Theorem cvgratnnlemseq 11288

Description: Lemma for cvgratnn 11293. (Contributed by Jim Kingdon, 21-Nov-2022.)

Hypotheses

Ref	Expression
cvgratnn.3	|- ( ph -> A e. RR )
cvgratnn.4	|- ( ph -> A < 1 )
cvgratnn.gt0	|- ( ph -> 0 < A )
cvgratnn.6	|- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
cvgratnn.7	|- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )
cvgratnn.m	|- ( ph -> M e. NN )
cvgratnn.n	|- ( ph -> N e. ( ZZ>= ` M ) )

Assertion

Ref	Expression
cvgratnnlemseq	|- ( ph -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )

Distinct variable groups:   A, k   k, F   k, N   ph, k   i, F, k   i, M   i, N   ph, i

Allowed substitution hints:   A( i)   M( k)

Proof of Theorem cvgratnnlemseq

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 9354	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
2		1zzd 9074	. . . . . . 7 |- ( ph -> 1 e. ZZ )
3		cvgratnn.6	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
4	1, 2, 3	serf 10240	. . . . . 6 |- ( ph -> seq 1 ( + , F ) : NN --> CC )
5	4	adantr 274	. . . . 5 |- ( ( ph /\ M < N ) -> seq 1 ( + , F ) : NN --> CC )
6		cvgratnn.m	. . . . . 6 |- ( ph -> M e. NN )
7	6	adantr 274	. . . . 5 |- ( ( ph /\ M < N ) -> M e. NN )
8	5, 7	ffvelrnd 5549	. . . 4 |- ( ( ph /\ M < N ) -> ( seq 1 ( + , F ) ` M ) e. CC )
9		eqid 2137	. . . . . . 7 |- ( ZZ>= ` ( M + 1 ) ) = ( ZZ>= ` ( M + 1 ) )
10	6	nnzd 9165	. . . . . . . 8 |- ( ph -> M e. ZZ )
11	10	peano2zd 9169	. . . . . . 7 |- ( ph -> ( M + 1 ) e. ZZ )
12		fveq2 5414	. . . . . . . . 9 |- ( k = x -> ( F ` k ) = ( F ` x ) )
13	12	eleq1d 2206	. . . . . . . 8 |- ( k = x -> ( ( F ` k ) e. CC <-> ( F ` x ) e. CC ) )
14	3	ralrimiva 2503	. . . . . . . . 9 |- ( ph -> A. k e. NN ( F ` k ) e. CC )
15	14	adantr 274	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> A. k e. NN ( F ` k ) e. CC )
16	6	peano2nnd 8728	. . . . . . . . 9 |- ( ph -> ( M + 1 ) e. NN )
17		eluznn 9387	. . . . . . . . 9 |- ( ( ( M + 1 ) e. NN /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> x e. NN )
18	16, 17	sylan 281	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> x e. NN )
19	13, 15, 18	rspcdva 2789	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. CC )
20	9, 11, 19	serf 10240	. . . . . 6 |- ( ph -> seq ( M + 1 ) ( + , F ) : ( ZZ>= ` ( M + 1 ) ) --> CC )
21	20	adantr 274	. . . . 5 |- ( ( ph /\ M < N ) -> seq ( M + 1 ) ( + , F ) : ( ZZ>= ` ( M + 1 ) ) --> CC )
22	11	adantr 274	. . . . . 6 |- ( ( ph /\ M < N ) -> ( M + 1 ) e. ZZ )
23		cvgratnn.n	. . . . . . . 8 |- ( ph -> N e. ( ZZ>= ` M ) )
24		eluzelz 9328	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
25	23, 24	syl 14	. . . . . . 7 |- ( ph -> N e. ZZ )
26	25	adantr 274	. . . . . 6 |- ( ( ph /\ M < N ) -> N e. ZZ )
27		zltp1le 9101	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) )
28	10, 25, 27	syl2anc 408	. . . . . . 7 |- ( ph -> ( M < N <-> ( M + 1 ) <_ N ) )
29	28	biimpa 294	. . . . . 6 |- ( ( ph /\ M < N ) -> ( M + 1 ) <_ N )
30		eluz2 9325	. . . . . 6 |- ( N e. ( ZZ>= ` ( M + 1 ) ) <-> ( ( M + 1 ) e. ZZ /\ N e. ZZ /\ ( M + 1 ) <_ N ) )
31	22, 26, 29, 30	syl3anbrc 1165	. . . . 5 |- ( ( ph /\ M < N ) -> N e. ( ZZ>= ` ( M + 1 ) ) )
32	21, 31	ffvelrnd 5549	. . . 4 |- ( ( ph /\ M < N ) -> ( seq ( M + 1 ) ( + , F ) ` N ) e. CC )
33	8, 32	pncan2d 8068	. . 3 |- ( ( ph /\ M < N ) -> ( ( ( seq 1 ( + , F ) ` M ) + ( seq ( M + 1 ) ( + , F ) ` N ) ) - ( seq 1 ( + , F ) ` M ) ) = ( seq ( M + 1 ) ( + , F ) ` N ) )
34		addcl 7738	. . . . . 6 |- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
35	34	adantl 275	. . . . 5 |- ( ( ( ph /\ M < N ) /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
36		addass 7743	. . . . . 6 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( ( x + y ) + z ) = ( x + ( y + z ) ) )
37	36	adantl 275	. . . . 5 |- ( ( ( ph /\ M < N ) /\ ( x e. CC /\ y e. CC /\ z e. CC ) ) -> ( ( x + y ) + z ) = ( x + ( y + z ) ) )
38	6, 1	eleqtrdi 2230	. . . . . 6 |- ( ph -> M e. ( ZZ>= ` 1 ) )
39	38	adantr 274	. . . . 5 |- ( ( ph /\ M < N ) -> M e. ( ZZ>= ` 1 ) )
40	14	ad2antrr 479	. . . . . 6 |- ( ( ( ph /\ M < N ) /\ x e. ( ZZ>= ` 1 ) ) -> A. k e. NN ( F ` k ) e. CC )
41		simpr 109	. . . . . . 7 |- ( ( ( ph /\ M < N ) /\ x e. ( ZZ>= ` 1 ) ) -> x e. ( ZZ>= ` 1 ) )
42	41, 1	eleqtrrdi 2231	. . . . . 6 |- ( ( ( ph /\ M < N ) /\ x e. ( ZZ>= ` 1 ) ) -> x e. NN )
43	13, 40, 42	rspcdva 2789	. . . . 5 |- ( ( ( ph /\ M < N ) /\ x e. ( ZZ>= ` 1 ) ) -> ( F ` x ) e. CC )
44	35, 37, 31, 39, 43	seq3split 10245	. . . 4 |- ( ( ph /\ M < N ) -> ( seq 1 ( + , F ) ` N ) = ( ( seq 1 ( + , F ) ` M ) + ( seq ( M + 1 ) ( + , F ) ` N ) ) )
45	44	oveq1d 5782	. . 3 |- ( ( ph /\ M < N ) -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = ( ( ( seq 1 ( + , F ) ` M ) + ( seq ( M + 1 ) ( + , F ) ` N ) ) - ( seq 1 ( + , F ) ` M ) ) )
46		eqidd 2138	. . . 4 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` i ) = ( F ` i ) )
47		fveq2 5414	. . . . . 6 |- ( k = i -> ( F ` k ) = ( F ` i ) )
48	47	eleq1d 2206	. . . . 5 |- ( k = i -> ( ( F ` k ) e. CC <-> ( F ` i ) e. CC ) )
49	14	ad2antrr 479	. . . . 5 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> A. k e. NN ( F ` k ) e. CC )
50	16	ad2antrr 479	. . . . . 6 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> ( M + 1 ) e. NN )
51		simpr 109	. . . . . 6 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> i e. ( ZZ>= ` ( M + 1 ) ) )
52		eluznn 9387	. . . . . 6 |- ( ( ( M + 1 ) e. NN /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> i e. NN )
53	50, 51, 52	syl2anc 408	. . . . 5 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> i e. NN )
54	48, 49, 53	rspcdva 2789	. . . 4 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` i ) e. CC )
55	46, 31, 54	fsum3ser 11159	. . 3 |- ( ( ph /\ M < N ) -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) = ( seq ( M + 1 ) ( + , F ) ` N ) )
56	33, 45, 55	3eqtr4d 2180	. 2 |- ( ( ph /\ M < N ) -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )
57		simpr 109	. . . . . . 7 |- ( ( ph /\ M = N ) -> M = N )
58	6	nnred 8726	. . . . . . . . 9 |- ( ph -> M e. RR )
59	58	ltp1d 8681	. . . . . . . 8 |- ( ph -> M < ( M + 1 ) )
60	59	adantr 274	. . . . . . 7 |- ( ( ph /\ M = N ) -> M < ( M + 1 ) )
61	57, 60	eqbrtrrd 3947	. . . . . 6 |- ( ( ph /\ M = N ) -> N < ( M + 1 ) )
62	11	adantr 274	. . . . . . 7 |- ( ( ph /\ M = N ) -> ( M + 1 ) e. ZZ )
63	25	adantr 274	. . . . . . 7 |- ( ( ph /\ M = N ) -> N e. ZZ )
64		fzn 9815	. . . . . . 7 |- ( ( ( M + 1 ) e. ZZ /\ N e. ZZ ) -> ( N < ( M + 1 ) <-> ( ( M + 1 ) ... N ) = (/) ) )
65	62, 63, 64	syl2anc 408	. . . . . 6 |- ( ( ph /\ M = N ) -> ( N < ( M + 1 ) <-> ( ( M + 1 ) ... N ) = (/) ) )
66	61, 65	mpbid 146	. . . . 5 |- ( ( ph /\ M = N ) -> ( ( M + 1 ) ... N ) = (/) )
67	66	sumeq1d 11128	. . . 4 |- ( ( ph /\ M = N ) -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) = sum_ i e. (/) ( F ` i ) )
68		sum0 11150	. . . 4 |- sum_ i e. (/) ( F ` i ) = 0
69	67, 68	syl6eq 2186	. . 3 |- ( ( ph /\ M = N ) -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) = 0 )
70	4, 6	ffvelrnd 5549	. . . . 5 |- ( ph -> ( seq 1 ( + , F ) ` M ) e. CC )
71	70	adantr 274	. . . 4 |- ( ( ph /\ M = N ) -> ( seq 1 ( + , F ) ` M ) e. CC )
72	71	subidd 8054	. . 3 |- ( ( ph /\ M = N ) -> ( ( seq 1 ( + , F ) ` M ) - ( seq 1 ( + , F ) ` M ) ) = 0 )
73	57	fveq2d 5418	. . . 4 |- ( ( ph /\ M = N ) -> ( seq 1 ( + , F ) ` M ) = ( seq 1 ( + , F ) ` N ) )
74	73	oveq1d 5782	. . 3 |- ( ( ph /\ M = N ) -> ( ( seq 1 ( + , F ) ` M ) - ( seq 1 ( + , F ) ` M ) ) = ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) )
75	69, 72, 74	3eqtr2rd 2177	. 2 |- ( ( ph /\ M = N ) -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )
76		eluzle 9331	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
77	23, 76	syl 14	. . 3 |- ( ph -> M <_ N )
78		zleloe 9094	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( M < N \/ M = N ) ) )
79	10, 25, 78	syl2anc 408	. . 3 |- ( ph -> ( M <_ N <-> ( M < N \/ M = N ) ) )
80	77, 79	mpbid 146	. 2 |- ( ph -> ( M < N \/ M = N ) )
81	56, 75, 80	mpjaodan 787	1 |- ( ph -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   /\ w3a 962   = wceq 1331   e. wcel 1480   A.wral 2414   (/)c0 3358   class class class wbr 3924   -->wf 5114   ` cfv 5118  (class class class)co 5767   CCcc 7611   RRcr 7612   0cc0 7613   1c1 7614   + caddc 7616   x. cmul 7618   < clt 7793   <_ cle 7794   - cmin 7926   NNcn 8713   ZZcz 9047   ZZ>=cuz 9319   ...cfz 9783   seqcseq 10211   abscabs 10762   sum_csu 11115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-isom 5127  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-frec 6281  df-1o 6306  df-oadd 6310  df-er 6422  df-en 6628  df-dom 6629  df-fin 6630  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-fz 9784  df-fzo 9913  df-seqfrec 10212  df-exp 10286  df-ihash 10515  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-clim 11041  df-sumdc 11116

This theorem is referenced by:  cvgratnnlemrate  11292