Theorem harmonic 14591

Description: The harmonic series H diverges. This fact follows from the stronger emcl 24729, 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.

Step	Hyp	Ref	Expression
1		nn0uz 11722	. . . 4 |- NN0 = ( ZZ>= ` 0 )
2		0zd 11389	. . . 4 |- ( H e. dom ~~> -> 0 e. ZZ )
3		1ex 10035	. . . . . 6 |- 1 e. _V
4	3	fvconst2 6469	. . . . 5 |- ( k e. NN0 -> ( ( NN0 X. { 1 } ) ` k ) = 1 )
5	4	adantl 482	. . . 4 |- ( ( H e. dom ~~> /\ k e. NN0 ) -> ( ( NN0 X. { 1 } ) ` k ) = 1 )
6		1red 10055	. . . 4 |- ( ( H e. dom ~~> /\ k e. NN0 ) -> 1 e. RR )
7		harmonic.2	. . . . . . 7 |- H = seq 1 ( + , F )
8	7	eleq1i 2692	. . . . . 6 |- ( H e. dom ~~> <-> seq 1 ( + , F ) e. dom ~~> )
9	8	biimpi 206	. . . . 5 |- ( H e. dom ~~> -> seq 1 ( + , F ) e. dom ~~> )
10		oveq2 6658	. . . . . . . . 9 |- ( n = k -> ( 1 / n ) = ( 1 / k ) )
11		harmonic.1	. . . . . . . . 9 |- F = ( n e. NN |-> ( 1 / n ) )
12		ovex 6678	. . . . . . . . 9 |- ( 1 / k ) e. _V
13	10, 11, 12	fvmpt 6282	. . . . . . . 8 |- ( k e. NN -> ( F ` k ) = ( 1 / k ) )
14		nnrecre 11057	. . . . . . . 8 |- ( k e. NN -> ( 1 / k ) e. RR )
15	13, 14	eqeltrd 2701	. . . . . . 7 |- ( k e. NN -> ( F ` k ) e. RR )
16	15	adantl 482	. . . . . 6 |- ( ( H e. dom ~~> /\ k e. NN ) -> ( F ` k ) e. RR )
17		nnrp 11842	. . . . . . . . . 10 |- ( k e. NN -> k e. RR+ )
18	17	rpreccld 11882	. . . . . . . . 9 |- ( k e. NN -> ( 1 / k ) e. RR+ )
19	18	rpge0d 11876	. . . . . . . 8 |- ( k e. NN -> 0 <_ ( 1 / k ) )
20	19, 13	breqtrrd 4681	. . . . . . 7 |- ( k e. NN -> 0 <_ ( F ` k ) )
21	20	adantl 482	. . . . . 6 |- ( ( H e. dom ~~> /\ k e. NN ) -> 0 <_ ( F ` k ) )
22		nnre 11027	. . . . . . . . . 10 |- ( k e. NN -> k e. RR )
23	22	lep1d 10955	. . . . . . . . 9 |- ( k e. NN -> k <_ ( k + 1 ) )
24		nngt0 11049	. . . . . . . . . 10 |- ( k e. NN -> 0 < k )
25		peano2re 10209	. . . . . . . . . . 11 |- ( k e. RR -> ( k + 1 ) e. RR )
26	22, 25	syl 17	. . . . . . . . . 10 |- ( k e. NN -> ( k + 1 ) e. RR )
27		peano2nn 11032	. . . . . . . . . . 11 |- ( k e. NN -> ( k + 1 ) e. NN )
28	27	nngt0d 11064	. . . . . . . . . 10 |- ( k e. NN -> 0 < ( k + 1 ) )
29		lerec 10906	. . . . . . . . . 10 |- ( ( ( k e. RR /\ 0 < k ) /\ ( ( k + 1 ) e. RR /\ 0 < ( k + 1 ) ) ) -> ( k <_ ( k + 1 ) <-> ( 1 / ( k + 1 ) ) <_ ( 1 / k ) ) )
30	22, 24, 26, 28, 29	syl22anc 1327	. . . . . . . . 9 |- ( k e. NN -> ( k <_ ( k + 1 ) <-> ( 1 / ( k + 1 ) ) <_ ( 1 / k ) ) )
31	23, 30	mpbid 222	. . . . . . . 8 |- ( k e. NN -> ( 1 / ( k + 1 ) ) <_ ( 1 / k ) )
32		oveq2 6658	. . . . . . . . . 10 |- ( n = ( k + 1 ) -> ( 1 / n ) = ( 1 / ( k + 1 ) ) )
33		ovex 6678	. . . . . . . . . 10 |- ( 1 / ( k + 1 ) ) e. _V
34	32, 11, 33	fvmpt 6282	. . . . . . . . 9 |- ( ( k + 1 ) e. NN -> ( F ` ( k + 1 ) ) = ( 1 / ( k + 1 ) ) )
35	27, 34	syl 17	. . . . . . . 8 |- ( k e. NN -> ( F ` ( k + 1 ) ) = ( 1 / ( k + 1 ) ) )
36	31, 35, 13	3brtr4d 4685	. . . . . . 7 |- ( k e. NN -> ( F ` ( k + 1 ) ) <_ ( F ` k ) )
37	36	adantl 482	. . . . . 6 |- ( ( H e. dom ~~> /\ k e. NN ) -> ( F ` ( k + 1 ) ) <_ ( F ` k ) )
38		oveq2 6658	. . . . . . . . 9 |- ( k = j -> ( 2 ^ k ) = ( 2 ^ j ) )
39	38	fveq2d 6195	. . . . . . . . 9 |- ( k = j -> ( F ` ( 2 ^ k ) ) = ( F ` ( 2 ^ j ) ) )
40	38, 39	oveq12d 6668	. . . . . . . 8 |- ( k = j -> ( ( 2 ^ k ) x. ( F ` ( 2 ^ k ) ) ) = ( ( 2 ^ j ) x. ( F ` ( 2 ^ j ) ) ) )
41		fconstmpt 5163	. . . . . . . . 9 |- ( NN0 X. { 1 } ) = ( k e. NN0 |-> 1 )
42		2nn 11185	. . . . . . . . . . . . . 14 |- 2 e. NN
43		nnexpcl 12873	. . . . . . . . . . . . . 14 |- ( ( 2 e. NN /\ k e. NN0 ) -> ( 2 ^ k ) e. NN )
44	42, 43	mpan 706	. . . . . . . . . . . . 13 |- ( k e. NN0 -> ( 2 ^ k ) e. NN )
45		oveq2 6658	. . . . . . . . . . . . . 14 |- ( n = ( 2 ^ k ) -> ( 1 / n ) = ( 1 / ( 2 ^ k ) ) )
46		ovex 6678	. . . . . . . . . . . . . 14 |- ( 1 / ( 2 ^ k ) ) e. _V
47	45, 11, 46	fvmpt 6282	. . . . . . . . . . . . 13 |- ( ( 2 ^ k ) e. NN -> ( F ` ( 2 ^ k ) ) = ( 1 / ( 2 ^ k ) ) )
48	44, 47	syl 17	. . . . . . . . . . . 12 |- ( k e. NN0 -> ( F ` ( 2 ^ k ) ) = ( 1 / ( 2 ^ k ) ) )
49	48	oveq2d 6666	. . . . . . . . . . 11 |- ( k e. NN0 -> ( ( 2 ^ k ) x. ( F ` ( 2 ^ k ) ) ) = ( ( 2 ^ k ) x. ( 1 / ( 2 ^ k ) ) ) )
50		nncn 11028	. . . . . . . . . . . . 13 |- ( ( 2 ^ k ) e. NN -> ( 2 ^ k ) e. CC )
51		nnne0 11053	. . . . . . . . . . . . 13 |- ( ( 2 ^ k ) e. NN -> ( 2 ^ k ) =/= 0 )
52	50, 51	recidd 10796	. . . . . . . . . . . 12 |- ( ( 2 ^ k ) e. NN -> ( ( 2 ^ k ) x. ( 1 / ( 2 ^ k ) ) ) = 1 )
53	44, 52	syl 17	. . . . . . . . . . 11 |- ( k e. NN0 -> ( ( 2 ^ k ) x. ( 1 / ( 2 ^ k ) ) ) = 1 )
54	49, 53	eqtrd 2656	. . . . . . . . . 10 |- ( k e. NN0 -> ( ( 2 ^ k ) x. ( F ` ( 2 ^ k ) ) ) = 1 )
55	54	mpteq2ia 4740	. . . . . . . . 9 |- ( k e. NN0 |-> ( ( 2 ^ k ) x. ( F ` ( 2 ^ k ) ) ) ) = ( k e. NN0 |-> 1 )
56	41, 55	eqtr4i 2647	. . . . . . . 8 |- ( NN0 X. { 1 } ) = ( k e. NN0 |-> ( ( 2 ^ k ) x. ( F ` ( 2 ^ k ) ) ) )
57		ovex 6678	. . . . . . . 8 |- ( ( 2 ^ j ) x. ( F ` ( 2 ^ j ) ) ) e. _V
58	40, 56, 57	fvmpt 6282	. . . . . . 7 |- ( j e. NN0 -> ( ( NN0 X. { 1 } ) ` j ) = ( ( 2 ^ j ) x. ( F ` ( 2 ^ j ) ) ) )
59	58	adantl 482	. . . . . 6 |- ( ( H e. dom ~~> /\ j e. NN0 ) -> ( ( NN0 X. { 1 } ) ` j ) = ( ( 2 ^ j ) x. ( F ` ( 2 ^ j ) ) ) )
60	16, 21, 37, 59	climcnds 14583	. . . . 5 |- ( H e. dom ~~> -> ( seq 1 ( + , F ) e. dom ~~> <-> seq 0 ( + , ( NN0 X. { 1 } ) ) e. dom ~~> ) )
61	9, 60	mpbid 222	. . . 4 |- ( H e. dom ~~> -> seq 0 ( + , ( NN0 X. { 1 } ) ) e. dom ~~> )
62	1, 2, 5, 6, 61	isumrecl 14496	. . 3 |- ( H e. dom ~~> -> sum_ k e. NN0 1 e. RR )
63		arch 11289	. . 3 |- ( sum_ k e. NN0 1 e. RR -> E. j e. NN sum_ k e. NN0 1 < j )
64	62, 63	syl 17	. 2 |- ( H e. dom ~~> -> E. j e. NN sum_ k e. NN0 1 < j )
65		fzfid 12772	. . . . . . 7 |- ( ( H e. dom ~~> /\ j e. NN ) -> ( 1 ... j ) e. Fin )
66		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
67		fsumconst 14522	. . . . . . 7 |- ( ( ( 1 ... j ) e. Fin /\ 1 e. CC ) -> sum_ k e. ( 1 ... j ) 1 = ( ( # ` ( 1 ... j ) ) x. 1 ) )
68	65, 66, 67	sylancl 694	. . . . . 6 |- ( ( H e. dom ~~> /\ j e. NN ) -> sum_ k e. ( 1 ... j ) 1 = ( ( # ` ( 1 ... j ) ) x. 1 ) )
69		nnnn0 11299	. . . . . . . . 9 |- ( j e. NN -> j e. NN0 )
70	69	adantl 482	. . . . . . . 8 |- ( ( H e. dom ~~> /\ j e. NN ) -> j e. NN0 )
71		hashfz1 13134	. . . . . . . 8 |- ( j e. NN0 -> ( # ` ( 1 ... j ) ) = j )
72	70, 71	syl 17	. . . . . . 7 |- ( ( H e. dom ~~> /\ j e. NN ) -> ( # ` ( 1 ... j ) ) = j )
73	72	oveq1d 6665	. . . . . 6 |- ( ( H e. dom ~~> /\ j e. NN ) -> ( ( # ` ( 1 ... j ) ) x. 1 ) = ( j x. 1 ) )
74		nncn 11028	. . . . . . . 8 |- ( j e. NN -> j e. CC )
75	74	adantl 482	. . . . . . 7 |- ( ( H e. dom ~~> /\ j e. NN ) -> j e. CC )
76	75	mulid1d 10057	. . . . . 6 |- ( ( H e. dom ~~> /\ j e. NN ) -> ( j x. 1 ) = j )
77	68, 73, 76	3eqtrd 2660	. . . . 5 |- ( ( H e. dom ~~> /\ j e. NN ) -> sum_ k e. ( 1 ... j ) 1 = j )
78		0zd 11389	. . . . . 6 |- ( ( H e. dom ~~> /\ j e. NN ) -> 0 e. ZZ )
79		elfznn 12370	. . . . . . . . 9 |- ( k e. ( 1 ... j ) -> k e. NN )
80		nnnn0 11299	. . . . . . . . 9 |- ( k e. NN -> k e. NN0 )
81	79, 80	syl 17	. . . . . . . 8 |- ( k e. ( 1 ... j ) -> k e. NN0 )
82	81	ssriv 3607	. . . . . . 7 |- ( 1 ... j ) C_ NN0
83	82	a1i 11	. . . . . 6 |- ( ( H e. dom ~~> /\ j e. NN ) -> ( 1 ... j ) C_ NN0 )
84	4	adantl 482	. . . . . 6 |- ( ( ( H e. dom ~~> /\ j e. NN ) /\ k e. NN0 ) -> ( ( NN0 X. { 1 } ) ` k ) = 1 )
85		1red 10055	. . . . . 6 |- ( ( ( H e. dom ~~> /\ j e. NN ) /\ k e. NN0 ) -> 1 e. RR )
86		0le1 10551	. . . . . . 7 |- 0 <_ 1
87	86	a1i 11	. . . . . 6 |- ( ( ( H e. dom ~~> /\ j e. NN ) /\ k e. NN0 ) -> 0 <_ 1 )
88	61	adantr 481	. . . . . 6 |- ( ( H e. dom ~~> /\ j e. NN ) -> seq 0 ( + , ( NN0 X. { 1 } ) ) e. dom ~~> )
89	1, 78, 65, 83, 84, 85, 87, 88	isumless 14577	. . . . 5 |- ( ( H e. dom ~~> /\ j e. NN ) -> sum_ k e. ( 1 ... j ) 1 <_ sum_ k e. NN0 1 )
90	77, 89	eqbrtrrd 4677	. . . 4 |- ( ( H e. dom ~~> /\ j e. NN ) -> j <_ sum_ k e. NN0 1 )
91		nnre 11027	. . . . 5 |- ( j e. NN -> j e. RR )
92		lenlt 10116	. . . . 5 |- ( ( j e. RR /\ sum_ k e. NN0 1 e. RR ) -> ( j <_ sum_ k e. NN0 1 <-> -. sum_ k e. NN0 1 < j ) )
93	91, 62, 92	syl2anr 495	. . . 4 |- ( ( H e. dom ~~> /\ j e. NN ) -> ( j <_ sum_ k e. NN0 1 <-> -. sum_ k e. NN0 1 < j ) )
94	90, 93	mpbid 222	. . 3 |- ( ( H e. dom ~~> /\ j e. NN ) -> -. sum_ k e. NN0 1 < j )
95	94	nrexdv 3001	. 2 |- ( H e. dom ~~> -> -. E. j e. NN sum_ k e. NN0 1 < j )
96	64, 95	pm2.65i 185	1 |- -. H e. dom ~~>

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ...cfz 12326   seqcseq 12801   ^cexp 12860   #chash 13117   ~~> cli 14215   sum_csu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417

This theorem is referenced by: (None)