Theorem ser3mono 10873

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 2234	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
3		sermono.1	. . . . . 6 |- ( ph -> K e. ( ZZ>= ` M ) )
4		eluzel2 9876	. . . . . 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 10870	. . 3 |- ( ( ph /\ k e. ( K ... N ) ) -> seq M ( + , F ) : ( ZZ>= ` M ) --> RR )
10		elfzuz 10374	. . . 4 |- ( k e. ( K ... N ) -> k e. ( ZZ>= ` K ) )
11		uztrn 9889	. . . 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 5818	. 2 |- ( ( ph /\ k e. ( K ... N ) ) -> ( seq M ( + , F ) ` k ) e. RR )
14		fveq2 5675	. . . . . 6 |- ( x = ( k + 1 ) -> ( F ` x ) = ( F ` ( k + 1 ) ) )
15	14	breq2d 4126	. . . . 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 2617	. . . . . 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 9881	. . . . . . . . 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 9881	. . . . . . . . . 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 9632	. . . . . . . . 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 10378	. . . . . . . . 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 9621	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> 1 e. ZZ )
31		fzaddel 10414	. . . . . . . 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 1275	. . . . . . 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 9599	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
35		ax-1cn 8236	. . . . . . . . 9 |- 1 e. CC
36		npcan 8498	. . . . . . . . 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 6074	. . . . . 6 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( ( K + 1 ) ... ( ( N - 1 ) + 1 ) ) = ( ( K + 1 ) ... N ) )
40	33, 39	eleqtrd 2313	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( ( K + 1 ) ... N ) )
41	15, 18, 40	rspcdva 2928	. . . 4 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> 0 <_ ( F ` ( k + 1 ) ) )
42		fzelp1 10430	. . . . . . . 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 6074	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( K ... ( ( N - 1 ) + 1 ) ) = ( K ... N ) )
45	43, 44	eleqtrd 2313	. . . . . 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 2303	. . . . . 6 |- ( x = ( k + 1 ) -> ( ( F ` x ) e. RR <-> ( F ` ( k + 1 ) ) e. RR ) )
48	7	ralrimiva 2617	. . . . . . 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 10418	. . . . . . . . 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 10431	. . . . . . . . . 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 2313	. . . . . . . 8 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( K ... N ) )
55	51, 54	sseldd 3243	. . . . . . 7 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( k + 1 ) e. ( M ... N ) )
56		elfzuz 10374	. . . . . . 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 2928	. . . . 5 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( F ` ( k + 1 ) ) e. RR )
59	46, 58	addge01d 8824	. . . 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 8269	. . . . 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 10851	. . 3 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F ) ` ( k + 1 ) ) = ( ( seq M ( + , F ) ` k ) + ( F ` ( k + 1 ) ) ) )
66	60, 65	breqtrrd 4142	. 2 |- ( ( ph /\ k e. ( K ... ( N - 1 ) ) ) -> ( seq M ( + , F ) ` k ) <_ ( seq M ( + , F ) ` ( k + 1 ) ) )
67	1, 13, 66	monoord 10871	1 |- ( ph -> ( seq M ( + , F ) ` K ) <_ ( seq M ( + , F ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   C_ wss 3214   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   <_ cle 8325   - cmin 8460   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361   seqcseq 10833

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-seqfrec 10834

This theorem is referenced by: (None)