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Theorem emcllem7 20311
Description: Lemma for emcl 20312 and harmonicbnd 20313. 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 10279 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
2 1z 10069 . . . . . 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 20310 . . . . . . 7  |-  ( F  ~~> 
gamma  /\  G  ~~>  gamma )
98simpri 448 . . . . . 6  |-  G  ~~>  gamma
109a1i 10 . . . . 5  |-  (  T. 
->  G  ~~>  gamma )
114, 5emcllem1 20305 . . . . . . . 8  |-  ( F : NN --> RR  /\  G : NN --> RR )
1211simpri 448 . . . . . . 7  |-  G : NN
--> RR
1312ffvelrni 5680 . . . . . 6  |-  ( k  e.  NN  ->  ( G `  k )  e.  RR )
1413adantl 452 . . . . 5  |-  ( (  T.  /\  k  e.  NN )  ->  ( G `  k )  e.  RR )
151, 3, 10, 14climrecl 12073 . . . 4  |-  (  T. 
->  gamma  e.  RR )
16 1nn 9773 . . . . 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 20306 . . . . . . . . 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 12151 . . . . . 6  |-  ( (  T.  /\  i  e.  NN )  ->  ( G `  i )  <_ 
gamma )
2423ralrimiva 2639 . . . . 5  |-  (  T. 
->  A. i  e.  NN  ( G `  i )  <_  gamma )
25 fveq2 5541 . . . . . . . 8  |-  ( i  =  1  ->  ( G `  i )  =  ( G ` 
1 ) )
26 oveq2 5882 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  (
1 ... n )  =  ( 1 ... 1
) )
2726sumeq1d 12190 . . . . . . . . . . . 12  |-  ( n  =  1  ->  sum_ m  e.  ( 1 ... n
) ( 1  /  m )  =  sum_ m  e.  ( 1 ... 1 ) ( 1  /  m ) )
28 ax-1cn 8811 . . . . . . . . . . . . 13  |-  1  e.  CC
29 oveq2 5882 . . . . . . . . . . . . . . 15  |-  ( m  =  1  ->  (
1  /  m )  =  ( 1  / 
1 ) )
3028div1i 9504 . . . . . . . . . . . . . . 15  |-  ( 1  /  1 )  =  1
3129, 30syl6eq 2344 . . . . . . . . . . . . . 14  |-  ( m  =  1  ->  (
1  /  m )  =  1 )
3231fsum1 12230 . . . . . . . . . . . . 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 2344 . . . . . . . . . . 11  |-  ( n  =  1  ->  sum_ m  e.  ( 1 ... n
) ( 1  /  m )  =  1 )
35 oveq1 5881 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  (
n  +  1 )  =  ( 1  +  1 ) )
36 df-2 9820 . . . . . . . . . . . . 13  |-  2  =  ( 1  +  1 )
3735, 36syl6eqr 2346 . . . . . . . . . . . 12  |-  ( n  =  1  ->  (
n  +  1 )  =  2 )
3837fveq2d 5545 . . . . . . . . . . 11  |-  ( n  =  1  ->  ( log `  ( n  + 
1 ) )  =  ( log `  2
) )
3934, 38oveq12d 5892 . . . . . . . . . 10  |-  ( n  =  1  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  (
n  +  1 ) ) )  =  ( 1  -  ( log `  2 ) ) )
40 1re 8853 . . . . . . . . . . . 12  |-  1  e.  RR
41 2rp 10375 . . . . . . . . . . . . 13  |-  2  e.  RR+
42 relogcl 19948 . . . . . . . . . . . . 13  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
4341, 42ax-mp 8 . . . . . . . . . . . 12  |-  ( log `  2 )  e.  RR
4440, 43resubcli 9125 . . . . . . . . . . 11  |-  ( 1  -  ( log `  2
) )  e.  RR
4544elexi 2810 . . . . . . . . . 10  |-  ( 1  -  ( log `  2
) )  e.  _V
4639, 5, 45fvmpt 5618 . . . . . . . . 9  |-  ( 1  e.  NN  ->  ( G `  1 )  =  ( 1  -  ( log `  2
) ) )
4716, 46ax-mp 8 . . . . . . . 8  |-  ( G `
 1 )  =  ( 1  -  ( log `  2 ) )
4825, 47syl6eq 2344 . . . . . . 7  |-  ( i  =  1  ->  ( G `  i )  =  ( 1  -  ( log `  2
) ) )
4948breq1d 4049 . . . . . 6  |-  ( i  =  1  ->  (
( G `  i
)  <_  gamma  <->  ( 1  -  ( log `  2
) )  <_  gamma )
)
5049rspcva 2895 . . . . 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 5541 . . . . . . . . . . . 12  |-  ( x  =  i  ->  ( F `  x )  =  ( F `  i ) )
5352negeqd 9062 . . . . . . . . . . 11  |-  ( x  =  i  ->  -u ( F `  x )  =  -u ( F `  i ) )
54 eqid 2296 . . . . . . . . . . 11  |-  ( x  e.  NN  |->  -u ( F `  x )
)  =  ( x  e.  NN  |->  -u ( F `  x )
)
55 negex 9066 . . . . . . . . . . 11  |-  -u ( F `  i )  e.  _V
5653, 54, 55fvmpt 5618 . . . . . . . . . 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 8847 . . . . . . . . . . . . 13  |-  0  e.  CC
6160a1i 10 . . . . . . . . . . . 12  |-  (  T. 
->  0  e.  CC )
62 nnex 9768 . . . . . . . . . . . . . 14  |-  NN  e.  _V
6362mptex 5762 . . . . . . . . . . . . 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 5680 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  ( F `  k )  e.  RR )
6766adantl 452 . . . . . . . . . . . . 13  |-  ( (  T.  /\  k  e.  NN )  ->  ( F `  k )  e.  RR )
6867recnd 8877 . . . . . . . . . . . 12  |-  ( (  T.  /\  k  e.  NN )  ->  ( F `  k )  e.  CC )
69 fveq2 5541 . . . . . . . . . . . . . . . 16  |-  ( x  =  k  ->  ( F `  x )  =  ( F `  k ) )
7069negeqd 9062 . . . . . . . . . . . . . . 15  |-  ( x  =  k  ->  -u ( F `  x )  =  -u ( F `  k ) )
71 negex 9066 . . . . . . . . . . . . . . 15  |-  -u ( F `  k )  e.  _V
7270, 54, 71fvmpt 5618 . . . . . . . . . . . . . 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 9056 . . . . . . . . . . . . 13  |-  -u ( F `  k )  =  ( 0  -  ( F `  k
) )
7573, 74syl6eq 2344 . . . . . . . . . . . 12  |-  ( (  T.  /\  k  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  k
)  =  ( 0  -  ( F `  k ) ) )
761, 3, 59, 61, 64, 68, 75climsubc2 12128 . . . . . . . . . . 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 9226 . . . . . . . . . . . 12  |-  ( (  T.  /\  k  e.  NN )  ->  -u ( F `  k )  e.  RR )
7973, 78eqeltrd 2370 . . . . . . . . . . 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 9774 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
8483adantl 452 . . . . . . . . . . . . . . 15  |-  ( (  T.  /\  k  e.  NN )  ->  (
k  +  1 )  e.  NN )
8565ffvelrni 5680 . . . . . . . . . . . . . . 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 9367 . . . . . . . . . . . . 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 5541 . . . . . . . . . . . . . . 15  |-  ( x  =  ( k  +  1 )  ->  ( F `  x )  =  ( F `  ( k  +  1 ) ) )
9089negeqd 9062 . . . . . . . . . . . . . 14  |-  ( x  =  ( k  +  1 )  ->  -u ( F `  x )  =  -u ( F `  ( k  +  1 ) ) )
91 negex 9066 . . . . . . . . . . . . . 14  |-  -u ( F `  ( k  +  1 ) )  e.  _V
9290, 54, 91fvmpt 5618 . . . . . . . . . . . . 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 4069 . . . . . . . . . . 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 12151 . . . . . . . . 9  |-  ( (  T.  /\  i  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  i
)  <_  ( 0  -  gamma ) )
9757, 96eqbrtrrd 4061 . . . . . . . 8  |-  ( (  T.  /\  i  e.  NN )  ->  -u ( F `  i )  <_  ( 0  -  gamma ) )
98 df-neg 9056 . . . . . . . 8  |-  -u gamma  =  ( 0  -  gamma )
9997, 98syl6breqr 4079 . . . . . . 7  |-  ( (  T.  /\  i  e.  NN )  ->  -u ( F `  i )  <_ 
-u gamma )
10015trud 1314 . . . . . . . 8  |-  gamma  e.  RR
10165ffvelrni 5680 . . . . . . . . 9  |-  ( i  e.  NN  ->  ( F `  i )  e.  RR )
102101adantl 452 . . . . . . . 8  |-  ( (  T.  /\  i  e.  NN )  ->  ( F `  i )  e.  RR )
103 leneg 9293 . . . . . . . 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 2639 . . . . 5  |-  (  T. 
->  A. i  e.  NN  gamma  <_  ( F `  i
) )
107 fveq2 5541 . . . . . . . 8  |-  ( i  =  1  ->  ( F `  i )  =  ( F ` 
1 ) )
108 fveq2 5541 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  ( log `  n )  =  ( log `  1
) )
109 log1 19955 . . . . . . . . . . . . 13  |-  ( log `  1 )  =  0
110108, 109syl6eq 2344 . . . . . . . . . . . 12  |-  ( n  =  1  ->  ( log `  n )  =  0 )
11134, 110oveq12d 5892 . . . . . . . . . . 11  |-  ( n  =  1  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  n
) )  =  ( 1  -  0 ) )
11228subid1i 9134 . . . . . . . . . . 11  |-  ( 1  -  0 )  =  1
113111, 112syl6eq 2344 . . . . . . . . . 10  |-  ( n  =  1  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  n
) )  =  1 )
11440elexi 2810 . . . . . . . . . 10  |-  1  e.  _V
115113, 4, 114fvmpt 5618 . . . . . . . . 9  |-  ( 1  e.  NN  ->  ( F `  1 )  =  1 )
11616, 115ax-mp 8 . . . . . . . 8  |-  ( F `
 1 )  =  1
117107, 116syl6eq 2344 . . . . . . 7  |-  ( i  =  1  ->  ( F `  i )  =  1 )
118117breq2d 4051 . . . . . 6  |-  ( i  =  1  ->  ( gamma  <_  ( F `  i )  <->  gamma  <_  1
) )
119118rspcva 2895 . . . . 5  |-  ( ( 1  e.  NN  /\  A. i  e.  NN  gamma  <_ 
( F `  i
) )  ->  gamma  <_  1
)
12016, 106, 119sylancr 644 . . . 4  |-  (  T. 
->  gamma  <_  1 )
12144, 40elicc2i 10732 . . . 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 5405 . . . . 5  |-  ( F : NN --> RR  ->  F  Fn  NN )
12465, 123mp1i 11 . . . 4  |-  (  T. 
->  F  Fn  NN )
12517, 1syl6eleq 2386 . . . . . . . 8  |-  ( (  T.  /\  i  e.  NN )  ->  i  e.  ( ZZ>= `  1 )
)
126 elfznn 10835 . . . . . . . . . 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 10835 . . . . . . . . . 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 11093 . . . . . . 7  |-  ( (  T.  /\  i  e.  NN )  ->  ( F `  i )  <_  ( F `  1
) )
133132, 116syl6breq 4078 . . . . . 6  |-  ( (  T.  /\  i  e.  NN )  ->  ( F `  i )  <_  1 )
134100, 40elicc2i 10732 . . . . . 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 2639 . . . 4  |-  (  T. 
->  A. i  e.  NN  ( F `  i )  e.  ( gamma [,] 1
) )
137 ffnfv 5701 . . . 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 5405 . . . . 5  |-  ( G : NN --> RR  ->  G  Fn  NN )
14012, 139mp1i 11 . . . 4  |-  (  T. 
->  G  Fn  NN )
14112ffvelrni 5680 . . . . . . 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 11092 . . . . . . 7  |-  ( (  T.  /\  i  e.  NN )  ->  ( G `  1 )  <_  ( G `  i
) )
14647, 145syl5eqbrr 4073 . . . . . 6  |-  ( (  T.  /\  i  e.  NN )  ->  (
1  -  ( log `  2 ) )  <_  ( G `  i ) )
14744, 100elicc2i 10732 . . . . . 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 2639 . . . 4  |-  (  T. 
->  A. i  e.  NN  ( G `  i )  e.  ( ( 1  -  ( log `  2
) ) [,] gamma ) )
150 ffnfv 5701 . . . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   class class class wbr 4039    e. cmpt 4093    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   [,]cicc 10675   ...cfz 10798    ~~> cli 11974   sum_csu 12174   logclog 19928   gammacem 20302
This theorem is referenced by:  emcl  20312  harmonicbnd  20313  harmonicbnd2  20314
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-em 20303
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