F  e.  CC
5855, 56, 57adddii 9388 . . 3  |-  ( 2  x.  ( B  +  F ) )  =  ( ( 2  x.  B )  +  ( 2  x.  F ) )
59 log2ublem2.6 . . . 4  |-  ( B  +  F )  =  G
6059oveq2i 6097 . . 3  |-  ( 2  x.  ( B  +  F ) )  =  ( 2  x.  G
)
6158, 60eqtr3i 2460 . 2  |-  ( ( 2  x.  B )  +  ( 2  x.  F ) )  =  ( 2  x.  G
)
62 7nn 10476 . . . . . . . . 9  |-  7  e.  NN
6362nnnn0i 10579 . . . . . . . 8  |-  7  e.  NN0
64 nnexpcl 11870 . . . . . . . 8  |-  ( ( 3  e.  NN  /\  7  e.  NN0 )  -> 
( 3 ^ 7 )  e.  NN )
656, 63, 64mp2an 672 . . . . . . 7  |-  ( 3 ^ 7 )  e.  NN
66 5nn 10474 . . . . . . . 8  |-  5  e.  NN
6766, 62nnmulcli 10338 . . . . . . 7  |-  ( 5  x.  7 )  e.  NN
6865, 67nnmulcli 10338 . . . . . 6  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  NN
6968nnrei 10323 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  RR
7069, 5remulcli 9392 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )  e.  RR
7170leidi 9866 . . 3  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )  <_ 
( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )
726nnnn0i 10579 . . . . . . . . . . . 12  |-  3  e.  NN0
73 nnexpcl 11870 . . . . . . . . . . . 12  |-  ( ( 9  e.  NN  /\  3  e.  NN0 )  -> 
( 9 ^ 3 )  e.  NN )
7414, 72, 73mp2an 672 . . . . . . . . . . 11  |-  ( 9 ^ 3 )  e.  NN
7574nncni 10324 . . . . . . . . . 10  |-  ( 9 ^ 3 )  e.  CC
7667nncni 10324 . . . . . . . . . 10  |-  ( 5  x.  7 )  e.  CC
7775, 76mulcomi 9384 . . . . . . . . 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 10583 . . . . . . . . . . . . . 14  |-  M  e.  CC
8123nn0cni 10583 . . . . . . . . . . . . . 14  |-  N  e.  CC
8280, 81addcomi 9552 . . . . . . . . . . . . 13  |-  ( M  +  N )  =  ( N  +  M
)
8378, 82eqtr3i 2460 . . . . . . . . . . . 12  |-  3  =  ( N  +  M )
8483oveq2i 6097 . . . . . . . . . . 11  |-  ( 9 ^ 3 )  =  ( 9 ^ ( N  +  M )
)
8514nncni 10324 . . . . . . . . . . . 12  |-  9  e.  CC
86 expadd 11898 . . . . . . . . . . . 12  |-  ( ( 9  e.  CC  /\  N  e.  NN0  /\  M  e.  NN0 )  ->  (
9 ^ ( N  +  M ) )  =  ( ( 9 ^ N )  x.  ( 9 ^ M
) ) )
8785, 23, 79, 86mp3an 1314 . . . . . . . . . . 11  |-  ( 9 ^ ( N  +  M ) )  =  ( ( 9 ^ N )  x.  (
9 ^ M ) )
8884, 87eqtri 2458 . . . . . . . . . 10  |-  ( 9 ^ 3 )  =  ( ( 9 ^ N )  x.  (
9 ^ M ) )
8988oveq2i 6097 . . . . . . . . 9  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 3 ) )  =  ( ( 5  x.  7 )  x.  (
( 9 ^ N
)  x.  ( 9 ^ M ) ) )
9029nncni 10324 . . . . . . . . . 10  |-  ( 9 ^ N )  e.  CC
91 nnexpcl 11870 . . . . . . . . . . . 12  |-  ( ( 9  e.  NN  /\  M  e.  NN0 )  -> 
( 9 ^ M
)  e.  NN )
9214, 79, 91mp2an 672 . . . . . . . . . . 11  |-  ( 9 ^ M )  e.  NN
9392nncni 10324 . . . . . . . . . 10  |-  ( 9 ^ M )  e.  CC
9476, 90, 93mul12i 9556 . . . . . . . . 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 6097 . . . . . . . 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 6097 . . . . . 6  |-  ( 3  x.  ( ( 9 ^ 3 )  x.  ( 5  x.  7 ) ) )  =  ( 3  x.  (
( 9 ^ N
)  x.  ( ( ( 2  x.  N
)  +  1 )  x.  F ) ) )
100 df-7 10377 . . . . . . . . . 10  |-  7  =  ( 6  +  1 )
101100oveq2i 6097 . . . . . . . . 9  |-  ( 3 ^ 7 )  =  ( 3 ^ (
6  +  1 ) )
102 3cn 10388 . . . . . . . . . . 11  |-  3  e.  CC
103 6nn0 10592 . . . . . . . . . . 11  |-  6  e.  NN0
104 expp1 11864 . . . . . . . . . . 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 11901 . . . . . . . . . . . . 13  |-  ( ( 3  e.  CC  /\  2  e.  NN0  /\  3  e.  NN0 )  ->  (
3 ^ ( 2  x.  3 ) )  =  ( ( 3 ^ 2 ) ^
3 ) )
107102, 7, 72, 106mp3an 1314 . . . . . . . . . . . 12  |-  ( 3 ^ ( 2  x.  3 ) )  =  ( ( 3 ^ 2 ) ^ 3 )
10855, 102mulcomi 9384 . . . . . . . . . . . . . 14  |-  ( 2  x.  3 )  =  ( 3  x.  2 )
109 3t2e6 10465 . . . . . . . . . . . . . 14  |-  ( 3  x.  2 )  =  6
110108, 109eqtri 2458 . . . . . . . . . . . . 13  |-  ( 2  x.  3 )  =  6
111110oveq2i 6097 . . . . . . . . . . . 12  |-  ( 3 ^ ( 2  x.  3 ) )  =  ( 3 ^ 6 )
112 sq3 11955 . . . . . . . . . . . . 13  |-  ( 3 ^ 2 )  =  9
113112oveq1i 6096 . . . . . . . . . . . 12  |-  ( ( 3 ^ 2 ) ^ 3 )  =  ( 9 ^ 3 )
114107, 111, 1133eqtr3i 2466 . . . . . . . . . . 11  |-  ( 3 ^ 6 )  =  ( 9 ^ 3 )
115114oveq1i 6096 . . . . . . . . . 10  |-  ( ( 3 ^ 6 )  x.  3 )  =  ( ( 9 ^ 3 )  x.  3 )
116105, 115eqtri 2458 . . . . . . . . 9  |-  ( 3 ^ ( 6  +  1 ) )  =  ( ( 9 ^ 3 )  x.  3 )
11775, 102mulcomi 9384 . . . . . . . . 9  |-  ( ( 9 ^ 3 )  x.  3 )  =  ( 3  x.  (
9 ^ 3 ) )
118101, 116, 1173eqtri 2462 . . . . . . . 8  |-  ( 3 ^ 7 )  =  ( 3  x.  (
9 ^ 3 ) )
119118oveq1i 6096 . . . . . . 7  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  =  ( ( 3  x.  ( 9 ^ 3 ) )  x.  (
5  x.  7 ) )
120102, 75, 76mulassi 9387 . . . . . . 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 10324 . . . . . . . . 9  |-  ( ( 2  x.  N )  +  1 )  e.  CC
123102, 122, 90mul32i 9557 . . . . . . . 8  |-  ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N ) )  =  ( ( 3  x.  ( 9 ^ N
) )  x.  (
( 2  x.  N
)  +  1 ) )
124123oveq1i 6096 . . . . . . 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 9383 . . . . . . . 8  |-  ( 3  x.  ( 9 ^ N ) )  e.  CC
126125, 122, 57mulassi 9387 . . . . . . 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 9383 . . . . . . . 8  |-  ( ( ( 2  x.  N
)  +  1 )  x.  F )  e.  CC
128102, 90, 127mulassi 9387 . . . . . . 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 6097 . . . 4  |-  ( 2  x.  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )  =  ( 2  x.  (
( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  (
9 ^ N ) )  x.  F ) )
13265nncni 10324 . . . . . 6  |-  ( 3 ^ 7 )  e.  CC
133132, 76mulcli 9383 . . . . 5  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  CC
134133, 55mulcomi 9384 . . . 4  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  2 )  =  ( 2  x.  (
( 3 ^ 7 )  x.  ( 5  x.  7 ) ) )
13530nncni 10324 . . . . 5  |-  ( ( 3  x.  ( ( 2  x.  N )  +  1 ) )  x.  ( 9 ^ N ) )  e.  CC
136135, 55, 57mul12i 9556 . . . 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 4310 . 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 22316 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 1369   T. wtru 1370    e. wcel 1756   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    <_ cle 9411    - cmin 9587    / cdiv 9985   NNcn 10314   2c2 10363   3c3 10364   5c5 10366   6c6 10367   7c7 10368   9c9 10370   NN0cn0 10571   ZZ>=cuz 10853   ...cfz 11429   ^cexp 11857   sum_csu 13155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156
This theorem is referenced by:  log2ublem3  22318
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