Theorem isermono 9237

Description: The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)

Hypotheses

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

Assertion

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

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

Proof of Theorem isermono

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

Step	Hyp	Ref	Expression
1		isermono.2	. 2 |- ( ph -> N e. ( ZZ>= ` K ) )
2		elfzuz 8886	. . . 4 |- ( k e. ( K ... N ) -> k e. ( ZZ>= ` K ) )
3		isermono.1	. . . 4 |- ( ph -> K e. ( ZZ>= ` M ) )
4		uztrn 8489	. . . 4 |- ( ( k e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
5	2, 3, 4	syl2anr 274	. . 3 |- ( ( ph /\ k e. ( K ... N ) ) -> k e. ( ZZ>= ` M ) )
6		reex 7015	. . . 4 |- RR e. _V
7	6	a1i 9	. . 3 |- ( ( ph /\ k e. ( K ... N ) ) -> RR e. _V )
8		isermono.3	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. RR )
9	8	adantlr 446	. . 3 |- ( ( ( ph /\ k e. ( K ... N ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. RR )
10		readdcl 7007	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( x + y ) e. RR )
11	10	adantl 262	. . 3 |- ( ( ( ph /\ k e. ( K ... N ) ) /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR )
12	5, 7, 9, 11	iseqcl 9223	. 2 |- ( ( ph /\ k e. ( K ... N ) ) -> ( seq M ( + , F , RR ) ` k ) e. RR )
13		simpr 103	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ( K ... ( N - 1 ) ) )
14	3	adantr 261	. . . . . . . . 9 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> K e. ( ZZ>= ` M ) )
15		eluzelz 8482	. . . . . . . . 9 |- ( K e. ( ZZ>= ` M ) -> K e. ZZ )
16	14, 15	syl 14	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> K e. ZZ )
17	1	adantr 261	. . . . . . . . . 10 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> N e. ( ZZ>= ` K ) )
18		eluzelz 8482	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` K ) -> N e. ZZ )
19	17, 18	syl 14	. . . . . . . . 9 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> N e. ZZ )
20		peano2zm 8283	. . . . . . . . 9 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
21	19, 20	syl 14	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( N - 1 ) e. ZZ )
22		elfzelz 8890	. . . . . . . . 9 |- ( k e. ( K ... ( N - 1 ) ) -> k e. ZZ )
23	22	adantl 262	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ZZ )
24		1zzd 8272	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> 1 e. ZZ )
25		fzaddel 8922	. . . . . . . 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 ) ) ) )
26	16, 21, 23, 24, 25	syl22anc 1136	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k e. ( K ... ( N - 1 ) ) <-> ( k + 1 ) e. ( ( K + 1 ) ... ( ( N - 1 ) + 1 ) ) ) )
27	13, 26	mpbid 135	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( ( K + 1 ) ... ( ( N - 1 ) + 1 ) ) )
28		zcn 8250	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
29		ax-1cn 6977	. . . . . . . . 9 |- 1 e. CC
30		npcan 7220	. . . . . . . . 9 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
31	28, 29, 30	sylancl 392	. . . . . . . 8 |- ( N e. ZZ -> ( ( N - 1 ) + 1 ) = N )
32	19, 31	syl 14	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( ( N - 1 ) + 1 ) = N )
33	32	oveq2d 5528	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( ( K + 1 ) ... ( ( N - 1 ) + 1 ) ) = ( ( K + 1 ) ... N ) )
34	27, 33	eleqtrd 2116	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( ( K + 1 ) ... N ) )
35		isermono.4	. . . . . . 7 |- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> 0 <_ ( F ` x ) )
36	35	ralrimiva 2392	. . . . . 6 |- ( ph -> A. x e. ( ( K + 1 ) ... N ) 0 <_ ( F ` x ) )
37	36	adantr 261	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> A. x e. ( ( K + 1 ) ... N ) 0 <_ ( F ` x ) )
38		fveq2 5178	. . . . . . 7 |- ( x = ( k + 1 ) -> ( F ` x ) = ( F ` ( k + 1 ) ) )
39	38	breq2d 3776	. . . . . 6 |- ( x = ( k + 1 ) -> ( 0 <_ ( F ` x ) <-> 0 <_ ( F ` ( k + 1 ) ) ) )
40	39	rspcv 2652	. . . . 5 |- ( ( k + 1 ) e. ( ( K + 1 ) ... N ) -> ( A. x e. ( ( K + 1 ) ... N ) 0 <_ ( F ` x ) -> 0 <_ ( F ` ( k + 1 ) ) ) )
41	34, 37, 40	sylc 56	. . . 4 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> 0 <_ ( F ` ( k + 1 ) ) )
42		fzelp1 8936	. . . . . . . 8 |- ( k e. ( K ... ( N - 1 ) ) -> k e. ( K ... ( ( N - 1 ) + 1 ) ) )
43	42	adantl 262	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ( K ... ( ( N - 1 ) + 1 ) ) )
44	32	oveq2d 5528	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( K ... ( ( N - 1 ) + 1 ) ) = ( K ... N ) )
45	43, 44	eleqtrd 2116	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ( K ... N ) )
46	45, 12	syldan 266	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F , RR ) ` k ) e. RR )
47		fzss1 8926	. . . . . . . . 9 |- ( K e. ( ZZ>= ` M ) -> ( K ... N ) C_ ( M ... N ) )
48	14, 47	syl 14	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( K ... N ) C_ ( M ... N ) )
49		fzp1elp1 8937	. . . . . . . . . 10 |- ( k e. ( K ... ( N - 1 ) ) -> ( k + 1 ) e. ( K ... ( ( N - 1 ) + 1 ) ) )
50	49	adantl 262	. . . . . . . . 9 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( K ... ( ( N - 1 ) + 1 ) ) )
51	50, 44	eleqtrd 2116	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( K ... N ) )
52	48, 51	sseldd 2946	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( M ... N ) )
53		elfzuz 8886	. . . . . . 7 |- ( ( k + 1 ) e. ( M ... N ) -> ( k + 1 ) e. ( ZZ>= ` M ) )
54	52, 53	syl 14	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( ZZ>= ` M ) )
55	8	ralrimiva 2392	. . . . . . 7 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. RR )
56	55	adantr 261	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. RR )
57	38	eleq1d 2106	. . . . . . 7 |- ( x = ( k + 1 ) -> ( ( F ` x ) e. RR <-> ( F ` ( k + 1 ) ) e. RR ) )
58	57	rspcv 2652	. . . . . 6 |- ( ( k + 1 ) e. ( ZZ>= ` M ) -> ( A. x e. ( ZZ>= ` M ) ( F ` x ) e. RR -> ( F ` ( k + 1 ) ) e. RR ) )
59	54, 56, 58	sylc 56	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( F ` ( k + 1 ) ) e. RR )
60	46, 59	addge01d 7524	. . . 4 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( 0 <_ ( F ` ( k + 1 ) ) <-> ( seq M ( + , F , RR ) ` k ) <_ ( ( seq M ( + , F , RR ) ` k ) + ( F ` ( k + 1 ) ) ) ) )
61	41, 60	mpbid 135	. . 3 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F , RR ) ` k ) <_ ( ( seq M ( + , F , RR ) ` k ) + ( F ` ( k + 1 ) ) ) )
62	45, 5	syldan 266	. . . 4 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ( ZZ>= ` M ) )
63	6	a1i 9	. . . 4 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> RR e. _V )
64	8	adantlr 446	. . . 4 |- ( ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. RR )
65	10	adantl 262	. . . 4 |- ( ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR )
66	62, 63, 64, 65	iseqp1 9225	. . 3 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F , RR ) ` ( k + 1 ) ) = ( ( seq M ( + , F , RR ) ` k ) + ( F ` ( k + 1 ) ) ) )
67	61, 66	breqtrrd 3790	. 2 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F , RR ) ` k ) <_ ( seq M ( + , F , RR ) ` ( k + 1 ) ) )
68	1, 12, 67	monoord 9235	1 |- ( ph -> ( seq M ( + , F , RR ) ` K ) <_ ( seq M ( + , F , RR ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   A.wral 2306   _Vcvv 2557   C_ wss 2917   class class class wbr 3764   ` cfv 4902  (class class class)co 5512   CCcc 6887   RRcr 6888   0cc0 6889   1c1 6890   + caddc 6892   <_ cle 7061   - cmin 7182   ZZcz 8245   ZZ>=cuz 8473   ...cfz 8874   seqcseq 9211

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474  df-fz 8875  df-iseq 9212

This theorem is referenced by: (None)