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Theorem harmonic 13321
Description: The harmonic series  H diverges. This fact follows from the stronger emcl 22396, 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). This is Metamath 100 proof #34. (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
Dummy variables  k 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0uz 10895 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
2 0zd 10658 . . . 4  |-  ( H  e.  dom  ~~>  ->  0  e.  ZZ )
3 1ex 9381 . . . . . 6  |-  1  e.  _V
43fvconst2 5933 . . . . 5  |-  ( k  e.  NN0  ->  ( ( NN0  X.  { 1 } ) `  k
)  =  1 )
54adantl 466 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN0 )  ->  (
( NN0  X.  { 1 } ) `  k
)  =  1 )
6 1red 9401 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN0 )  ->  1  e.  RR )
7 harmonic.2 . . . . . . 7  |-  H  =  seq 1 (  +  ,  F )
87eleq1i 2506 . . . . . 6  |-  ( H  e.  dom  ~~>  <->  seq 1
(  +  ,  F
)  e.  dom  ~~>  )
98biimpi 194 . . . . 5  |-  ( H  e.  dom  ~~>  ->  seq 1 (  +  ,  F )  e.  dom  ~~>  )
10 oveq2 6099 . . . . . . . . 9  |-  ( n  =  k  ->  (
1  /  n )  =  ( 1  / 
k ) )
11 harmonic.1 . . . . . . . . 9  |-  F  =  ( n  e.  NN  |->  ( 1  /  n
) )
12 ovex 6116 . . . . . . . . 9  |-  ( 1  /  k )  e. 
_V
1310, 11, 12fvmpt 5774 . . . . . . . 8  |-  ( k  e.  NN  ->  ( F `  k )  =  ( 1  / 
k ) )
14 nnrecre 10358 . . . . . . . 8  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
1513, 14eqeltrd 2517 . . . . . . 7  |-  ( k  e.  NN  ->  ( F `  k )  e.  RR )
1615adantl 466 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN )  ->  ( F `  k )  e.  RR )
17 nnrp 11000 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR+ )
1817rpreccld 11037 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR+ )
1918rpge0d 11031 . . . . . . . 8  |-  ( k  e.  NN  ->  0  <_  ( 1  /  k
) )
2019, 13breqtrrd 4318 . . . . . . 7  |-  ( k  e.  NN  ->  0  <_  ( F `  k
) )
2120adantl 466 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN )  ->  0  <_  ( F `  k
) )
22 nnre 10329 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR )
2322lep1d 10264 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  <_  ( k  +  1 ) )
24 nngt0 10351 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  k )
25 peano2re 9542 . . . . . . . . . . 11  |-  ( k  e.  RR  ->  (
k  +  1 )  e.  RR )
2622, 25syl 16 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  RR )
27 peano2nn 10334 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
2827nngt0d 10365 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  ( k  +  1 ) )
29 lerec 10214 . . . . . . . . . 10  |-  ( ( ( k  e.  RR  /\  0  <  k )  /\  ( ( k  +  1 )  e.  RR  /\  0  < 
( k  +  1 ) ) )  -> 
( k  <_  (
k  +  1 )  <-> 
( 1  /  (
k  +  1 ) )  <_  ( 1  /  k ) ) )
3022, 24, 26, 28, 29syl22anc 1219 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
k  <_  ( k  +  1 )  <->  ( 1  /  ( k  +  1 ) )  <_ 
( 1  /  k
) ) )
3123, 30mpbid 210 . . . . . . . 8  |-  ( k  e.  NN  ->  (
1  /  ( k  +  1 ) )  <_  ( 1  / 
k ) )
32 oveq2 6099 . . . . . . . . . 10  |-  ( n  =  ( k  +  1 )  ->  (
1  /  n )  =  ( 1  / 
( k  +  1 ) ) )
33 ovex 6116 . . . . . . . . . 10  |-  ( 1  /  ( k  +  1 ) )  e. 
_V
3432, 11, 33fvmpt 5774 . . . . . . . . 9  |-  ( ( k  +  1 )  e.  NN  ->  ( F `  ( k  +  1 ) )  =  ( 1  / 
( k  +  1 ) ) )
3527, 34syl 16 . . . . . . . 8  |-  ( k  e.  NN  ->  ( F `  ( k  +  1 ) )  =  ( 1  / 
( k  +  1 ) ) )
3631, 35, 133brtr4d 4322 . . . . . . 7  |-  ( k  e.  NN  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
3736adantl 466 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
38 oveq2 6099 . . . . . . . . 9  |-  ( k  =  j  ->  (
2 ^ k )  =  ( 2 ^ j ) )
3938fveq2d 5695 . . . . . . . . 9  |-  ( k  =  j  ->  ( F `  ( 2 ^ k ) )  =  ( F `  ( 2 ^ j
) ) )
4038, 39oveq12d 6109 . . . . . . . 8  |-  ( k  =  j  ->  (
( 2 ^ k
)  x.  ( F `
 ( 2 ^ k ) ) )  =  ( ( 2 ^ j )  x.  ( F `  (
2 ^ j ) ) ) )
41 fconstmpt 4882 . . . . . . . . 9  |-  ( NN0 
X.  { 1 } )  =  ( k  e.  NN0  |->  1 )
42 2nn 10479 . . . . . . . . . . . . . 14  |-  2  e.  NN
43 nnexpcl 11878 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
4442, 43mpan 670 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( 2 ^ k )  e.  NN )
45 oveq2 6099 . . . . . . . . . . . . . 14  |-  ( n  =  ( 2 ^ k )  ->  (
1  /  n )  =  ( 1  / 
( 2 ^ k
) ) )
46 ovex 6116 . . . . . . . . . . . . . 14  |-  ( 1  /  ( 2 ^ k ) )  e. 
_V
4745, 11, 46fvmpt 5774 . . . . . . . . . . . . 13  |-  ( ( 2 ^ k )  e.  NN  ->  ( F `  ( 2 ^ k ) )  =  ( 1  / 
( 2 ^ k
) ) )
4844, 47syl 16 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( F `
 ( 2 ^ k ) )  =  ( 1  /  (
2 ^ k ) ) )
4948oveq2d 6107 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) )  =  ( ( 2 ^ k )  x.  (
1  /  ( 2 ^ k ) ) ) )
50 nncn 10330 . . . . . . . . . . . . 13  |-  ( ( 2 ^ k )  e.  NN  ->  (
2 ^ k )  e.  CC )
51 nnne0 10354 . . . . . . . . . . . . 13  |-  ( ( 2 ^ k )  e.  NN  ->  (
2 ^ k )  =/=  0 )
5250, 51recidd 10102 . . . . . . . . . . . 12  |-  ( ( 2 ^ k )  e.  NN  ->  (
( 2 ^ k
)  x.  ( 1  /  ( 2 ^ k ) ) )  =  1 )
5344, 52syl 16 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( 2 ^ k )  x.  ( 1  / 
( 2 ^ k
) ) )  =  1 )
5449, 53eqtrd 2475 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) )  =  1 )
5554mpteq2ia 4374 . . . . . . . . 9  |-  ( k  e.  NN0  |->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) ) )  =  ( k  e. 
NN0  |->  1 )
5641, 55eqtr4i 2466 . . . . . . . 8  |-  ( NN0 
X.  { 1 } )  =  ( k  e.  NN0  |->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) ) )
57 ovex 6116 . . . . . . . 8  |-  ( ( 2 ^ j )  x.  ( F `  ( 2 ^ j
) ) )  e. 
_V
5840, 56, 57fvmpt 5774 . . . . . . 7  |-  ( j  e.  NN0  ->  ( ( NN0  X.  { 1 } ) `  j
)  =  ( ( 2 ^ j )  x.  ( F `  ( 2 ^ j
) ) ) )
5958adantl 466 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN0 )  ->  (
( NN0  X.  { 1 } ) `  j
)  =  ( ( 2 ^ j )  x.  ( F `  ( 2 ^ j
) ) ) )
6016, 21, 37, 59climcnds 13314 . . . . 5  |-  ( H  e.  dom  ~~>  ->  (  seq 1 (  +  ,  F )  e.  dom  ~~>  <->  seq 0 (  +  , 
( NN0  X.  { 1 } ) )  e. 
dom 
~~>  ) )
619, 60mpbid 210 . . . 4  |-  ( H  e.  dom  ~~>  ->  seq 0 (  +  , 
( NN0  X.  { 1 } ) )  e. 
dom 
~~>  )
621, 2, 5, 6, 61isumrecl 13232 . . 3  |-  ( H  e.  dom  ~~>  ->  sum_ k  e.  NN0  1  e.  RR )
63 arch 10576 . . 3  |-  ( sum_ k  e.  NN0  1  e.  RR  ->  E. j  e.  NN  sum_ k  e.  NN0  1  <  j )
6462, 63syl 16 . 2  |-  ( H  e.  dom  ~~>  ->  E. j  e.  NN  sum_ k  e.  NN0  1  <  j )
65 fzfid 11795 . . . . . . 7  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
1 ... j )  e. 
Fin )
66 ax-1cn 9340 . . . . . . 7  |-  1  e.  CC
67 fsumconst 13257 . . . . . . 7  |-  ( ( ( 1 ... j
)  e.  Fin  /\  1  e.  CC )  -> 
sum_ k  e.  ( 1 ... j ) 1  =  ( (
# `  ( 1 ... j ) )  x.  1 ) )
6865, 66, 67sylancl 662 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  sum_ k  e.  ( 1 ... j
) 1  =  ( ( # `  (
1 ... j ) )  x.  1 ) )
69 nnnn0 10586 . . . . . . . . 9  |-  ( j  e.  NN  ->  j  e.  NN0 )
7069adantl 466 . . . . . . . 8  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  j  e.  NN0 )
71 hashfz1 12117 . . . . . . . 8  |-  ( j  e.  NN0  ->  ( # `  ( 1 ... j
) )  =  j )
7270, 71syl 16 . . . . . . 7  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  ( # `
 ( 1 ... j ) )  =  j )
7372oveq1d 6106 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
( # `  ( 1 ... j ) )  x.  1 )  =  ( j  x.  1 ) )
74 nncn 10330 . . . . . . . 8  |-  ( j  e.  NN  ->  j  e.  CC )
7574adantl 466 . . . . . . 7  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  j  e.  CC )
7675mulid1d 9403 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
j  x.  1 )  =  j )
7768, 73, 763eqtrd 2479 . . . . 5  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  sum_ k  e.  ( 1 ... j
) 1  =  j )
78 0zd 10658 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  0  e.  ZZ )
79 elfznn 11478 . . . . . . . . 9  |-  ( k  e.  ( 1 ... j )  ->  k  e.  NN )
80 nnnn0 10586 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
8179, 80syl 16 . . . . . . . 8  |-  ( k  e.  ( 1 ... j )  ->  k  e.  NN0 )
8281ssriv 3360 . . . . . . 7  |-  ( 1 ... j )  C_  NN0
8382a1i 11 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
1 ... j )  C_  NN0 )
844adantl 466 . . . . . 6  |-  ( ( ( H  e.  dom  ~~>  /\  j  e.  NN )  /\  k  e.  NN0 )  ->  ( ( NN0 
X.  { 1 } ) `  k )  =  1 )
85 1red 9401 . . . . . 6  |-  ( ( ( H  e.  dom  ~~>  /\  j  e.  NN )  /\  k  e.  NN0 )  ->  1  e.  RR )
86 0le1 9863 . . . . . . 7  |-  0  <_  1
8786a1i 11 . . . . . 6  |-  ( ( ( H  e.  dom  ~~>  /\  j  e.  NN )  /\  k  e.  NN0 )  ->  0  <_  1
)
8861adantr 465 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  seq 0 (  +  , 
( NN0  X.  { 1 } ) )  e. 
dom 
~~>  )
891, 78, 65, 83, 84, 85, 87, 88isumless 13308 . . . . 5  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  sum_ k  e.  ( 1 ... j
) 1  <_  sum_ k  e.  NN0  1 )
9077, 89eqbrtrrd 4314 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  j  <_ 
sum_ k  e.  NN0  1 )
91 nnre 10329 . . . . 5  |-  ( j  e.  NN  ->  j  e.  RR )
92 lenlt 9453 . . . . 5  |-  ( ( j  e.  RR  /\  sum_ k  e.  NN0  1  e.  RR )  ->  (
j  <_  sum_ k  e. 
NN0  1  <->  -.  sum_ k  e.  NN0  1  <  j
) )
9391, 62, 92syl2anr 478 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
j  <_  sum_ k  e. 
NN0  1  <->  -.  sum_ k  e.  NN0  1  <  j
) )
9490, 93mpbid 210 . . 3  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  -.  sum_ k  e.  NN0  1  <  j )
9594nrexdv 2819 . 2  |-  ( H  e.  dom  ~~>  ->  -.  E. j  e.  NN  sum_ k  e.  NN0  1  < 
j )
9664, 95pm2.65i 173 1  |-  -.  H  e.  dom  ~~>
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2716    C_ wss 3328   {csn 3877   class class class wbr 4292    e. cmpt 4350    X. cxp 4838   dom cdm 4840   ` cfv 5418  (class class class)co 6091   Fincfn 7310   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287    < clt 9418    <_ cle 9419    / cdiv 9993   NNcn 10322   2c2 10371   NN0cn0 10579   ...cfz 11437    seqcseq 11806   ^cexp 11865   #chash 12103    ~~> cli 12962   sum_csu 13163
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-ico 11306  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-rlim 12967  df-sum 13164
This theorem is referenced by: (None)
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