Theorem ser3mono 10596

Description: The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)

Hypotheses

Ref	Expression
sermono.1	|- ( ph -> K e. ( ZZ>= ` M ) )
sermono.2	|- ( ph -> N e. ( ZZ>= ` K ) )
ser3mono.3	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. RR )
sermono.4	|- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> 0 <_ ( F ` x ) )

Assertion

Ref	Expression
ser3mono	|- ( 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 ser3mono

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		eqid 2196	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
3		sermono.1	. . . . . 6 |- ( ph -> K e. ( ZZ>= ` M ) )
4		eluzel2 9623	. . . . . 6 |- ( K e. ( ZZ>= ` M ) -> M e. ZZ )
5	3, 4	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
6	5	adantr 276	. . . 4 |- ( ( ph /\ k e. ( K ... N ) ) -> M e. ZZ )
7		ser3mono.3	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. RR )
8	7	adantlr 477	. . . 4 |- ( ( ( ph /\ k e. ( K ... N ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. RR )
9	2, 6, 8	serfre 10593	. . 3 |- ( ( ph /\ k e. ( K ... N ) ) -> seq M ( + , F ) : ( ZZ>= ` M ) --> RR )
10		elfzuz 10113	. . . 4 |- ( k e. ( K ... N ) -> k e. ( ZZ>= ` K ) )
11		uztrn 9635	. . . 4 |- ( ( k e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
12	10, 3, 11	syl2anr 290	. . 3 |- ( ( ph /\ k e. ( K ... N ) ) -> k e. ( ZZ>= ` M ) )
13	9, 12	ffvelcdmd 5701	. 2 |- ( ( ph /\ k e. ( K ... N ) ) -> ( seq M ( + , F ) ` k ) e. RR )
14		fveq2 5561	. . . . . 6 |- ( x = ( k + 1 ) -> ( F ` x ) = ( F ` ( k + 1 ) ) )
15	14	breq2d 4046	. . . . 5 |- ( x = ( k + 1 ) -> ( 0 <_ ( F ` x ) <-> 0 <_ ( F ` ( k + 1 ) ) ) )
16		sermono.4	. . . . . . 7 |- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> 0 <_ ( F ` x ) )
17	16	ralrimiva 2570	. . . . . 6 |- ( ph -> A. x e. ( ( K + 1 ) ... N ) 0 <_ ( F ` x ) )
18	17	adantr 276	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> A. x e. ( ( K + 1 ) ... N ) 0 <_ ( F ` x ) )
19		simpr 110	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ( K ... ( N - 1 ) ) )
20	3	adantr 276	. . . . . . . . 9 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> K e. ( ZZ>= ` M ) )
21		eluzelz 9627	. . . . . . . . 9 |- ( K e. ( ZZ>= ` M ) -> K e. ZZ )
22	20, 21	syl 14	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> K e. ZZ )
23	1	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> N e. ( ZZ>= ` K ) )
24		eluzelz 9627	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` K ) -> N e. ZZ )
25	23, 24	syl 14	. . . . . . . . 9 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> N e. ZZ )
26		peano2zm 9381	. . . . . . . . 9 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
27	25, 26	syl 14	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( N - 1 ) e. ZZ )
28		elfzelz 10117	. . . . . . . . 9 |- ( k e. ( K ... ( N - 1 ) ) -> k e. ZZ )
29	28	adantl 277	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ZZ )
30		1zzd 9370	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> 1 e. ZZ )
31		fzaddel 10151	. . . . . . . 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 ) ) ) )
32	22, 27, 29, 30, 31	syl22anc 1250	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k e. ( K ... ( N - 1 ) ) <-> ( k + 1 ) e. ( ( K + 1 ) ... ( ( N - 1 ) + 1 ) ) ) )
33	19, 32	mpbid 147	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( ( K + 1 ) ... ( ( N - 1 ) + 1 ) ) )
34		zcn 9348	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
35		ax-1cn 7989	. . . . . . . . 9 |- 1 e. CC
36		npcan 8252	. . . . . . . . 9 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
37	34, 35, 36	sylancl 413	. . . . . . . 8 |- ( N e. ZZ -> ( ( N - 1 ) + 1 ) = N )
38	25, 37	syl 14	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( ( N - 1 ) + 1 ) = N )
39	38	oveq2d 5941	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( ( K + 1 ) ... ( ( N - 1 ) + 1 ) ) = ( ( K + 1 ) ... N ) )
40	33, 39	eleqtrd 2275	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( ( K + 1 ) ... N ) )
41	15, 18, 40	rspcdva 2873	. . . 4 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> 0 <_ ( F ` ( k + 1 ) ) )
42		fzelp1 10166	. . . . . . . 8 |- ( k e. ( K ... ( N - 1 ) ) -> k e. ( K ... ( ( N - 1 ) + 1 ) ) )
43	42	adantl 277	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ( K ... ( ( N - 1 ) + 1 ) ) )
44	38	oveq2d 5941	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( K ... ( ( N - 1 ) + 1 ) ) = ( K ... N ) )
45	43, 44	eleqtrd 2275	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ( K ... N ) )
46	45, 13	syldan 282	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F ) ` k ) e. RR )
47	14	eleq1d 2265	. . . . . 6 |- ( x = ( k + 1 ) -> ( ( F ` x ) e. RR <-> ( F ` ( k + 1 ) ) e. RR ) )
48	7	ralrimiva 2570	. . . . . . 7 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. RR )
49	48	adantr 276	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. RR )
50		fzss1 10155	. . . . . . . . 9 |- ( K e. ( ZZ>= ` M ) -> ( K ... N ) C_ ( M ... N ) )
51	20, 50	syl 14	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( K ... N ) C_ ( M ... N ) )
52		fzp1elp1 10167	. . . . . . . . . 10 |- ( k e. ( K ... ( N - 1 ) ) -> ( k + 1 ) e. ( K ... ( ( N - 1 ) + 1 ) ) )
53	52	adantl 277	. . . . . . . . 9 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( K ... ( ( N - 1 ) + 1 ) ) )
54	53, 44	eleqtrd 2275	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( K ... N ) )
55	51, 54	sseldd 3185	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( M ... N ) )
56		elfzuz 10113	. . . . . . 7 |- ( ( k + 1 ) e. ( M ... N ) -> ( k + 1 ) e. ( ZZ>= ` M ) )
57	55, 56	syl 14	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( ZZ>= ` M ) )
58	47, 49, 57	rspcdva 2873	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( F ` ( k + 1 ) ) e. RR )
59	46, 58	addge01d 8577	. . . 4 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( 0 <_ ( F ` ( k + 1 ) ) <-> ( seq M ( + , F ) ` k ) <_ ( ( seq M ( + , F ) ` k ) + ( F ` ( k + 1 ) ) ) ) )
60	41, 59	mpbid 147	. . 3 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F ) ` k ) <_ ( ( seq M ( + , F ) ` k ) + ( F ` ( k + 1 ) ) ) )
61	45, 12	syldan 282	. . . 4 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> k e. ( ZZ>= ` M ) )
62	7	adantlr 477	. . . 4 |- ( ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. RR )
63		readdcl 8022	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> ( x + y ) e. RR )
64	63	adantl 277	. . . 4 |- ( ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR )
65	61, 62, 64	seq3p1 10574	. . 3 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F ) ` ( k + 1 ) ) = ( ( seq M ( + , F ) ` k ) + ( F ` ( k + 1 ) ) ) )
66	60, 65	breqtrrd 4062	. 2 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F ) ` k ) <_ ( seq M ( + , F ) ` ( k + 1 ) ) )
67	1, 13, 66	monoord 10594	1 |- ( ph -> ( seq M ( + , F ) ` K ) <_ ( seq M ( + , F ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   C_ wss 3157   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   CCcc 7894   RRcr 7895   0cc0 7896   1c1 7897   + caddc 7899   <_ cle 8079   - cmin 8214   ZZcz 9343   ZZ>=cuz 9618   ...cfz 10100   seqcseq 10556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-fz 10101  df-seqfrec 10557

This theorem is referenced by: (None)