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Theorem emcllem7 22521
Description: Lemma for emcl 22522 and harmonicbnd 22523. 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 11000 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 10781 . . . . 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 22520 . . . . . . 7  |-  ( F  ~~> 
gamma  /\  G  ~~>  gamma )
87simpri 462 . . . . . 6  |-  G  ~~>  gamma
98a1i 11 . . . . 5  |-  ( T. 
->  G  ~~>  gamma )
103, 4emcllem1 22515 . . . . . . . 8  |-  ( F : NN --> RR  /\  G : NN --> RR )
1110simpri 462 . . . . . . 7  |-  G : NN
--> RR
1211ffvelrni 5944 . . . . . 6  |-  ( k  e.  NN  ->  ( G `  k )  e.  RR )
1312adantl 466 . . . . 5  |-  ( ( T.  /\  k  e.  NN )  ->  ( G `  k )  e.  RR )
141, 2, 9, 13climrecl 13172 . . . 4  |-  ( T. 
->  gamma  e.  RR )
15 1nn 10437 . . . . 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 22516 . . . . . . . . 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 13250 . . . . . 6  |-  ( ( T.  /\  i  e.  NN )  ->  ( G `  i )  <_ 
gamma )
2322ralrimiva 2825 . . . . 5  |-  ( T. 
->  A. i  e.  NN  ( G `  i )  <_  gamma )
24 fveq2 5792 . . . . . . . 8  |-  ( i  =  1  ->  ( G `  i )  =  ( G ` 
1 ) )
25 oveq2 6201 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  (
1 ... n )  =  ( 1 ... 1
) )
2625sumeq1d 13289 . . . . . . . . . . . 12  |-  ( n  =  1  ->  sum_ m  e.  ( 1 ... n
) ( 1  /  m )  =  sum_ m  e.  ( 1 ... 1 ) ( 1  /  m ) )
27 1z 10780 . . . . . . . . . . . . 13  |-  1  e.  ZZ
28 ax-1cn 9444 . . . . . . . . . . . . 13  |-  1  e.  CC
29 oveq2 6201 . . . . . . . . . . . . . . 15  |-  ( m  =  1  ->  (
1  /  m )  =  ( 1  / 
1 ) )
30 1div1e1 10128 . . . . . . . . . . . . . . 15  |-  ( 1  /  1 )  =  1
3129, 30syl6eq 2508 . . . . . . . . . . . . . 14  |-  ( m  =  1  ->  (
1  /  m )  =  1 )
3231fsum1 13329 . . . . . . . . . . . . 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 2508 . . . . . . . . . . 11  |-  ( n  =  1  ->  sum_ m  e.  ( 1 ... n
) ( 1  /  m )  =  1 )
35 oveq1 6200 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  (
n  +  1 )  =  ( 1  +  1 ) )
36 df-2 10484 . . . . . . . . . . . . 13  |-  2  =  ( 1  +  1 )
3735, 36syl6eqr 2510 . . . . . . . . . . . 12  |-  ( n  =  1  ->  (
n  +  1 )  =  2 )
3837fveq2d 5796 . . . . . . . . . . 11  |-  ( n  =  1  ->  ( log `  ( n  + 
1 ) )  =  ( log `  2
) )
3934, 38oveq12d 6211 . . . . . . . . . 10  |-  ( n  =  1  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  (
n  +  1 ) ) )  =  ( 1  -  ( log `  2 ) ) )
40 1re 9489 . . . . . . . . . . . 12  |-  1  e.  RR
41 2rp 11100 . . . . . . . . . . . . 13  |-  2  e.  RR+
42 relogcl 22153 . . . . . . . . . . . . 13  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
4341, 42ax-mp 5 . . . . . . . . . . . 12  |-  ( log `  2 )  e.  RR
4440, 43resubcli 9775 . . . . . . . . . . 11  |-  ( 1  -  ( log `  2
) )  e.  RR
4544elexi 3081 . . . . . . . . . 10  |-  ( 1  -  ( log `  2
) )  e.  _V
4639, 4, 45fvmpt 5876 . . . . . . . . 9  |-  ( 1  e.  NN  ->  ( G `  1 )  =  ( 1  -  ( log `  2
) ) )
4715, 46ax-mp 5 . . . . . . . 8  |-  ( G `
 1 )  =  ( 1  -  ( log `  2 ) )
4824, 47syl6eq 2508 . . . . . . 7  |-  ( i  =  1  ->  ( G `  i )  =  ( 1  -  ( log `  2
) ) )
4948breq1d 4403 . . . . . 6  |-  ( i  =  1  ->  (
( G `  i
)  <_  gamma  <->  ( 1  -  ( log `  2
) )  <_  gamma )
)
5049rspcva 3170 . . . . 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 5792 . . . . . . . . . . . 12  |-  ( x  =  i  ->  ( F `  x )  =  ( F `  i ) )
5352negeqd 9708 . . . . . . . . . . 11  |-  ( x  =  i  ->  -u ( F `  x )  =  -u ( F `  i ) )
54 eqid 2451 . . . . . . . . . . 11  |-  ( x  e.  NN  |->  -u ( F `  x )
)  =  ( x  e.  NN  |->  -u ( F `  x )
)
55 negex 9712 . . . . . . . . . . 11  |-  -u ( F `  i )  e.  _V
5653, 54, 55fvmpt 5876 . . . . . . . . . 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 9483 . . . . . . . . . . . 12  |-  ( T. 
->  0  e.  CC )
61 nnex 10432 . . . . . . . . . . . . . 14  |-  NN  e.  _V
6261mptex 6050 . . . . . . . . . . . . 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 5944 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  ( F `  k )  e.  RR )
6665adantl 466 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  ( F `  k )  e.  RR )
6766recnd 9516 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  ( F `  k )  e.  CC )
68 fveq2 5792 . . . . . . . . . . . . . . . 16  |-  ( x  =  k  ->  ( F `  x )  =  ( F `  k ) )
6968negeqd 9708 . . . . . . . . . . . . . . 15  |-  ( x  =  k  ->  -u ( F `  x )  =  -u ( F `  k ) )
70 negex 9712 . . . . . . . . . . . . . . 15  |-  -u ( F `  k )  e.  _V
7169, 54, 70fvmpt 5876 . . . . . . . . . . . . . 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 9702 . . . . . . . . . . . . 13  |-  -u ( F `  k )  =  ( 0  -  ( F `  k
) )
7472, 73syl6eq 2508 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  k
)  =  ( 0  -  ( F `  k ) ) )
751, 2, 59, 60, 63, 67, 74climsubc2 13227 . . . . . . . . . . 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 9879 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  -u ( F `  k )  e.  RR )
7872, 77eqeltrd 2539 . . . . . . . . . . 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 10438 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
8382adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( T.  /\  k  e.  NN )  ->  (
k  +  1 )  e.  NN )
8464ffvelrni 5944 . . . . . . . . . . . . . . 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 10022 . . . . . . . . . . . . 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 5792 . . . . . . . . . . . . . . 15  |-  ( x  =  ( k  +  1 )  ->  ( F `  x )  =  ( F `  ( k  +  1 ) ) )
8988negeqd 9708 . . . . . . . . . . . . . 14  |-  ( x  =  ( k  +  1 )  ->  -u ( F `  x )  =  -u ( F `  ( k  +  1 ) ) )
90 negex 9712 . . . . . . . . . . . . . 14  |-  -u ( F `  ( k  +  1 ) )  e.  _V
9189, 54, 90fvmpt 5876 . . . . . . . . . . . . 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 4423 . . . . . . . . . . 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 13250 . . . . . . . . 9  |-  ( ( T.  /\  i  e.  NN )  ->  (
( x  e.  NN  |->  -u ( F `  x
) ) `  i
)  <_  ( 0  -  gamma ) )
9657, 95eqbrtrrd 4415 . . . . . . . 8  |-  ( ( T.  /\  i  e.  NN )  ->  -u ( F `  i )  <_  ( 0  -  gamma ) )
97 df-neg 9702 . . . . . . . 8  |-  -u gamma  =  ( 0  -  gamma )
9896, 97syl6breqr 4433 . . . . . . 7  |-  ( ( T.  /\  i  e.  NN )  ->  -u ( F `  i )  <_ 
-u gamma )
9914trud 1379 . . . . . . . 8  |-  gamma  e.  RR
10064ffvelrni 5944 . . . . . . . . 9  |-  ( i  e.  NN  ->  ( F `  i )  e.  RR )
101100adantl 466 . . . . . . . 8  |-  ( ( T.  /\  i  e.  NN )  ->  ( F `  i )  e.  RR )
102 leneg 9946 . . . . . . . 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 2825 . . . . 5  |-  ( T. 
->  A. i  e.  NN  gamma  <_  ( F `  i
) )
106 fveq2 5792 . . . . . . . 8  |-  ( i  =  1  ->  ( F `  i )  =  ( F ` 
1 ) )
107 fveq2 5792 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  ( log `  n )  =  ( log `  1
) )
108 log1 22160 . . . . . . . . . . . . 13  |-  ( log `  1 )  =  0
109107, 108syl6eq 2508 . . . . . . . . . . . 12  |-  ( n  =  1  ->  ( log `  n )  =  0 )
11034, 109oveq12d 6211 . . . . . . . . . . 11  |-  ( n  =  1  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  n
) )  =  ( 1  -  0 ) )
111 1m0e1 10536 . . . . . . . . . . 11  |-  ( 1  -  0 )  =  1
112110, 111syl6eq 2508 . . . . . . . . . 10  |-  ( n  =  1  ->  ( sum_ m  e.  ( 1 ... n ) ( 1  /  m )  -  ( log `  n
) )  =  1 )
11340elexi 3081 . . . . . . . . . 10  |-  1  e.  _V
114112, 3, 113fvmpt 5876 . . . . . . . . 9  |-  ( 1  e.  NN  ->  ( F `  1 )  =  1 )
11515, 114ax-mp 5 . . . . . . . 8  |-  ( F `
 1 )  =  1
116106, 115syl6eq 2508 . . . . . . 7  |-  ( i  =  1  ->  ( F `  i )  =  1 )
117116breq2d 4405 . . . . . 6  |-  ( i  =  1  ->  ( gamma  <_  ( F `  i )  <->  gamma  <_  1
) )
118117rspcva 3170 . . . . 5  |-  ( ( 1  e.  NN  /\  A. i  e.  NN  gamma  <_ 
( F `  i
) )  ->  gamma  <_  1
)
11915, 105, 118sylancr 663 . . . 4  |-  ( T. 
->  gamma  <_  1 )
12044, 40elicc2i 11465 . . . 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 5660 . . . . 5  |-  ( F : NN --> RR  ->  F  Fn  NN )
12364, 122mp1i 12 . . . 4  |-  ( T. 
->  F  Fn  NN )
12416, 1syl6eleq 2549 . . . . . . . 8  |-  ( ( T.  /\  i  e.  NN )  ->  i  e.  ( ZZ>= `  1 )
)
125 elfznn 11588 . . . . . . . . . 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 11588 . . . . . . . . . 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 11947 . . . . . . 7  |-  ( ( T.  /\  i  e.  NN )  ->  ( F `  i )  <_  ( F `  1
) )
132131, 115syl6breq 4432 . . . . . 6  |-  ( ( T.  /\  i  e.  NN )  ->  ( F `  i )  <_  1 )
13399, 40elicc2i 11465 . . . . . 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 2825 . . . 4  |-  ( T. 
->  A. i  e.  NN  ( F `  i )  e.  ( gamma [,] 1
) )
136 ffnfv 5971 . . . 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 5660 . . . . 5  |-  ( G : NN --> RR  ->  G  Fn  NN )
13911, 138mp1i 12 . . . 4  |-  ( T. 
->  G  Fn  NN )
14011ffvelrni 5944 . . . . . . 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 11946 . . . . . . 7  |-  ( ( T.  /\  i  e.  NN )  ->  ( G `  1 )  <_  ( G `  i
) )
14547, 144syl5eqbrr 4427 . . . . . 6  |-  ( ( T.  /\  i  e.  NN )  ->  (
1  -  ( log `  2 ) )  <_  ( G `  i ) )
14644, 99elicc2i 11465 . . . . . 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 2825 . . . 4  |-  ( T. 
->  A. i  e.  NN  ( G `  i )  e.  ( ( 1  -  ( log `  2
) ) [,] gamma ) )
149 ffnfv 5971 . . . 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 1379 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 1370   T. wtru 1371    e. wcel 1758   A.wral 2795   _Vcvv 3071   class class class wbr 4393    |-> cmpt 4451    Fn wfn 5514   -->wf 5515   ` cfv 5519  (class class class)co 6193   CCcc 9384   RRcr 9385   0cc0 9386   1c1 9387    + caddc 9389    <_ cle 9523    - cmin 9699   -ucneg 9700    / cdiv 10097   NNcn 10426   2c2 10475   ZZcz 10750   ZZ>=cuz 10965   RR+crp 11095   [,]cicc 11407   ...cfz 11547    ~~> cli 13073   sum_csu 13274   logclog 22132   gammacem 22511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-fi 7765  df-sup 7795  df-oi 7828  df-card 8213  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-ioo 11408  df-ioc 11409  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-fl 11752  df-mod 11819  df-seq 11917  df-exp 11976  df-fac 12162  df-bc 12189  df-hash 12214  df-shft 12667  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-limsup 13060  df-clim 13077  df-rlim 13078  df-sum 13275  df-ef 13464  df-sin 13466  df-cos 13467  df-pi 13469  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-rest 14472  df-topn 14473  df-0g 14491  df-gsum 14492  df-topgen 14493  df-pt 14494  df-prds 14497  df-xrs 14551  df-qtop 14556  df-imas 14557  df-xps 14559  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-submnd 15576  df-mulg 15659  df-cntz 15946  df-cmn 16392  df-psmet 17927  df-xmet 17928  df-met 17929  df-bl 17930  df-mopn 17931  df-fbas 17932  df-fg 17933  df-cnfld 17937  df-top 18628  df-bases 18630  df-topon 18631  df-topsp 18632  df-cld 18748  df-ntr 18749  df-cls 18750  df-nei 18827  df-lp 18865  df-perf 18866  df-cn 18956  df-cnp 18957  df-haus 19044  df-tx 19260  df-hmeo 19453  df-fil 19544  df-fm 19636  df-flim 19637  df-flf 19638  df-xms 20020  df-ms 20021  df-tms 20022  df-cncf 20579  df-limc 21467  df-dv 21468  df-log 22134  df-em 22512
This theorem is referenced by:  emcl  22522  harmonicbnd  22523  harmonicbnd2  22524
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