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Theorem harmonic 13632
Description: The harmonic series  H diverges. This fact follows from the stronger emcl 23076, 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 11115 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
2 0zd 10875 . . . 4  |-  ( H  e.  dom  ~~>  ->  0  e.  ZZ )
3 1ex 9590 . . . . . 6  |-  1  e.  _V
43fvconst2 6115 . . . . 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 9610 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN0 )  ->  1  e.  RR )
7 harmonic.2 . . . . . . 7  |-  H  =  seq 1 (  +  ,  F )
87eleq1i 2544 . . . . . 6  |-  ( H  e.  dom  ~~>  <->  seq 1
(  +  ,  F
)  e.  dom  ~~>  )
98biimpi 194 . . . . 5  |-  ( H  e.  dom  ~~>  ->  seq 1 (  +  ,  F )  e.  dom  ~~>  )
10 oveq2 6291 . . . . . . . . 9  |-  ( n  =  k  ->  (
1  /  n )  =  ( 1  / 
k ) )
11 harmonic.1 . . . . . . . . 9  |-  F  =  ( n  e.  NN  |->  ( 1  /  n
) )
12 ovex 6308 . . . . . . . . 9  |-  ( 1  /  k )  e. 
_V
1310, 11, 12fvmpt 5949 . . . . . . . 8  |-  ( k  e.  NN  ->  ( F `  k )  =  ( 1  / 
k ) )
14 nnrecre 10571 . . . . . . . 8  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
1513, 14eqeltrd 2555 . . . . . . 7  |-  ( k  e.  NN  ->  ( F `  k )  e.  RR )
1615adantl 466 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN )  ->  ( F `  k )  e.  RR )
17 nnrp 11228 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR+ )
1817rpreccld 11265 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR+ )
1918rpge0d 11259 . . . . . . . 8  |-  ( k  e.  NN  ->  0  <_  ( 1  /  k
) )
2019, 13breqtrrd 4473 . . . . . . 7  |-  ( k  e.  NN  ->  0  <_  ( F `  k
) )
2120adantl 466 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN )  ->  0  <_  ( F `  k
) )
22 nnre 10542 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR )
2322lep1d 10476 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  <_  ( k  +  1 ) )
24 nngt0 10564 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  k )
25 peano2re 9751 . . . . . . . . . . 11  |-  ( k  e.  RR  ->  (
k  +  1 )  e.  RR )
2622, 25syl 16 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  RR )
27 peano2nn 10547 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
2827nngt0d 10578 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  ( k  +  1 ) )
29 lerec 10426 . . . . . . . . . 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 1229 . . . . . . . . 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 6291 . . . . . . . . . 10  |-  ( n  =  ( k  +  1 )  ->  (
1  /  n )  =  ( 1  / 
( k  +  1 ) ) )
33 ovex 6308 . . . . . . . . . 10  |-  ( 1  /  ( k  +  1 ) )  e. 
_V
3432, 11, 33fvmpt 5949 . . . . . . . . 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 4477 . . . . . . 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 6291 . . . . . . . . 9  |-  ( k  =  j  ->  (
2 ^ k )  =  ( 2 ^ j ) )
3938fveq2d 5869 . . . . . . . . 9  |-  ( k  =  j  ->  ( F `  ( 2 ^ k ) )  =  ( F `  ( 2 ^ j
) ) )
4038, 39oveq12d 6301 . . . . . . . 8  |-  ( k  =  j  ->  (
( 2 ^ k
)  x.  ( F `
 ( 2 ^ k ) ) )  =  ( ( 2 ^ j )  x.  ( F `  (
2 ^ j ) ) ) )
41 fconstmpt 5042 . . . . . . . . 9  |-  ( NN0 
X.  { 1 } )  =  ( k  e.  NN0  |->  1 )
42 2nn 10692 . . . . . . . . . . . . . 14  |-  2  e.  NN
43 nnexpcl 12146 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
4442, 43mpan 670 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( 2 ^ k )  e.  NN )
45 oveq2 6291 . . . . . . . . . . . . . 14  |-  ( n  =  ( 2 ^ k )  ->  (
1  /  n )  =  ( 1  / 
( 2 ^ k
) ) )
46 ovex 6308 . . . . . . . . . . . . . 14  |-  ( 1  /  ( 2 ^ k ) )  e. 
_V
4745, 11, 46fvmpt 5949 . . . . . . . . . . . . 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 6299 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) )  =  ( ( 2 ^ k )  x.  (
1  /  ( 2 ^ k ) ) ) )
50 nncn 10543 . . . . . . . . . . . . 13  |-  ( ( 2 ^ k )  e.  NN  ->  (
2 ^ k )  e.  CC )
51 nnne0 10567 . . . . . . . . . . . . 13  |-  ( ( 2 ^ k )  e.  NN  ->  (
2 ^ k )  =/=  0 )
5250, 51recidd 10314 . . . . . . . . . . . 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 2508 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) )  =  1 )
5554mpteq2ia 4529 . . . . . . . . 9  |-  ( k  e.  NN0  |->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) ) )  =  ( k  e. 
NN0  |->  1 )
5641, 55eqtr4i 2499 . . . . . . . 8  |-  ( NN0 
X.  { 1 } )  =  ( k  e.  NN0  |->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) ) )
57 ovex 6308 . . . . . . . 8  |-  ( ( 2 ^ j )  x.  ( F `  ( 2 ^ j
) ) )  e. 
_V
5840, 56, 57fvmpt 5949 . . . . . . 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 13625 . . . . 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 13542 . . 3  |-  ( H  e.  dom  ~~>  ->  sum_ k  e.  NN0  1  e.  RR )
63 arch 10791 . . 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 12050 . . . . . . 7  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
1 ... j )  e. 
Fin )
66 ax-1cn 9549 . . . . . . 7  |-  1  e.  CC
67 fsumconst 13567 . . . . . . 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 10801 . . . . . . . . 9  |-  ( j  e.  NN  ->  j  e.  NN0 )
7069adantl 466 . . . . . . . 8  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  j  e.  NN0 )
71 hashfz1 12386 . . . . . . . 8  |-  ( j  e.  NN0  ->  ( # `  ( 1 ... j
) )  =  j )
7270, 71syl 16 . . . . . . 7  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  ( # `
 ( 1 ... j ) )  =  j )
7372oveq1d 6298 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
( # `  ( 1 ... j ) )  x.  1 )  =  ( j  x.  1 ) )
74 nncn 10543 . . . . . . . 8  |-  ( j  e.  NN  ->  j  e.  CC )
7574adantl 466 . . . . . . 7  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  j  e.  CC )
7675mulid1d 9612 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
j  x.  1 )  =  j )
7768, 73, 763eqtrd 2512 . . . . 5  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  sum_ k  e.  ( 1 ... j
) 1  =  j )
78 0zd 10875 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  0  e.  ZZ )
79 elfznn 11713 . . . . . . . . 9  |-  ( k  e.  ( 1 ... j )  ->  k  e.  NN )
80 nnnn0 10801 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
8179, 80syl 16 . . . . . . . 8  |-  ( k  e.  ( 1 ... j )  ->  k  e.  NN0 )
8281ssriv 3508 . . . . . . 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 9610 . . . . . 6  |-  ( ( ( H  e.  dom  ~~>  /\  j  e.  NN )  /\  k  e.  NN0 )  ->  1  e.  RR )
86 0le1 10075 . . . . . . 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 13619 . . . . 5  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  sum_ k  e.  ( 1 ... j
) 1  <_  sum_ k  e.  NN0  1 )
9077, 89eqbrtrrd 4469 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  j  <_ 
sum_ k  e.  NN0  1 )
91 nnre 10542 . . . . 5  |-  ( j  e.  NN  ->  j  e.  RR )
92 lenlt 9662 . . . . 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 2920 . 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 1379    e. wcel 1767   E.wrex 2815    C_ wss 3476   {csn 4027   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   dom cdm 4999   ` cfv 5587  (class class class)co 6283   Fincfn 7516   CCcc 9489   RRcr 9490   0cc0 9491   1c1 9492    + caddc 9494    x. cmul 9496    < clt 9627    <_ cle 9628    / cdiv 10205   NNcn 10535   2c2 10584   NN0cn0 10794   ...cfz 11671    seqcseq 12074   ^cexp 12133   #chash 12372    ~~> cli 13269   sum_csu 13470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-n0 10795  df-z 10864  df-uz 11082  df-rp 11220  df-ico 11534  df-fz 11672  df-fzo 11792  df-fl 11896  df-seq 12075  df-exp 12134  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273  df-rlim 13274  df-sum 13471
This theorem is referenced by: (None)
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