Theorem isermono 9457

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 9041	. . . 4 |- ( k e. ( K ... N ) -> k e. ( ZZ>= ` K ) )
3		isermono.1	. . . 4 |- ( ph -> K e. ( ZZ>= ` M ) )
4		uztrn 8635	. . . 4 |- ( ( k e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
5	2, 3, 4	syl2anr 284	. . 3 |- ( ( ph /\ k e. ( K ... N ) ) -> k e. ( ZZ>= ` M ) )
6		reex 7107	. . . 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 460	. . 3 |- ( ( ( ph /\ k e. ( K ... N ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. RR )
10		readdcl 7099	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( x + y ) e. RR )
11	10	adantl 271	. . 3 |- ( ( ( ph /\ k e. ( K ... N ) ) /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR )
12	5, 7, 9, 11	iseqcl 9443	. 2 |- ( ( ph /\ k e. ( K ... N ) ) -> ( seq M ( + , F , RR ) ` k ) e. RR )
13		simpr 108	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ( K ... ( N - 1 ) ) )
14	3	adantr 270	. . . . . . . . 9 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> K e. ( ZZ>= ` M ) )
15		eluzelz 8628	. . . . . . . . 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 270	. . . . . . . . . 10 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> N e. ( ZZ>= ` K ) )
18		eluzelz 8628	. . . . . . . . . 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 8389	. . . . . . . . 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 9045	. . . . . . . . 9 |- ( k e. ( K ... ( N - 1 ) ) -> k e. ZZ )
23	22	adantl 271	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ZZ )
24		1zzd 8378	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> 1 e. ZZ )
25		fzaddel 9077	. . . . . . . 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 1170	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k e. ( K ... ( N - 1 ) ) <-> ( k + 1 ) e. ( ( K + 1 ) ... ( ( N - 1 ) + 1 ) ) ) )
27	13, 26	mpbid 145	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( ( K + 1 ) ... ( ( N - 1 ) + 1 ) ) )
28		zcn 8356	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
29		ax-1cn 7069	. . . . . . . . 9 |- 1 e. CC
30		npcan 7317	. . . . . . . . 9 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
31	28, 29, 30	sylancl 404	. . . . . . . 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 5548	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( ( K + 1 ) ... ( ( N - 1 ) + 1 ) ) = ( ( K + 1 ) ... N ) )
34	27, 33	eleqtrd 2157	. . . . 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 2434	. . . . . 6 |- ( ph -> A. x e. ( ( K + 1 ) ... N ) 0 <_ ( F ` x ) )
37	36	adantr 270	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> A. x e. ( ( K + 1 ) ... N ) 0 <_ ( F ` x ) )
38		fveq2 5198	. . . . . . 7 |- ( x = ( k + 1 ) -> ( F ` x ) = ( F ` ( k + 1 ) ) )
39	38	breq2d 3797	. . . . . 6 |- ( x = ( k + 1 ) -> ( 0 <_ ( F ` x ) <-> 0 <_ ( F ` ( k + 1 ) ) ) )
40	39	rspcv 2697	. . . . 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 61	. . . 4 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> 0 <_ ( F ` ( k + 1 ) ) )
42		fzelp1 9091	. . . . . . . 8 |- ( k e. ( K ... ( N - 1 ) ) -> k e. ( K ... ( ( N - 1 ) + 1 ) ) )
43	42	adantl 271	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ( K ... ( ( N - 1 ) + 1 ) ) )
44	32	oveq2d 5548	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( K ... ( ( N - 1 ) + 1 ) ) = ( K ... N ) )
45	43, 44	eleqtrd 2157	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ( K ... N ) )
46	45, 12	syldan 276	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F , RR ) ` k ) e. RR )
47		fzss1 9081	. . . . . . . . 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 9092	. . . . . . . . . 10 |- ( k e. ( K ... ( N - 1 ) ) -> ( k + 1 ) e. ( K ... ( ( N - 1 ) + 1 ) ) )
50	49	adantl 271	. . . . . . . . 9 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( K ... ( ( N - 1 ) + 1 ) ) )
51	50, 44	eleqtrd 2157	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( K ... N ) )
52	48, 51	sseldd 3000	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( M ... N ) )
53		elfzuz 9041	. . . . . . 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 2434	. . . . . . 7 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. RR )
56	55	adantr 270	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. RR )
57	38	eleq1d 2147	. . . . . . 7 |- ( x = ( k + 1 ) -> ( ( F ` x ) e. RR <-> ( F ` ( k + 1 ) ) e. RR ) )
58	57	rspcv 2697	. . . . . 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 61	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( F ` ( k + 1 ) ) e. RR )
60	46, 59	addge01d 7633	. . . 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 145	. . 3 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F , RR ) ` k ) <_ ( ( seq M ( + , F , RR ) ` k ) + ( F ` ( k + 1 ) ) ) )
62	45, 5	syldan 276	. . . 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 460	. . . 4 |- ( ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. RR )
65	10	adantl 271	. . . 4 |- ( ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR )
66	62, 63, 64, 65	iseqp1 9445	. . 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 3811	. 2 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F , RR ) ` k ) <_ ( seq M ( + , F , RR ) ` ( k + 1 ) ) )
68	1, 12, 67	monoord 9455	1 |- ( ph -> ( seq M ( + , F , RR ) ` K ) <_ ( seq M ( + , F , RR ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   _Vcvv 2601   C_ wss 2973   class class class wbr 3785   ` cfv 4922  (class class class)co 5532   CCcc 6979   RRcr 6980   0cc0 6981   1c1 6982   + caddc 6984   <_ cle 7154   - cmin 7279   ZZcz 8351   ZZ>=cuz 8619   ...cfz 9029   seqcseq 9431

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-fz 9030  df-iseq 9432

This theorem is referenced by: (None)