Theorem ser3mono 10704

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 2229	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
3		sermono.1	. . . . . 6 |- ( ph -> K e. ( ZZ>= ` M ) )
4		eluzel2 9723	. . . . . 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 10701	. . 3 |- ( ( ph /\ k e. ( K ... N ) ) -> seq M ( + , F ) : ( ZZ>= ` M ) --> RR )
10		elfzuz 10213	. . . 4 |- ( k e. ( K ... N ) -> k e. ( ZZ>= ` K ) )
11		uztrn 9735	. . . 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 5770	. 2 |- ( ( ph /\ k e. ( K ... N ) ) -> ( seq M ( + , F ) ` k ) e. RR )
14		fveq2 5626	. . . . . 6 |- ( x = ( k + 1 ) -> ( F ` x ) = ( F ` ( k + 1 ) ) )
15	14	breq2d 4094	. . . . 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 2603	. . . . . 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 9727	. . . . . . . . 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 9727	. . . . . . . . . 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 9480	. . . . . . . . 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 10217	. . . . . . . . 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 9469	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> 1 e. ZZ )
31		fzaddel 10251	. . . . . . . 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 1272	. . . . . . 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 9447	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
35		ax-1cn 8088	. . . . . . . . 9 |- 1 e. CC
36		npcan 8351	. . . . . . . . 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 6016	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( ( K + 1 ) ... ( ( N - 1 ) + 1 ) ) = ( ( K + 1 ) ... N ) )
40	33, 39	eleqtrd 2308	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( ( K + 1 ) ... N ) )
41	15, 18, 40	rspcdva 2912	. . . 4 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> 0 <_ ( F ` ( k + 1 ) ) )
42		fzelp1 10266	. . . . . . . 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 6016	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( K ... ( ( N - 1 ) + 1 ) ) = ( K ... N ) )
45	43, 44	eleqtrd 2308	. . . . . 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 2298	. . . . . 6 |- ( x = ( k + 1 ) -> ( ( F ` x ) e. RR <-> ( F ` ( k + 1 ) ) e. RR ) )
48	7	ralrimiva 2603	. . . . . . 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 10255	. . . . . . . . 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 10267	. . . . . . . . . 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 2308	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( K ... N ) )
55	51, 54	sseldd 3225	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( M ... N ) )
56		elfzuz 10213	. . . . . . 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 2912	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( F ` ( k + 1 ) ) e. RR )
59	46, 58	addge01d 8676	. . . 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 8121	. . . . 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 10682	. . 3 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F ) ` ( k + 1 ) ) = ( ( seq M ( + , F ) ` k ) + ( F ` ( k + 1 ) ) ) )
66	60, 65	breqtrrd 4110	. 2 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F ) ` k ) <_ ( seq M ( + , F ) ` ( k + 1 ) ) )
67	1, 13, 66	monoord 10702	1 |- ( ph -> ( seq M ( + , F ) ` K ) <_ ( seq M ( + , F ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   class class class wbr 4082   ` cfv 5317  (class class class)co 6000   CCcc 7993   RRcr 7994   0cc0 7995   1c1 7996   + caddc 7998   <_ cle 8178   - cmin 8313   ZZcz 9442   ZZ>=cuz 9718   ...cfz 10200   seqcseq 10664

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-seqfrec 10665

This theorem is referenced by: (None)