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Theorem emcllem7 22370
Description: Lemma for emcl 22371 and harmonicbnd 22372. 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 10888 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 10669 . . . . 5  |-  ( T. 
->  1  e.  ZZ )
3 emcl.1 . . . . . . . 8  |-  F  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m
)  -  ( log `  n ) ) )
4 emcl.2 . . . . . . . 8  |-  G  =  ( n  e.  NN  |->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m
)  -  ( log `  ( n  +  1 ) ) ) )
5 emcl.3 . . . . . . . 8  |-  H  =  ( n  e.  NN  |->  ( log `  ( 1  +  ( 1  /  n ) ) ) )
6 emcl.4 . . . . . . . 8  |-  T  =  ( n  e.  NN  |->  ( ( 1  /  n )  -  ( log `  ( 1  +  ( 1  /  n
) ) ) ) )
73, 4, 5, 6emcllem6 22369 . . . . . . 7  |-  ( F  ~~> 
gamma  /\  G  ~~>  gamma )
87simpri 462 . . . . . 6  |-  G  ~~>  gamma
98a1i 11 . . . . 5  |-  ( T. 
->  G  ~~>  gamma )
103, 4emcllem1 22364 . . . . . . . 8  |-  ( F : NN --> RR  /\  G : NN --> RR )
1110simpri 462 . . . . . . 7  |-  G : NN
--> RR
1211ffvelrni 5837 . . . . . 6  |-  ( k  e.  NN  ->  ( G `  k )  e.  RR )
1312adantl 466 . . . . 5  |-  ( ( T.  /\  k  e.  NN )  ->  ( G `  k )  e.  RR )
141, 2, 9, 13climrecl 13053 . . . 4  |-  ( T. 
->  gamma  e.  RR )
15 1nn 10325 . . . . 5  |-  1  e.  NN
16 simpr 461 . . . . . . 7  |-  ( ( T.  /\  i  e.  NN )  ->  i  e.  NN )
178a1i 11 . . . . . . 7  |-  ( ( T.  /\  i  e.  NN )  ->  G  ~~>  gamma )
1812adantl 466 . . . . . . 7  |-  ( ( ( T.  /\  i  e.  NN )  /\  k  e.  NN )  ->  ( G `  k )  e.  RR )
193, 4emcllem2 22365 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
( F `  (
k  +  1 ) )  <_  ( F `  k )  /\  ( G `  k )  <_  ( G `  (
k  +  1 ) ) ) )
2019simprd 463 . . . . . . . 8  |-  ( k  e.  NN  ->  ( G `  k )  <_  ( G `  (
k  +  1 ) ) )
2120adantl 466 . . . . . . 7  |-  ( ( ( T.  /\  i  e.  NN )  /\  k  e.  NN )  ->  ( G `  k )  <_  ( G `  (
k  +  1 ) ) )
221, 16, 17, 18, 21climub 13131 . . . . . 6  |-  ( ( T.  /\  i  e.  NN )  ->  ( G `  i )  <_ 
gamma )
2322ralrimiva 2794 . . . . 5  |-  ( T. 
->  A. i  e.  NN  ( G `  i )  <_  gamma )
24 fveq2 5686 . . . . . . . 8  |-  ( i  =  1  ->  ( G `  i )  =  ( G ` 
1 ) )
25 oveq2 6094 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  (
1 ... n )  =  ( 1 ... 1
) )
2625sumeq1d 13170 . . . . . . . . . . . 12  |-  ( n  =  1  ->  sum_ m  e.  ( 1 ... n
) ( 1  /  m )  =  sum_ m  e.  ( 1 ... 1 ) ( 1  /  m ) )
27 1z 10668 . . . . . . . . . . . . 13  |-  1  e.  ZZ
28 ax-1cn 9332 . . . . . . . . . . . . 13  |-  1  e.  CC
29 oveq2 6094 . . . . . . . . . . . . . . 15  |-  ( m  =  1  ->  (
1  /  m )  =  ( 1  / 
1 ) )
30 1div1e1 10016 . . . . . . . . . . . . . . 15  |-  ( 1  /  1 )  =  1
3129, 30syl6eq 2486 . . . . . . . . . . . . . 14  |-  ( m  =  1  ->  (
1  /  m )  =  1 )
3231fsum1 13210 . . . . . . . . . . . . 13  |-  ( ( 1  e.  ZZ  /\  1  e.  CC )  -> 
sum_ m  e.  (
1 ... 1 ) ( 1  /  m )  =  1 )
3327, 28, 32mp2an 672 . . . . . . . . . . . 12  |-  sum_ m  e.  ( 1 ... 1
) ( 1  /  m )  =  1
3426, 33syl6eq 2486 . . . . . . . . . . 11  |-  ( n  =  1  ->  sum_ m  e.  ( 1 ... n
) ( 1  /  m )  =  1 )
35 oveq1 6093 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  (
n  +  1 )  =  ( 1  +  1 ) )
36 df-2 10372 . . . . . . . . . . . . 13  |-  2  =  ( 1  +  1 )
3735, 36syl6eqr 2488 . . . . . . . . . . . 12  |-  ( n  =  1  ->  (
n  +  1 )  =  2 )
3837fveq2d 5690 . . . . . . . . . . 11  |-  ( n  =  1  ->  ( log `  ( n  + 
1 ) )  =  ( log `  2
) )
3934, 38oveq12d 6104 . . . . . . . . . 10  |-  ( n  =  1  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  (
n  +  1 ) ) )  =  ( 1  -  ( log `  2 ) ) )
40 1re 9377 . . . . . . . . . . . 12  |-  1  e.  RR
41 2rp 10988 . . . . . . . . . . . . 13  |-  2  e.  RR+
42 relogcl 22002 . . . . . . . . . . . . 13  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
4341, 42ax-mp 5 . . . . . . . . . . . 12  |-  ( log `  2 )  e.  RR
4440, 43resubcli 9663 . . . . . . . . . . 11  |-  ( 1  -  ( log `  2
) )  e.  RR
4544elexi 2977 . . . . . . . . . 10  |-  ( 1  -  ( log `  2
) )  e.  _V
4639, 4, 45fvmpt 5769 . . . . . . . . 9  |-  ( 1  e.  NN  ->  ( G `  1 )  =  ( 1  -  ( log `  2
) ) )
4715, 46ax-mp 5 . . . . . . . 8  |-  ( G `
 1 )  =  ( 1  -  ( log `  2 ) )
4824, 47syl6eq 2486 . . . . . . 7  |-  ( i  =  1  ->  ( G `  i )  =  ( 1  -  ( log `  2
) ) )
4948breq1d 4297 . . . . . 6  |-  ( i  =  1  ->  (
( G `  i
)  <_  gamma  <->  ( 1  -  ( log `  2
) )  <_  gamma )
)
5049rspcva 3066 . . . . 5  |-  ( ( 1  e.  NN  /\  A. i  e.  NN  ( G `  i )  <_ 
gamma )  ->  ( 1  -  ( log `  2
) )  <_  gamma )
5115, 23, 50sylancr 663 . . . 4  |-  ( T. 
->  ( 1  -  ( log `  2 ) )  <_  gamma )
52 fveq2 5686 . . . . . . . . . . . 12  |-  ( x  =  i  ->  ( F `  x )  =  ( F `  i ) )
5352negeqd 9596 . . . . . . . . . . 11  |-  ( x  =  i  ->  -u ( F `  x )  =  -u ( F `  i ) )
54 eqid 2438 . . . . . . . . . . 11  |-  ( x  e.  NN  |->  -u ( F `  x )
)  =  ( x  e.  NN  |->  -u ( F `  x )
)
55 negex 9600 . . . . . . . . . . 11  |-  -u ( F `  i )  e.  _V
5653, 54, 55fvmpt 5769 . . . . . . . . . 10  |-  ( i  e.  NN  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  i
)  =  -u ( F `  i )
)
5756adantl 466 . . . . . . . . 9  |-  ( ( T.  /\  i  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  i
)  =  -u ( F `  i )
)
587simpli 458 . . . . . . . . . . . . 13  |-  F  ~~>  gamma
5958a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  F  ~~>  gamma )
60 0cnd 9371 . . . . . . . . . . . 12  |-  ( T. 
->  0  e.  CC )
61 nnex 10320 . . . . . . . . . . . . . 14  |-  NN  e.  _V
6261mptex 5943 . . . . . . . . . . . . 13  |-  ( x  e.  NN  |->  -u ( F `  x )
)  e.  _V
6362a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  ( x  e.  NN  |->  -u ( F `  x
) )  e.  _V )
6410simpli 458 . . . . . . . . . . . . . . 15  |-  F : NN
--> RR
6564ffvelrni 5837 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  ( F `  k )  e.  RR )
6665adantl 466 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  ( F `  k )  e.  RR )
6766recnd 9404 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  ( F `  k )  e.  CC )
68 fveq2 5686 . . . . . . . . . . . . . . . 16  |-  ( x  =  k  ->  ( F `  x )  =  ( F `  k ) )
6968negeqd 9596 . . . . . . . . . . . . . . 15  |-  ( x  =  k  ->  -u ( F `  x )  =  -u ( F `  k ) )
70 negex 9600 . . . . . . . . . . . . . . 15  |-  -u ( F `  k )  e.  _V
7169, 54, 70fvmpt 5769 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  k
)  =  -u ( F `  k )
)
7271adantl 466 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  k
)  =  -u ( F `  k )
)
73 df-neg 9590 . . . . . . . . . . . . 13  |-  -u ( F `  k )  =  ( 0  -  ( F `  k
) )
7472, 73syl6eq 2486 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  k
)  =  ( 0  -  ( F `  k ) ) )
751, 2, 59, 60, 63, 67, 74climsubc2 13108 . . . . . . . . . . 11  |-  ( T. 
->  ( x  e.  NN  |->  -u ( F `  x
) )  ~~>  ( 0  -  gamma ) )
7675adantr 465 . . . . . . . . . 10  |-  ( ( T.  /\  i  e.  NN )  ->  (
x  e.  NN  |->  -u ( F `  x ) )  ~~>  ( 0  - 
gamma ) )
7766renegcld 9767 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  -u ( F `  k )  e.  RR )
7872, 77eqeltrd 2512 . . . . . . . . . . 11  |-  ( ( T.  /\  k  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  k
)  e.  RR )
7978adantlr 714 . . . . . . . . . 10  |-  ( ( ( T.  /\  i  e.  NN )  /\  k  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  k
)  e.  RR )
8019simpld 459 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
8180adantl 466 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
82 peano2nn 10326 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
8382adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( T.  /\  k  e.  NN )  ->  (
k  +  1 )  e.  NN )
8464ffvelrni 5837 . . . . . . . . . . . . . . 15  |-  ( ( k  +  1 )  e.  NN  ->  ( F `  ( k  +  1 ) )  e.  RR )
8583, 84syl 16 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  k  e.  NN )  ->  ( F `  ( k  +  1 ) )  e.  RR )
8685, 66lenegd 9910 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( F `  (
k  +  1 ) )  <_  ( F `  k )  <->  -u ( F `
 k )  <_  -u ( F `  (
k  +  1 ) ) ) )
8781, 86mpbid 210 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  -u ( F `  k )  <_ 
-u ( F `  ( k  +  1 ) ) )
88 fveq2 5686 . . . . . . . . . . . . . . 15  |-  ( x  =  ( k  +  1 )  ->  ( F `  x )  =  ( F `  ( k  +  1 ) ) )
8988negeqd 9596 . . . . . . . . . . . . . 14  |-  ( x  =  ( k  +  1 )  ->  -u ( F `  x )  =  -u ( F `  ( k  +  1 ) ) )
90 negex 9600 . . . . . . . . . . . . . 14  |-  -u ( F `  ( k  +  1 ) )  e.  _V
9189, 54, 90fvmpt 5769 . . . . . . . . . . . . 13  |-  ( ( k  +  1 )  e.  NN  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  (
k  +  1 ) )  =  -u ( F `  ( k  +  1 ) ) )
9283, 91syl 16 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  (
k  +  1 ) )  =  -u ( F `  ( k  +  1 ) ) )
9387, 72, 923brtr4d 4317 . . . . . . . . . . 11  |-  ( ( T.  /\  k  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  k
)  <_  ( (
x  e.  NN  |->  -u ( F `  x ) ) `  ( k  +  1 ) ) )
9493adantlr 714 . . . . . . . . . 10  |-  ( ( ( T.  /\  i  e.  NN )  /\  k  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  k
)  <_  ( (
x  e.  NN  |->  -u ( F `  x ) ) `  ( k  +  1 ) ) )
951, 16, 76, 79, 94climub 13131 . . . . . . . . 9  |-  ( ( T.  /\  i  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  i
)  <_  ( 0  -  gamma ) )
9657, 95eqbrtrrd 4309 . . . . . . . 8  |-  ( ( T.  /\  i  e.  NN )  ->  -u ( F `  i )  <_  ( 0  -  gamma ) )
97 df-neg 9590 . . . . . . . 8  |-  -u gamma  =  ( 0  -  gamma )
9896, 97syl6breqr 4327 . . . . . . 7  |-  ( ( T.  /\  i  e.  NN )  ->  -u ( F `  i )  <_ 
-u gamma )
9914trud 1378 . . . . . . . 8  |-  gamma  e.  RR
10064ffvelrni 5837 . . . . . . . . 9  |-  ( i  e.  NN  ->  ( F `  i )  e.  RR )
101100adantl 466 . . . . . . . 8  |-  ( ( T.  /\  i  e.  NN )  ->  ( F `  i )  e.  RR )
102 leneg 9834 . . . . . . . 8  |-  ( (
gamma  e.  RR  /\  ( F `  i )  e.  RR )  ->  ( gamma  <_  ( F `  i )  <->  -u ( F `
 i )  <_  -u
gamma ) )
10399, 101, 102sylancr 663 . . . . . . 7  |-  ( ( T.  /\  i  e.  NN )  ->  ( gamma  <_  ( F `  i )  <->  -u ( F `
 i )  <_  -u
gamma ) )
10498, 103mpbird 232 . . . . . 6  |-  ( ( T.  /\  i  e.  NN )  ->  gamma  <_  ( F `  i )
)
105104ralrimiva 2794 . . . . 5  |-  ( T. 
->  A. i  e.  NN  gamma  <_  ( F `  i
) )
106 fveq2 5686 . . . . . . . 8  |-  ( i  =  1  ->  ( F `  i )  =  ( F ` 
1 ) )
107 fveq2 5686 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  ( log `  n )  =  ( log `  1
) )
108 log1 22009 . . . . . . . . . . . . 13  |-  ( log `  1 )  =  0
109107, 108syl6eq 2486 . . . . . . . . . . . 12  |-  ( n  =  1  ->  ( log `  n )  =  0 )
11034, 109oveq12d 6104 . . . . . . . . . . 11  |-  ( n  =  1  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  n
) )  =  ( 1  -  0 ) )
111 1m0e1 10424 . . . . . . . . . . 11  |-  ( 1  -  0 )  =  1
112110, 111syl6eq 2486 . . . . . . . . . 10  |-  ( n  =  1  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  n
) )  =  1 )
11340elexi 2977 . . . . . . . . . 10  |-  1  e.  _V
114112, 3, 113fvmpt 5769 . . . . . . . . 9  |-  ( 1  e.  NN  ->  ( F `  1 )  =  1 )
11515, 114ax-mp 5 . . . . . . . 8  |-  ( F `
 1 )  =  1
116106, 115syl6eq 2486 . . . . . . 7  |-  ( i  =  1  ->  ( F `  i )  =  1 )
117116breq2d 4299 . . . . . 6  |-  ( i  =  1  ->  ( gamma  <_  ( F `  i )  <->  gamma  <_  1
) )
118117rspcva 3066 . . . . 5  |-  ( ( 1  e.  NN  /\  A. i  e.  NN  gamma  <_ 
( F `  i
) )  ->  gamma  <_  1
)
11915, 105, 118sylancr 663 . . . 4  |-  ( T. 
->  gamma  <_  1 )
12044, 40elicc2i 11353 . . . 4  |-  ( gamma  e.  ( ( 1  -  ( log `  2
) ) [,] 1
)  <->  ( gamma  e.  RR  /\  ( 1  -  ( log `  2 ) )  <_  gamma  /\  gamma  <_  1
) )
12114, 51, 119, 120syl3anbrc 1172 . . 3  |-  ( T. 
->  gamma  e.  ( ( 1  -  ( log `  2 ) ) [,] 1 ) )
122 ffn 5554 . . . . 5  |-  ( F : NN --> RR  ->  F  Fn  NN )
12364, 122mp1i 12 . . . 4  |-  ( T. 
->  F  Fn  NN )
12416, 1syl6eleq 2528 . . . . . . . 8  |-  ( ( T.  /\  i  e.  NN )  ->  i  e.  ( ZZ>= `  1 )
)
125 elfznn 11470 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... i )  ->  k  e.  NN )
126125adantl 466 . . . . . . . . 9  |-  ( ( ( T.  /\  i  e.  NN )  /\  k  e.  ( 1 ... i
) )  ->  k  e.  NN )
127126, 65syl 16 . . . . . . . 8  |-  ( ( ( T.  /\  i  e.  NN )  /\  k  e.  ( 1 ... i
) )  ->  ( F `  k )  e.  RR )
128 elfznn 11470 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( i  -  1 ) )  ->  k  e.  NN )
129128adantl 466 . . . . . . . . 9  |-  ( ( ( T.  /\  i  e.  NN )  /\  k  e.  ( 1 ... (
i  -  1 ) ) )  ->  k  e.  NN )
130129, 80syl 16 . . . . . . . 8  |-  ( ( ( T.  /\  i  e.  NN )  /\  k  e.  ( 1 ... (
i  -  1 ) ) )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
131124, 127, 130monoord2 11829 . . . . . . 7  |-  ( ( T.  /\  i  e.  NN )  ->  ( F `  i )  <_  ( F `  1
) )
132131, 115syl6breq 4326 . . . . . 6  |-  ( ( T.  /\  i  e.  NN )  ->  ( F `  i )  <_  1 )
13399, 40elicc2i 11353 . . . . . 6  |-  ( ( F `  i )  e.  ( gamma [,] 1
)  <->  ( ( F `
 i )  e.  RR  /\  gamma  <_  ( F `  i )  /\  ( F `  i
)  <_  1 ) )
134101, 104, 132, 133syl3anbrc 1172 . . . . 5  |-  ( ( T.  /\  i  e.  NN )  ->  ( F `  i )  e.  ( gamma [,] 1 ) )
135134ralrimiva 2794 . . . 4  |-  ( T. 
->  A. i  e.  NN  ( F `  i )  e.  ( gamma [,] 1
) )
136 ffnfv 5864 . . . 4  |-  ( F : NN --> ( gamma [,] 1 )  <->  ( F  Fn  NN  /\  A. i  e.  NN  ( F `  i )  e.  (
gamma [,] 1 ) ) )
137123, 135, 136sylanbrc 664 . . 3  |-  ( T. 
->  F : NN --> ( gamma [,] 1 ) )
138 ffn 5554 . . . . 5  |-  ( G : NN --> RR  ->  G  Fn  NN )
13911, 138mp1i 12 . . . 4  |-  ( T. 
->  G  Fn  NN )
14011ffvelrni 5837 . . . . . . 7  |-  ( i  e.  NN  ->  ( G `  i )  e.  RR )
141140adantl 466 . . . . . 6  |-  ( ( T.  /\  i  e.  NN )  ->  ( G `  i )  e.  RR )
142126, 12syl 16 . . . . . . . 8  |-  ( ( ( T.  /\  i  e.  NN )  /\  k  e.  ( 1 ... i
) )  ->  ( G `  k )  e.  RR )
143129, 20syl 16 . . . . . . . 8  |-  ( ( ( T.  /\  i  e.  NN )  /\  k  e.  ( 1 ... (
i  -  1 ) ) )  ->  ( G `  k )  <_  ( G `  (
k  +  1 ) ) )
144124, 142, 143monoord 11828 . . . . . . 7  |-  ( ( T.  /\  i  e.  NN )  ->  ( G `  1 )  <_  ( G `  i
) )
14547, 144syl5eqbrr 4321 . . . . . 6  |-  ( ( T.  /\  i  e.  NN )  ->  (
1  -  ( log `  2 ) )  <_  ( G `  i ) )
14644, 99elicc2i 11353 . . . . . 6  |-  ( ( G `  i )  e.  ( ( 1  -  ( log `  2
) ) [,] gamma )  <-> 
( ( G `  i )  e.  RR  /\  ( 1  -  ( log `  2 ) )  <_  ( G `  i )  /\  ( G `  i )  <_ 
gamma ) )
147141, 145, 22, 146syl3anbrc 1172 . . . . 5  |-  ( ( T.  /\  i  e.  NN )  ->  ( G `  i )  e.  ( ( 1  -  ( log `  2
) ) [,] gamma ) )
148147ralrimiva 2794 . . . 4  |-  ( T. 
->  A. i  e.  NN  ( G `  i )  e.  ( ( 1  -  ( log `  2
) ) [,] gamma ) )
149 ffnfv 5864 . . . 4  |-  ( G : NN --> ( ( 1  -  ( log `  2 ) ) [,] gamma )  <->  ( G  Fn  NN  /\  A. i  e.  NN  ( G `  i )  e.  ( ( 1  -  ( log `  2 ) ) [,] gamma ) ) )
150139, 148, 149sylanbrc 664 . . 3  |-  ( T. 
->  G : NN --> ( ( 1  -  ( log `  2 ) ) [,] gamma ) )
151121, 137, 1503jca 1168 . 2  |-  ( T. 
->  ( gamma  e.  (
( 1  -  ( log `  2 ) ) [,] 1 )  /\  F : NN --> ( gamma [,] 1 )  /\  G : NN --> ( ( 1  -  ( log `  2
) ) [,] gamma ) ) )
152151trud 1378 1  |-  ( gamma  e.  ( ( 1  -  ( log `  2
) ) [,] 1
)  /\  F : NN
--> ( gamma [,] 1 )  /\  G : NN --> ( ( 1  -  ( log `  2 ) ) [,] gamma ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   T. wtru 1370    e. wcel 1756   A.wral 2710   _Vcvv 2967   class class class wbr 4287    e. cmpt 4345    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    <_ cle 9411    - cmin 9587   -ucneg 9588    / cdiv 9985   NNcn 10314   2c2 10363   ZZcz 10638   ZZ>=cuz 10853   RR+crp 10983   [,]cicc 11295   ...cfz 11429    ~~> cli 12954   sum_csu 13155   logclog 21981   gammacem 22360
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  ax-addf 9353  ax-mulf 9354
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-iin 4169  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-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  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-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-pi 13350  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-limc 21316  df-dv 21317  df-log 21983  df-em 22361
This theorem is referenced by:  emcl  22371  harmonicbnd  22372  harmonicbnd2  22373
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