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Theorem harmonic 12280
Description: The harmonic series  H diverges. This fact follows from the stronger emcl 20259, which establishes that the harmonic series grows as  log n  +  gamma  + o(1), but this uses a more elementary method, attributed to Nicole Oresme (1323-1382). (Contributed by Mario Carneiro, 11-Jul-2014.)
Hypotheses
Ref Expression
harmonic.1  |-  F  =  ( n  e.  NN  |->  ( 1  /  n
) )
harmonic.2  |-  H  =  seq  1 (  +  ,  F )
Assertion
Ref Expression
harmonic  |-  -.  H  e.  dom  ~~>

Proof of Theorem harmonic
StepHypRef Expression
1 nn0uz 10230 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
2 0z 10003 . . . . 5  |-  0  e.  ZZ
32a1i 12 . . . 4  |-  ( H  e.  dom  ~~>  ->  0  e.  ZZ )
4 1ex 8801 . . . . . 6  |-  1  e.  _V
54fvconst2 5663 . . . . 5  |-  ( k  e.  NN0  ->  ( ( NN0  X.  { 1 } ) `  k
)  =  1 )
65adantl 454 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN0 )  ->  (
( NN0  X.  { 1 } ) `  k
)  =  1 )
7 1re 8805 . . . . 5  |-  1  e.  RR
87a1i 12 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN0 )  ->  1  e.  RR )
9 harmonic.2 . . . . . . 7  |-  H  =  seq  1 (  +  ,  F )
109eleq1i 2321 . . . . . 6  |-  ( H  e.  dom  ~~>  <->  seq  1
(  +  ,  F
)  e.  dom  ~~>  )
1110biimpi 188 . . . . 5  |-  ( H  e.  dom  ~~>  ->  seq  1 (  +  ,  F )  e.  dom  ~~>  )
12 oveq2 5800 . . . . . . . . 9  |-  ( n  =  k  ->  (
1  /  n )  =  ( 1  / 
k ) )
13 harmonic.1 . . . . . . . . 9  |-  F  =  ( n  e.  NN  |->  ( 1  /  n
) )
14 ovex 5817 . . . . . . . . 9  |-  ( 1  /  k )  e. 
_V
1512, 13, 14fvmpt 5536 . . . . . . . 8  |-  ( k  e.  NN  ->  ( F `  k )  =  ( 1  / 
k ) )
16 nnrecre 9750 . . . . . . . 8  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
1715, 16eqeltrd 2332 . . . . . . 7  |-  ( k  e.  NN  ->  ( F `  k )  e.  RR )
1817adantl 454 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN )  ->  ( F `  k )  e.  RR )
19 nnrp 10331 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR+ )
2019rpreccld 10368 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR+ )
2120rpge0d 10362 . . . . . . . 8  |-  ( k  e.  NN  ->  0  <_  ( 1  /  k
) )
2221, 15breqtrrd 4023 . . . . . . 7  |-  ( k  e.  NN  ->  0  <_  ( F `  k
) )
2322adantl 454 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN )  ->  0  <_  ( F `  k
) )
24 nnre 9721 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR )
2524lep1d 9656 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  <_  ( k  +  1 ) )
26 nngt0 9743 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  k )
27 peano2re 8953 . . . . . . . . . . 11  |-  ( k  e.  RR  ->  (
k  +  1 )  e.  RR )
2824, 27syl 17 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  RR )
29 peano2nn 9726 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
3029nngt0d 9757 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  ( k  +  1 ) )
31 lerec 9606 . . . . . . . . . 10  |-  ( ( ( k  e.  RR  /\  0  <  k )  /\  ( ( k  +  1 )  e.  RR  /\  0  < 
( k  +  1 ) ) )  -> 
( k  <_  (
k  +  1 )  <-> 
( 1  /  (
k  +  1 ) )  <_  ( 1  /  k ) ) )
3224, 26, 28, 30, 31syl22anc 1188 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
k  <_  ( k  +  1 )  <->  ( 1  /  ( k  +  1 ) )  <_ 
( 1  /  k
) ) )
3325, 32mpbid 203 . . . . . . . 8  |-  ( k  e.  NN  ->  (
1  /  ( k  +  1 ) )  <_  ( 1  / 
k ) )
34 oveq2 5800 . . . . . . . . . 10  |-  ( n  =  ( k  +  1 )  ->  (
1  /  n )  =  ( 1  / 
( k  +  1 ) ) )
35 ovex 5817 . . . . . . . . . 10  |-  ( 1  /  ( k  +  1 ) )  e. 
_V
3634, 13, 35fvmpt 5536 . . . . . . . . 9  |-  ( ( k  +  1 )  e.  NN  ->  ( F `  ( k  +  1 ) )  =  ( 1  / 
( k  +  1 ) ) )
3729, 36syl 17 . . . . . . . 8  |-  ( k  e.  NN  ->  ( F `  ( k  +  1 ) )  =  ( 1  / 
( k  +  1 ) ) )
3833, 37, 153brtr4d 4027 . . . . . . 7  |-  ( k  e.  NN  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
3938adantl 454 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
40 oveq2 5800 . . . . . . . . 9  |-  ( k  =  j  ->  (
2 ^ k )  =  ( 2 ^ j ) )
4140fveq2d 5462 . . . . . . . . 9  |-  ( k  =  j  ->  ( F `  ( 2 ^ k ) )  =  ( F `  ( 2 ^ j
) ) )
4240, 41oveq12d 5810 . . . . . . . 8  |-  ( k  =  j  ->  (
( 2 ^ k
)  x.  ( F `
 ( 2 ^ k ) ) )  =  ( ( 2 ^ j )  x.  ( F `  (
2 ^ j ) ) ) )
43 fconstmpt 4720 . . . . . . . . 9  |-  ( NN0 
X.  { 1 } )  =  ( k  e.  NN0  |->  1 )
44 2nn 9845 . . . . . . . . . . . . . 14  |-  2  e.  NN
45 nnexpcl 11083 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
4644, 45mpan 654 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( 2 ^ k )  e.  NN )
47 oveq2 5800 . . . . . . . . . . . . . 14  |-  ( n  =  ( 2 ^ k )  ->  (
1  /  n )  =  ( 1  / 
( 2 ^ k
) ) )
48 ovex 5817 . . . . . . . . . . . . . 14  |-  ( 1  /  ( 2 ^ k ) )  e. 
_V
4947, 13, 48fvmpt 5536 . . . . . . . . . . . . 13  |-  ( ( 2 ^ k )  e.  NN  ->  ( F `  ( 2 ^ k ) )  =  ( 1  / 
( 2 ^ k
) ) )
5046, 49syl 17 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( F `
 ( 2 ^ k ) )  =  ( 1  /  (
2 ^ k ) ) )
5150oveq2d 5808 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) )  =  ( ( 2 ^ k )  x.  (
1  /  ( 2 ^ k ) ) ) )
52 nncn 9722 . . . . . . . . . . . . 13  |-  ( ( 2 ^ k )  e.  NN  ->  (
2 ^ k )  e.  CC )
53 nnne0 9746 . . . . . . . . . . . . 13  |-  ( ( 2 ^ k )  e.  NN  ->  (
2 ^ k )  =/=  0 )
5452, 53recidd 9499 . . . . . . . . . . . 12  |-  ( ( 2 ^ k )  e.  NN  ->  (
( 2 ^ k
)  x.  ( 1  /  ( 2 ^ k ) ) )  =  1 )
5546, 54syl 17 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( 2 ^ k )  x.  ( 1  / 
( 2 ^ k
) ) )  =  1 )
5651, 55eqtrd 2290 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) )  =  1 )
5756mpteq2ia 4076 . . . . . . . . 9  |-  ( k  e.  NN0  |->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) ) )  =  ( k  e. 
NN0  |->  1 )
5843, 57eqtr4i 2281 . . . . . . . 8  |-  ( NN0 
X.  { 1 } )  =  ( k  e.  NN0  |->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) ) )
59 ovex 5817 . . . . . . . 8  |-  ( ( 2 ^ j )  x.  ( F `  ( 2 ^ j
) ) )  e. 
_V
6042, 58, 59fvmpt 5536 . . . . . . 7  |-  ( j  e.  NN0  ->  ( ( NN0  X.  { 1 } ) `  j
)  =  ( ( 2 ^ j )  x.  ( F `  ( 2 ^ j
) ) ) )
6160adantl 454 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN0 )  ->  (
( NN0  X.  { 1 } ) `  j
)  =  ( ( 2 ^ j )  x.  ( F `  ( 2 ^ j
) ) ) )
6218, 23, 39, 61climcnds 12273 . . . . 5  |-  ( H  e.  dom  ~~>  ->  (  seq  1 (  +  ,  F )  e.  dom  ~~>  <->  seq  0 (  +  , 
( NN0  X.  { 1 } ) )  e. 
dom 
~~>  ) )
6311, 62mpbid 203 . . . 4  |-  ( H  e.  dom  ~~>  ->  seq  0 (  +  , 
( NN0  X.  { 1 } ) )  e. 
dom 
~~>  )
641, 3, 6, 8, 63isumrecl 12194 . . 3  |-  ( H  e.  dom  ~~>  ->  sum_ k  e.  NN0  1  e.  RR )
65 arch 9930 . . 3  |-  ( sum_ k  e.  NN0  1  e.  RR  ->  E. j  e.  NN  sum_ k  e.  NN0  1  <  j )
6664, 65syl 17 . 2  |-  ( H  e.  dom  ~~>  ->  E. j  e.  NN  sum_ k  e.  NN0  1  <  j )
67 fzfid 11002 . . . . . . 7  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
1 ... j )  e. 
Fin )
68 ax-1cn 8763 . . . . . . 7  |-  1  e.  CC
69 fsumconst 12218 . . . . . . 7  |-  ( ( ( 1 ... j
)  e.  Fin  /\  1  e.  CC )  -> 
sum_ k  e.  ( 1 ... j ) 1  =  ( (
# `  ( 1 ... j ) )  x.  1 ) )
7067, 68, 69sylancl 646 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  sum_ k  e.  ( 1 ... j
) 1  =  ( ( # `  (
1 ... j ) )  x.  1 ) )
71 nnnn0 9940 . . . . . . . . 9  |-  ( j  e.  NN  ->  j  e.  NN0 )
7271adantl 454 . . . . . . . 8  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  j  e.  NN0 )
73 hashfz1 11312 . . . . . . . 8  |-  ( j  e.  NN0  ->  ( # `  ( 1 ... j
) )  =  j )
7472, 73syl 17 . . . . . . 7  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  ( # `
 ( 1 ... j ) )  =  j )
7574oveq1d 5807 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
( # `  ( 1 ... j ) )  x.  1 )  =  ( j  x.  1 ) )
76 nncn 9722 . . . . . . . 8  |-  ( j  e.  NN  ->  j  e.  CC )
7776adantl 454 . . . . . . 7  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  j  e.  CC )
7877mulid1d 8820 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
j  x.  1 )  =  j )
7970, 75, 783eqtrd 2294 . . . . 5  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  sum_ k  e.  ( 1 ... j
) 1  =  j )
802a1i 12 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  0  e.  ZZ )
81 elfznn 10786 . . . . . . . . 9  |-  ( k  e.  ( 1 ... j )  ->  k  e.  NN )
82 nnnn0 9940 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
8381, 82syl 17 . . . . . . . 8  |-  ( k  e.  ( 1 ... j )  ->  k  e.  NN0 )
8483ssriv 3159 . . . . . . 7  |-  ( 1 ... j )  C_  NN0
8584a1i 12 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
1 ... j )  C_  NN0 )
865adantl 454 . . . . . 6  |-  ( ( ( H  e.  dom  ~~>  /\  j  e.  NN )  /\  k  e.  NN0 )  ->  ( ( NN0 
X.  { 1 } ) `  k )  =  1 )
877a1i 12 . . . . . 6  |-  ( ( ( H  e.  dom  ~~>  /\  j  e.  NN )  /\  k  e.  NN0 )  ->  1  e.  RR )
88 0le1 9265 . . . . . . 7  |-  0  <_  1
8988a1i 12 . . . . . 6  |-  ( ( ( H  e.  dom  ~~>  /\  j  e.  NN )  /\  k  e.  NN0 )  ->  0  <_  1
)
9063adantr 453 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  seq  0 (  +  , 
( NN0  X.  { 1 } ) )  e. 
dom 
~~>  )
911, 80, 67, 85, 86, 87, 89, 90isumless 12267 . . . . 5  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  sum_ k  e.  ( 1 ... j
) 1  <_  sum_ k  e.  NN0  1 )
9279, 91eqbrtrrd 4019 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  j  <_ 
sum_ k  e.  NN0  1 )
93 nnre 9721 . . . . 5  |-  ( j  e.  NN  ->  j  e.  RR )
94 lenlt 8869 . . . . 5  |-  ( ( j  e.  RR  /\  sum_ k  e.  NN0  1  e.  RR )  ->  (
j  <_  sum_ k  e. 
NN0  1  <->  -.  sum_ k  e.  NN0  1  <  j
) )
9593, 64, 94syl2anr 466 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
j  <_  sum_ k  e. 
NN0  1  <->  -.  sum_ k  e.  NN0  1  <  j
) )
9692, 95mpbid 203 . . 3  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  -.  sum_ k  e.  NN0  1  <  j )
9796nrexdv 2621 . 2  |-  ( H  e.  dom  ~~>  ->  -.  E. j  e.  NN  sum_ k  e.  NN0  1  < 
j )
9866, 97pm2.65i 167 1  |-  -.  H  e.  dom  ~~>
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2519    C_ wss 3127   {csn 3614   class class class wbr 3997    e. cmpt 4051    X. cxp 4659   dom cdm 4661   ` cfv 4673  (class class class)co 5792   Fincfn 6831   CCcc 8703   RRcr 8704   0cc0 8705   1c1 8706    + caddc 8708    x. cmul 8710    < clt 8835    <_ cle 8836    / cdiv 9391   NNcn 9714   2c2 9763   NN0cn0 9933   ZZcz 9992   ...cfz 10749    seq cseq 11013   ^cexp 11071   #chash 11304    ~~> cli 11924   sum_csu 12124
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-pm 6743  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9934  df-z 9993  df-uz 10199  df-rp 10323  df-ico 10629  df-fz 10750  df-fzo 10838  df-fl 10892  df-seq 11014  df-exp 11072  df-hash 11305  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-clim 11928  df-rlim 11929  df-sum 12125
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