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Theorem emcllem7 20295
Description: Lemma for emcl 20296 and harmonicbnd 20297. Derive bounds on  gamma as  F ( 1 ) and  G ( 1 ). (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 9-Apr-2016.)
Hypotheses
Ref Expression
emcl.1  |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m
)  -  ( log `  n ) ) )
emcl.2  |-  G  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m
)  -  ( log `  ( n  +  1 ) ) ) )
emcl.3  |-  H  =  ( n  e.  NN  |->  ( log `  ( 1  +  ( 1  /  n ) ) ) )
emcl.4  |-  T  =  ( n  e.  NN  |->  ( ( 1  /  n )  -  ( log `  ( 1  +  ( 1  /  n
) ) ) ) )
Assertion
Ref Expression
emcllem7  |-  ( gamma  e.  ( ( 1  -  ( log `  2
) ) [,] 1
)  /\  F : NN
--> ( gamma [,] 1 )  /\  G : NN --> ( ( 1  -  ( log `  2 ) ) [,] gamma ) )
Distinct variable groups:    m, H    m, n, T
Allowed substitution hints:    F( m, n)    G( m, n)    H( n)

Proof of Theorem emcllem7
Dummy variables  i 
k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnuz 10263 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
2 1z 10053 . . . . . 6  |-  1  e.  ZZ
32a1i 10 . . . . 5  |-  (  T. 
->  1  e.  ZZ )
4 emcl.1 . . . . . . . 8  |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m
)  -  ( log `  n ) ) )
5 emcl.2 . . . . . . . 8  |-  G  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m
)  -  ( log `  ( n  +  1 ) ) ) )
6 emcl.3 . . . . . . . 8  |-  H  =  ( n  e.  NN  |->  ( log `  ( 1  +  ( 1  /  n ) ) ) )
7 emcl.4 . . . . . . . 8  |-  T  =  ( n  e.  NN  |->  ( ( 1  /  n )  -  ( log `  ( 1  +  ( 1  /  n
) ) ) ) )
84, 5, 6, 7emcllem6 20294 . . . . . . 7  |-  ( F  ~~> 
gamma  /\  G  ~~>  gamma )
98simpri 448 . . . . . 6  |-  G  ~~>  gamma
109a1i 10 . . . . 5  |-  (  T. 
->  G  ~~>  gamma )
114, 5emcllem1 20289 . . . . . . . 8  |-  ( F : NN --> RR  /\  G : NN --> RR )
1211simpri 448 . . . . . . 7  |-  G : NN
--> RR
1312ffvelrni 5664 . . . . . 6  |-  ( k  e.  NN  ->  ( G `  k )  e.  RR )
1413adantl 452 . . . . 5  |-  ( (  T.  /\  k  e.  NN )  ->  ( G `  k )  e.  RR )
151, 3, 10, 14climrecl 12057 . . . 4  |-  (  T. 
->  gamma  e.  RR )
16 1nn 9757 . . . . 5  |-  1  e.  NN
17 simpr 447 . . . . . . 7  |-  ( (  T.  /\  i  e.  NN )  ->  i  e.  NN )
189a1i 10 . . . . . . 7  |-  ( (  T.  /\  i  e.  NN )  ->  G  ~~>  gamma )
1913adantl 452 . . . . . . 7  |-  ( ( (  T.  /\  i  e.  NN )  /\  k  e.  NN )  ->  ( G `  k )  e.  RR )
204, 5emcllem2 20290 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
( F `  (
k  +  1 ) )  <_  ( F `  k )  /\  ( G `  k )  <_  ( G `  (
k  +  1 ) ) ) )
2120simprd 449 . . . . . . . 8  |-  ( k  e.  NN  ->  ( G `  k )  <_  ( G `  (
k  +  1 ) ) )
2221adantl 452 . . . . . . 7  |-  ( ( (  T.  /\  i  e.  NN )  /\  k  e.  NN )  ->  ( G `  k )  <_  ( G `  (
k  +  1 ) ) )
231, 17, 18, 19, 22climub 12135 . . . . . 6  |-  ( (  T.  /\  i  e.  NN )  ->  ( G `  i )  <_ 
gamma )
2423ralrimiva 2626 . . . . 5  |-  (  T. 
->  A. i  e.  NN  ( G `  i )  <_  gamma )
25 fveq2 5525 . . . . . . . 8  |-  ( i  =  1  ->  ( G `  i )  =  ( G ` 
1 ) )
26 oveq2 5866 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  (
1 ... n )  =  ( 1 ... 1
) )
2726sumeq1d 12174 . . . . . . . . . . . 12  |-  ( n  =  1  ->  sum_ m  e.  ( 1 ... n
) ( 1  /  m )  =  sum_ m  e.  ( 1 ... 1 ) ( 1  /  m ) )
28 ax-1cn 8795 . . . . . . . . . . . . 13  |-  1  e.  CC
29 oveq2 5866 . . . . . . . . . . . . . . 15  |-  ( m  =  1  ->  (
1  /  m )  =  ( 1  / 
1 ) )
3028div1i 9488 . . . . . . . . . . . . . . 15  |-  ( 1  /  1 )  =  1
3129, 30syl6eq 2331 . . . . . . . . . . . . . 14  |-  ( m  =  1  ->  (
1  /  m )  =  1 )
3231fsum1 12214 . . . . . . . . . . . . 13  |-  ( ( 1  e.  ZZ  /\  1  e.  CC )  -> 
sum_ m  e.  (
1 ... 1 ) ( 1  /  m )  =  1 )
332, 28, 32mp2an 653 . . . . . . . . . . . 12  |-  sum_ m  e.  ( 1 ... 1
) ( 1  /  m )  =  1
3427, 33syl6eq 2331 . . . . . . . . . . 11  |-  ( n  =  1  ->  sum_ m  e.  ( 1 ... n
) ( 1  /  m )  =  1 )
35 oveq1 5865 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  (
n  +  1 )  =  ( 1  +  1 ) )
36 df-2 9804 . . . . . . . . . . . . 13  |-  2  =  ( 1  +  1 )
3735, 36syl6eqr 2333 . . . . . . . . . . . 12  |-  ( n  =  1  ->  (
n  +  1 )  =  2 )
3837fveq2d 5529 . . . . . . . . . . 11  |-  ( n  =  1  ->  ( log `  ( n  + 
1 ) )  =  ( log `  2
) )
3934, 38oveq12d 5876 . . . . . . . . . 10  |-  ( n  =  1  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  (
n  +  1 ) ) )  =  ( 1  -  ( log `  2 ) ) )
40 1re 8837 . . . . . . . . . . . 12  |-  1  e.  RR
41 2rp 10359 . . . . . . . . . . . . 13  |-  2  e.  RR+
42 relogcl 19932 . . . . . . . . . . . . 13  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
4341, 42ax-mp 8 . . . . . . . . . . . 12  |-  ( log `  2 )  e.  RR
4440, 43resubcli 9109 . . . . . . . . . . 11  |-  ( 1  -  ( log `  2
) )  e.  RR
4544elexi 2797 . . . . . . . . . 10  |-  ( 1  -  ( log `  2
) )  e.  _V
4639, 5, 45fvmpt 5602 . . . . . . . . 9  |-  ( 1  e.  NN  ->  ( G `  1 )  =  ( 1  -  ( log `  2
) ) )
4716, 46ax-mp 8 . . . . . . . 8  |-  ( G `
 1 )  =  ( 1  -  ( log `  2 ) )
4825, 47syl6eq 2331 . . . . . . 7  |-  ( i  =  1  ->  ( G `  i )  =  ( 1  -  ( log `  2
) ) )
4948breq1d 4033 . . . . . 6  |-  ( i  =  1  ->  (
( G `  i
)  <_  gamma  <->  ( 1  -  ( log `  2
) )  <_  gamma )
)
5049rspcva 2882 . . . . 5  |-  ( ( 1  e.  NN  /\  A. i  e.  NN  ( G `  i )  <_ 
gamma )  ->  ( 1  -  ( log `  2
) )  <_  gamma )
5116, 24, 50sylancr 644 . . . 4  |-  (  T. 
->  ( 1  -  ( log `  2 ) )  <_  gamma )
52 fveq2 5525 . . . . . . . . . . . 12  |-  ( x  =  i  ->  ( F `  x )  =  ( F `  i ) )
5352negeqd 9046 . . . . . . . . . . 11  |-  ( x  =  i  ->  -u ( F `  x )  =  -u ( F `  i ) )
54 eqid 2283 . . . . . . . . . . 11  |-  ( x  e.  NN  |->  -u ( F `  x )
)  =  ( x  e.  NN  |->  -u ( F `  x )
)
55 negex 9050 . . . . . . . . . . 11  |-  -u ( F `  i )  e.  _V
5653, 54, 55fvmpt 5602 . . . . . . . . . 10  |-  ( i  e.  NN  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  i
)  =  -u ( F `  i )
)
5756adantl 452 . . . . . . . . 9  |-  ( (  T.  /\  i  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  i
)  =  -u ( F `  i )
)
588simpli 444 . . . . . . . . . . . . 13  |-  F  ~~>  gamma
5958a1i 10 . . . . . . . . . . . 12  |-  (  T. 
->  F  ~~>  gamma )
60 0cn 8831 . . . . . . . . . . . . 13  |-  0  e.  CC
6160a1i 10 . . . . . . . . . . . 12  |-  (  T. 
->  0  e.  CC )
62 nnex 9752 . . . . . . . . . . . . . 14  |-  NN  e.  _V
6362mptex 5746 . . . . . . . . . . . . 13  |-  ( x  e.  NN  |->  -u ( F `  x )
)  e.  _V
6463a1i 10 . . . . . . . . . . . 12  |-  (  T. 
->  ( x  e.  NN  |->  -u ( F `  x
) )  e.  _V )
6511simpli 444 . . . . . . . . . . . . . . 15  |-  F : NN
--> RR
6665ffvelrni 5664 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  ( F `  k )  e.  RR )
6766adantl 452 . . . . . . . . . . . . 13  |-  ( (  T.  /\  k  e.  NN )  ->  ( F `  k )  e.  RR )
6867recnd 8861 . . . . . . . . . . . 12  |-  ( (  T.  /\  k  e.  NN )  ->  ( F `  k )  e.  CC )
69 fveq2 5525 . . . . . . . . . . . . . . . 16  |-  ( x  =  k  ->  ( F `  x )  =  ( F `  k ) )
7069negeqd 9046 . . . . . . . . . . . . . . 15  |-  ( x  =  k  ->  -u ( F `  x )  =  -u ( F `  k ) )
71 negex 9050 . . . . . . . . . . . . . . 15  |-  -u ( F `  k )  e.  _V
7270, 54, 71fvmpt 5602 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  k
)  =  -u ( F `  k )
)
7372adantl 452 . . . . . . . . . . . . 13  |-  ( (  T.  /\  k  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  k
)  =  -u ( F `  k )
)
74 df-neg 9040 . . . . . . . . . . . . 13  |-  -u ( F `  k )  =  ( 0  -  ( F `  k
) )
7573, 74syl6eq 2331 . . . . . . . . . . . 12  |-  ( (  T.  /\  k  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  k
)  =  ( 0  -  ( F `  k ) ) )
761, 3, 59, 61, 64, 68, 75climsubc2 12112 . . . . . . . . . . 11  |-  (  T. 
->  ( x  e.  NN  |->  -u ( F `  x
) )  ~~>  ( 0  -  gamma ) )
7776adantr 451 . . . . . . . . . 10  |-  ( (  T.  /\  i  e.  NN )  ->  (
x  e.  NN  |->  -u ( F `  x ) )  ~~>  ( 0  - 
gamma ) )
7867renegcld 9210 . . . . . . . . . . . 12  |-  ( (  T.  /\  k  e.  NN )  ->  -u ( F `  k )  e.  RR )
7973, 78eqeltrd 2357 . . . . . . . . . . 11  |-  ( (  T.  /\  k  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  k
)  e.  RR )
8079adantlr 695 . . . . . . . . . 10  |-  ( ( (  T.  /\  i  e.  NN )  /\  k  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  k
)  e.  RR )
8120simpld 445 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
8281adantl 452 . . . . . . . . . . . . 13  |-  ( (  T.  /\  k  e.  NN )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
83 peano2nn 9758 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
8483adantl 452 . . . . . . . . . . . . . . 15  |-  ( (  T.  /\  k  e.  NN )  ->  (
k  +  1 )  e.  NN )
8565ffvelrni 5664 . . . . . . . . . . . . . . 15  |-  ( ( k  +  1 )  e.  NN  ->  ( F `  ( k  +  1 ) )  e.  RR )
8684, 85syl 15 . . . . . . . . . . . . . 14  |-  ( (  T.  /\  k  e.  NN )  ->  ( F `  ( k  +  1 ) )  e.  RR )
8786, 67lenegd 9351 . . . . . . . . . . . . 13  |-  ( (  T.  /\  k  e.  NN )  ->  (
( F `  (
k  +  1 ) )  <_  ( F `  k )  <->  -u ( F `
 k )  <_  -u ( F `  (
k  +  1 ) ) ) )
8882, 87mpbid 201 . . . . . . . . . . . 12  |-  ( (  T.  /\  k  e.  NN )  ->  -u ( F `  k )  <_ 
-u ( F `  ( k  +  1 ) ) )
89 fveq2 5525 . . . . . . . . . . . . . . 15  |-  ( x  =  ( k  +  1 )  ->  ( F `  x )  =  ( F `  ( k  +  1 ) ) )
9089negeqd 9046 . . . . . . . . . . . . . 14  |-  ( x  =  ( k  +  1 )  ->  -u ( F `  x )  =  -u ( F `  ( k  +  1 ) ) )
91 negex 9050 . . . . . . . . . . . . . 14  |-  -u ( F `  ( k  +  1 ) )  e.  _V
9290, 54, 91fvmpt 5602 . . . . . . . . . . . . 13  |-  ( ( k  +  1 )  e.  NN  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  (
k  +  1 ) )  =  -u ( F `  ( k  +  1 ) ) )
9384, 92syl 15 . . . . . . . . . . . 12  |-  ( (  T.  /\  k  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  (
k  +  1 ) )  =  -u ( F `  ( k  +  1 ) ) )
9488, 73, 933brtr4d 4053 . . . . . . . . . . 11  |-  ( (  T.  /\  k  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  k
)  <_  ( (
x  e.  NN  |->  -u ( F `  x ) ) `  ( k  +  1 ) ) )
9594adantlr 695 . . . . . . . . . 10  |-  ( ( (  T.  /\  i  e.  NN )  /\  k  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  k
)  <_  ( (
x  e.  NN  |->  -u ( F `  x ) ) `  ( k  +  1 ) ) )
961, 17, 77, 80, 95climub 12135 . . . . . . . . 9  |-  ( (  T.  /\  i  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  i
)  <_  ( 0  -  gamma ) )
9757, 96eqbrtrrd 4045 . . . . . . . 8  |-  ( (  T.  /\  i  e.  NN )  ->  -u ( F `  i )  <_  ( 0  -  gamma ) )
98 df-neg 9040 . . . . . . . 8  |-  -u gamma  =  ( 0  -  gamma )
9997, 98syl6breqr 4063 . . . . . . 7  |-  ( (  T.  /\  i  e.  NN )  ->  -u ( F `  i )  <_ 
-u gamma )
10015trud 1314 . . . . . . . 8  |-  gamma  e.  RR
10165ffvelrni 5664 . . . . . . . . 9  |-  ( i  e.  NN  ->  ( F `  i )  e.  RR )
102101adantl 452 . . . . . . . 8  |-  ( (  T.  /\  i  e.  NN )  ->  ( F `  i )  e.  RR )
103 leneg 9277 . . . . . . . 8  |-  ( (
gamma  e.  RR  /\  ( F `  i )  e.  RR )  ->  ( gamma  <_  ( F `  i )  <->  -u ( F `
 i )  <_  -u
gamma ) )
104100, 102, 103sylancr 644 . . . . . . 7  |-  ( (  T.  /\  i  e.  NN )  ->  ( gamma  <_  ( F `  i )  <->  -u ( F `
 i )  <_  -u
gamma ) )
10599, 104mpbird 223 . . . . . 6  |-  ( (  T.  /\  i  e.  NN )  ->  gamma  <_  ( F `  i )
)
106105ralrimiva 2626 . . . . 5  |-  (  T. 
->  A. i  e.  NN  gamma  <_  ( F `  i
) )
107 fveq2 5525 . . . . . . . 8  |-  ( i  =  1  ->  ( F `  i )  =  ( F ` 
1 ) )
108 fveq2 5525 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  ( log `  n )  =  ( log `  1
) )
109 log1 19939 . . . . . . . . . . . . 13  |-  ( log `  1 )  =  0
110108, 109syl6eq 2331 . . . . . . . . . . . 12  |-  ( n  =  1  ->  ( log `  n )  =  0 )
11134, 110oveq12d 5876 . . . . . . . . . . 11  |-  ( n  =  1  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  n
) )  =  ( 1  -  0 ) )
11228subid1i 9118 . . . . . . . . . . 11  |-  ( 1  -  0 )  =  1
113111, 112syl6eq 2331 . . . . . . . . . 10  |-  ( n  =  1  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  n
) )  =  1 )
11440elexi 2797 . . . . . . . . . 10  |-  1  e.  _V
115113, 4, 114fvmpt 5602 . . . . . . . . 9  |-  ( 1  e.  NN  ->  ( F `  1 )  =  1 )
11616, 115ax-mp 8 . . . . . . . 8  |-  ( F `
 1 )  =  1
117107, 116syl6eq 2331 . . . . . . 7  |-  ( i  =  1  ->  ( F `  i )  =  1 )
118117breq2d 4035 . . . . . 6  |-  ( i  =  1  ->  ( gamma  <_  ( F `  i )  <->  gamma  <_  1
) )
119118rspcva 2882 . . . . 5  |-  ( ( 1  e.  NN  /\  A. i  e.  NN  gamma  <_ 
( F `  i
) )  ->  gamma  <_  1
)
12016, 106, 119sylancr 644 . . . 4  |-  (  T. 
->  gamma  <_  1 )
12144, 40elicc2i 10716 . . . 4  |-  ( gamma  e.  ( ( 1  -  ( log `  2
) ) [,] 1
)  <->  ( gamma  e.  RR  /\  ( 1  -  ( log `  2 ) )  <_  gamma  /\  gamma  <_  1
) )
12215, 51, 120, 121syl3anbrc 1136 . . 3  |-  (  T. 
->  gamma  e.  ( ( 1  -  ( log `  2 ) ) [,] 1 ) )
123 ffn 5389 . . . . 5  |-  ( F : NN --> RR  ->  F  Fn  NN )
12465, 123mp1i 11 . . . 4  |-  (  T. 
->  F  Fn  NN )
12517, 1syl6eleq 2373 . . . . . . . 8  |-  ( (  T.  /\  i  e.  NN )  ->  i  e.  ( ZZ>= `  1 )
)
126 elfznn 10819 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... i )  ->  k  e.  NN )
127126adantl 452 . . . . . . . . 9  |-  ( ( (  T.  /\  i  e.  NN )  /\  k  e.  ( 1 ... i
) )  ->  k  e.  NN )
128127, 66syl 15 . . . . . . . 8  |-  ( ( (  T.  /\  i  e.  NN )  /\  k  e.  ( 1 ... i
) )  ->  ( F `  k )  e.  RR )
129 elfznn 10819 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( i  -  1 ) )  ->  k  e.  NN )
130129adantl 452 . . . . . . . . 9  |-  ( ( (  T.  /\  i  e.  NN )  /\  k  e.  ( 1 ... (
i  -  1 ) ) )  ->  k  e.  NN )
131130, 81syl 15 . . . . . . . 8  |-  ( ( (  T.  /\  i  e.  NN )  /\  k  e.  ( 1 ... (
i  -  1 ) ) )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
132125, 128, 131monoord2 11077 . . . . . . 7  |-  ( (  T.  /\  i  e.  NN )  ->  ( F `  i )  <_  ( F `  1
) )
133132, 116syl6breq 4062 . . . . . 6  |-  ( (  T.  /\  i  e.  NN )  ->  ( F `  i )  <_  1 )
134100, 40elicc2i 10716 . . . . . 6  |-  ( ( F `  i )  e.  ( gamma [,] 1
)  <->  ( ( F `
 i )  e.  RR  /\  gamma  <_  ( F `  i )  /\  ( F `  i
)  <_  1 ) )
135102, 105, 133, 134syl3anbrc 1136 . . . . 5  |-  ( (  T.  /\  i  e.  NN )  ->  ( F `  i )  e.  ( gamma [,] 1 ) )
136135ralrimiva 2626 . . . 4  |-  (  T. 
->  A. i  e.  NN  ( F `  i )  e.  ( gamma [,] 1
) )
137 ffnfv 5685 . . . 4  |-  ( F : NN --> ( gamma [,] 1 )  <->  ( F  Fn  NN  /\  A. i  e.  NN  ( F `  i )  e.  (
gamma [,] 1 ) ) )
138124, 136, 137sylanbrc 645 . . 3  |-  (  T. 
->  F : NN --> ( gamma [,] 1 ) )
139 ffn 5389 . . . . 5  |-  ( G : NN --> RR  ->  G  Fn  NN )
14012, 139mp1i 11 . . . 4  |-  (  T. 
->  G  Fn  NN )
14112ffvelrni 5664 . . . . . . 7  |-  ( i  e.  NN  ->  ( G `  i )  e.  RR )
142141adantl 452 . . . . . 6  |-  ( (  T.  /\  i  e.  NN )  ->  ( G `  i )  e.  RR )
143127, 13syl 15 . . . . . . . 8  |-  ( ( (  T.  /\  i  e.  NN )  /\  k  e.  ( 1 ... i
) )  ->  ( G `  k )  e.  RR )
144130, 21syl 15 . . . . . . . 8  |-  ( ( (  T.  /\  i  e.  NN )  /\  k  e.  ( 1 ... (
i  -  1 ) ) )  ->  ( G `  k )  <_  ( G `  (
k  +  1 ) ) )
145125, 143, 144monoord 11076 . . . . . . 7  |-  ( (  T.  /\  i  e.  NN )  ->  ( G `  1 )  <_  ( G `  i
) )
14647, 145syl5eqbrr 4057 . . . . . 6  |-  ( (  T.  /\  i  e.  NN )  ->  (
1  -  ( log `  2 ) )  <_  ( G `  i ) )
14744, 100elicc2i 10716 . . . . . 6  |-  ( ( G `  i )  e.  ( ( 1  -  ( log `  2
) ) [,] gamma )  <-> 
( ( G `  i )  e.  RR  /\  ( 1  -  ( log `  2 ) )  <_  ( G `  i )  /\  ( G `  i )  <_ 
gamma ) )
148142, 146, 23, 147syl3anbrc 1136 . . . . 5  |-  ( (  T.  /\  i  e.  NN )  ->  ( G `  i )  e.  ( ( 1  -  ( log `  2
) ) [,] gamma ) )
149148ralrimiva 2626 . . . 4  |-  (  T. 
->  A. i  e.  NN  ( G `  i )  e.  ( ( 1  -  ( log `  2
) ) [,] gamma ) )
150 ffnfv 5685 . . . 4  |-  ( G : NN --> ( ( 1  -  ( log `  2 ) ) [,] gamma )  <->  ( G  Fn  NN  /\  A. i  e.  NN  ( G `  i )  e.  ( ( 1  -  ( log `  2 ) ) [,] gamma ) ) )
151140, 149, 150sylanbrc 645 . . 3  |-  (  T. 
->  G : NN --> ( ( 1  -  ( log `  2 ) ) [,] gamma ) )
152122, 138, 1513jca 1132 . 2  |-  (  T. 
->  ( gamma  e.  (
( 1  -  ( log `  2 ) ) [,] 1 )  /\  F : NN --> ( gamma [,] 1 )  /\  G : NN --> ( ( 1  -  ( log `  2
) ) [,] gamma ) ) )
153152trud 1314 1  |-  ( gamma  e.  ( ( 1  -  ( log `  2
) ) [,] 1
)  /\  F : NN
--> ( gamma [,] 1 )  /\  G : NN --> ( ( 1  -  ( log `  2 ) ) [,] gamma ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    T. wtru 1307    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    <_ cle 8868    - cmin 9037   -ucneg 9038    / cdiv 9423   NNcn 9746   2c2 9795   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   [,]cicc 10659   ...cfz 10782    ~~> cli 11958   sum_csu 12158   logclog 19912   gammacem 20286
This theorem is referenced by:  emcl  20296  harmonicbnd  20297  harmonicbnd2  20298
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-pi 12354  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-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-log 19914  df-em 20287
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