Theorem sermono 12833

Description: The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-Jun-2013.)

Hypotheses

Ref	Expression
sermono.1	|- ( ph -> K e. ( ZZ>= ` M ) )
sermono.2	|- ( ph -> N e. ( ZZ>= ` K ) )
sermono.3	|- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) e. RR )
sermono.4	|- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> 0 <_ ( F ` x ) )

Assertion

Ref	Expression
sermono	|- ( ph -> ( seq M ( + , F ) ` K ) <_ ( seq M ( + , F ) ` N ) )

Distinct variable groups:   x, F   x, K   x, M   x, N   ph, x

Proof of Theorem sermono

Dummy variables k y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sermono.2	. 2 |- ( ph -> N e. ( ZZ>= ` K ) )
2		elfzuz 12338	. . . 4 |- ( k e. ( K ... N ) -> k e. ( ZZ>= ` K ) )
3		sermono.1	. . . 4 |- ( ph -> K e. ( ZZ>= ` M ) )
4		uztrn 11704	. . . 4 |- ( ( k e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
5	2, 3, 4	syl2anr 495	. . 3 |- ( ( ph /\ k e. ( K ... N ) ) -> k e. ( ZZ>= ` M ) )
6		elfzuz3 12339	. . . . . . 7 |- ( k e. ( K ... N ) -> N e. ( ZZ>= ` k ) )
7	6	adantl 482	. . . . . 6 |- ( ( ph /\ k e. ( K ... N ) ) -> N e. ( ZZ>= ` k ) )
8		fzss2 12381	. . . . . 6 |- ( N e. ( ZZ>= ` k ) -> ( M ... k ) C_ ( M ... N ) )
9	7, 8	syl 17	. . . . 5 |- ( ( ph /\ k e. ( K ... N ) ) -> ( M ... k ) C_ ( M ... N ) )
10	9	sselda 3603	. . . 4 |- ( ( ( ph /\ k e. ( K ... N ) ) /\ x e. ( M ... k ) ) -> x e. ( M ... N ) )
11		sermono.3	. . . . 5 |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) e. RR )
12	11	adantlr 751	. . . 4 |- ( ( ( ph /\ k e. ( K ... N ) ) /\ x e. ( M ... N ) ) -> ( F ` x ) e. RR )
13	10, 12	syldan 487	. . 3 |- ( ( ( ph /\ k e. ( K ... N ) ) /\ x e. ( M ... k ) ) -> ( F ` x ) e. RR )
14		readdcl 10019	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( x + y ) e. RR )
15	14	adantl 482	. . 3 |- ( ( ( ph /\ k e. ( K ... N ) ) /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR )
16	5, 13, 15	seqcl 12821	. 2 |- ( ( ph /\ k e. ( K ... N ) ) -> ( seq M ( + , F ) ` k ) e. RR )
17		simpr 477	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ( K ... ( N - 1 ) ) )
18	3	adantr 481	. . . . . . . . 9 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> K e. ( ZZ>= ` M ) )
19		eluzelz 11697	. . . . . . . . 9 |- ( K e. ( ZZ>= ` M ) -> K e. ZZ )
20	18, 19	syl 17	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> K e. ZZ )
21	1	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> N e. ( ZZ>= ` K ) )
22		eluzelz 11697	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` K ) -> N e. ZZ )
23	21, 22	syl 17	. . . . . . . . 9 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> N e. ZZ )
24		peano2zm 11420	. . . . . . . . 9 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
25	23, 24	syl 17	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( N - 1 ) e. ZZ )
26		elfzelz 12342	. . . . . . . . 9 |- ( k e. ( K ... ( N - 1 ) ) -> k e. ZZ )
27	26	adantl 482	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ZZ )
28		1zzd 11408	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> 1 e. ZZ )
29		fzaddel 12375	. . . . . . . 8 |- ( ( ( K e. ZZ /\ ( N - 1 ) e. ZZ ) /\ ( k e. ZZ /\ 1 e. ZZ ) ) -> ( k e. ( K ... ( N - 1 ) ) <-> ( k + 1 ) e. ( ( K + 1 ) ... ( ( N - 1 ) + 1 ) ) ) )
30	20, 25, 27, 28, 29	syl22anc 1327	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k e. ( K ... ( N - 1 ) ) <-> ( k + 1 ) e. ( ( K + 1 ) ... ( ( N - 1 ) + 1 ) ) ) )
31	17, 30	mpbid 222	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( ( K + 1 ) ... ( ( N - 1 ) + 1 ) ) )
32		zcn 11382	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
33		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
34		npcan 10290	. . . . . . . . 9 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
35	32, 33, 34	sylancl 694	. . . . . . . 8 |- ( N e. ZZ -> ( ( N - 1 ) + 1 ) = N )
36	23, 35	syl 17	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( ( N - 1 ) + 1 ) = N )
37	36	oveq2d 6666	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( ( K + 1 ) ... ( ( N - 1 ) + 1 ) ) = ( ( K + 1 ) ... N ) )
38	31, 37	eleqtrd 2703	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( ( K + 1 ) ... N ) )
39		sermono.4	. . . . . . 7 |- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> 0 <_ ( F ` x ) )
40	39	ralrimiva 2966	. . . . . 6 |- ( ph -> A. x e. ( ( K + 1 ) ... N ) 0 <_ ( F ` x ) )
41	40	adantr 481	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> A. x e. ( ( K + 1 ) ... N ) 0 <_ ( F ` x ) )
42		fveq2 6191	. . . . . . 7 |- ( x = ( k + 1 ) -> ( F ` x ) = ( F ` ( k + 1 ) ) )
43	42	breq2d 4665	. . . . . 6 |- ( x = ( k + 1 ) -> ( 0 <_ ( F ` x ) <-> 0 <_ ( F ` ( k + 1 ) ) ) )
44	43	rspcv 3305	. . . . 5 |- ( ( k + 1 ) e. ( ( K + 1 ) ... N ) -> ( A. x e. ( ( K + 1 ) ... N ) 0 <_ ( F ` x ) -> 0 <_ ( F ` ( k + 1 ) ) ) )
45	38, 41, 44	sylc 65	. . . 4 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> 0 <_ ( F ` ( k + 1 ) ) )
46		fzelp1 12393	. . . . . . . 8 |- ( k e. ( K ... ( N - 1 ) ) -> k e. ( K ... ( ( N - 1 ) + 1 ) ) )
47	46	adantl 482	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ( K ... ( ( N - 1 ) + 1 ) ) )
48	36	oveq2d 6666	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( K ... ( ( N - 1 ) + 1 ) ) = ( K ... N ) )
49	47, 48	eleqtrd 2703	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ( K ... N ) )
50	49, 16	syldan 487	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F ) ` k ) e. RR )
51		fzss1 12380	. . . . . . . 8 |- ( K e. ( ZZ>= ` M ) -> ( K ... N ) C_ ( M ... N ) )
52	18, 51	syl 17	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( K ... N ) C_ ( M ... N ) )
53		fzp1elp1 12394	. . . . . . . . 9 |- ( k e. ( K ... ( N - 1 ) ) -> ( k + 1 ) e. ( K ... ( ( N - 1 ) + 1 ) ) )
54	53	adantl 482	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( K ... ( ( N - 1 ) + 1 ) ) )
55	54, 48	eleqtrd 2703	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( K ... N ) )
56	52, 55	sseldd 3604	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( M ... N ) )
57	11	ralrimiva 2966	. . . . . . 7 |- ( ph -> A. x e. ( M ... N ) ( F ` x ) e. RR )
58	57	adantr 481	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> A. x e. ( M ... N ) ( F ` x ) e. RR )
59	42	eleq1d 2686	. . . . . . 7 |- ( x = ( k + 1 ) -> ( ( F ` x ) e. RR <-> ( F ` ( k + 1 ) ) e. RR ) )
60	59	rspcv 3305	. . . . . 6 |- ( ( k + 1 ) e. ( M ... N ) -> ( A. x e. ( M ... N ) ( F ` x ) e. RR -> ( F ` ( k + 1 ) ) e. RR ) )
61	56, 58, 60	sylc 65	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( F ` ( k + 1 ) ) e. RR )
62	50, 61	addge01d 10615	. . . 4 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( 0 <_ ( F ` ( k + 1 ) ) <-> ( seq M ( + , F ) ` k ) <_ ( ( seq M ( + , F ) ` k ) + ( F ` ( k + 1 ) ) ) ) )
63	45, 62	mpbid 222	. . 3 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F ) ` k ) <_ ( ( seq M ( + , F ) ` k ) + ( F ` ( k + 1 ) ) ) )
64	49, 5	syldan 487	. . . 4 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ( ZZ>= ` M ) )
65		seqp1 12816	. . . 4 |- ( k e. ( ZZ>= ` M ) -> ( seq M ( + , F ) ` ( k + 1 ) ) = ( ( seq M ( + , F ) ` k ) + ( F ` ( k + 1 ) ) ) )
66	64, 65	syl 17	. . 3 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F ) ` ( k + 1 ) ) = ( ( seq M ( + , F ) ` k ) + ( F ` ( k + 1 ) ) ) )
67	63, 66	breqtrrd 4681	. 2 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F ) ` k ) <_ ( seq M ( + , F ) ` ( k + 1 ) ) )
68	1, 16, 67	monoord 12831	1 |- ( ph -> ( seq M ( + , F ) ` K ) <_ ( seq M ( + , F ) ` N ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802

This theorem is referenced by:  cvgcmp  14548  isumsup2  14578  climcnds  14583  ovolunlem1a  23264  mblfinlem2  33447