Theorem cvgratnnlemseq 11952

Description: Lemma for cvgratnn 11957. (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 9719	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
2		1zzd 9434	. . . . . . 7 |- ( ph -> 1 e. ZZ )
3		cvgratnn.6	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
4	1, 2, 3	serf 10665	. . . . . 6 |- ( ph -> seq 1 ( + , F ) : NN --> CC )
5	4	adantr 276	. . . . 5 |- ( ( ph /\ M < N ) -> seq 1 ( + , F ) : NN --> CC )
6		cvgratnn.m	. . . . . 6 |- ( ph -> M e. NN )
7	6	adantr 276	. . . . 5 |- ( ( ph /\ M < N ) -> M e. NN )
8	5, 7	ffvelcdmd 5739	. . . 4 |- ( ( ph /\ M < N ) -> ( seq 1 ( + , F ) ` M ) e. CC )
9		eqid 2207	. . . . . . 7 |- ( ZZ>= ` ( M + 1 ) ) = ( ZZ>= ` ( M + 1 ) )
10	6	nnzd 9529	. . . . . . . 8 |- ( ph -> M e. ZZ )
11	10	peano2zd 9533	. . . . . . 7 |- ( ph -> ( M + 1 ) e. ZZ )
12		fveq2 5599	. . . . . . . . 9 |- ( k = x -> ( F ` k ) = ( F ` x ) )
13	12	eleq1d 2276	. . . . . . . 8 |- ( k = x -> ( ( F ` k ) e. CC <-> ( F ` x ) e. CC ) )
14	3	ralrimiva 2581	. . . . . . . . 9 |- ( ph -> A. k e. NN ( F ` k ) e. CC )
15	14	adantr 276	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> A. k e. NN ( F ` k ) e. CC )
16	6	peano2nnd 9086	. . . . . . . . 9 |- ( ph -> ( M + 1 ) e. NN )
17		eluznn 9756	. . . . . . . . 9 |- ( ( ( M + 1 ) e. NN /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> x e. NN )
18	16, 17	sylan 283	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> x e. NN )
19	13, 15, 18	rspcdva 2889	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. CC )
20	9, 11, 19	serf 10665	. . . . . 6 |- ( ph -> seq ( M + 1 ) ( + , F ) : ( ZZ>= ` ( M + 1 ) ) --> CC )
21	20	adantr 276	. . . . 5 |- ( ( ph /\ M < N ) -> seq ( M + 1 ) ( + , F ) : ( ZZ>= ` ( M + 1 ) ) --> CC )
22	11	adantr 276	. . . . . 6 |- ( ( ph /\ M < N ) -> ( M + 1 ) e. ZZ )
23		cvgratnn.n	. . . . . . . 8 |- ( ph -> N e. ( ZZ>= ` M ) )
24		eluzelz 9692	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
25	23, 24	syl 14	. . . . . . 7 |- ( ph -> N e. ZZ )
26	25	adantr 276	. . . . . 6 |- ( ( ph /\ M < N ) -> N e. ZZ )
27		zltp1le 9462	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) )
28	10, 25, 27	syl2anc 411	. . . . . . 7 |- ( ph -> ( M < N <-> ( M + 1 ) <_ N ) )
29	28	biimpa 296	. . . . . 6 |- ( ( ph /\ M < N ) -> ( M + 1 ) <_ N )
30		eluz2 9689	. . . . . 6 |- ( N e. ( ZZ>= ` ( M + 1 ) ) <-> ( ( M + 1 ) e. ZZ /\ N e. ZZ /\ ( M + 1 ) <_ N ) )
31	22, 26, 29, 30	syl3anbrc 1184	. . . . 5 |- ( ( ph /\ M < N ) -> N e. ( ZZ>= ` ( M + 1 ) ) )
32	21, 31	ffvelcdmd 5739	. . . 4 |- ( ( ph /\ M < N ) -> ( seq ( M + 1 ) ( + , F ) ` N ) e. CC )
33	8, 32	pncan2d 8420	. . 3 |- ( ( ph /\ M < N ) -> ( ( ( seq 1 ( + , F ) ` M ) + ( seq ( M + 1 ) ( + , F ) ` N ) ) - ( seq 1 ( + , F ) ` M ) ) = ( seq ( M + 1 ) ( + , F ) ` N ) )
34		addcl 8085	. . . . . 6 |- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
35	34	adantl 277	. . . . 5 |- ( ( ( ph /\ M < N ) /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
36		addass 8090	. . . . . 6 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( ( x + y ) + z ) = ( x + ( y + z ) ) )
37	36	adantl 277	. . . . 5 |- ( ( ( ph /\ M < N ) /\ ( x e. CC /\ y e. CC /\ z e. CC ) ) -> ( ( x + y ) + z ) = ( x + ( y + z ) ) )
38	6, 1	eleqtrdi 2300	. . . . . 6 |- ( ph -> M e. ( ZZ>= ` 1 ) )
39	38	adantr 276	. . . . 5 |- ( ( ph /\ M < N ) -> M e. ( ZZ>= ` 1 ) )
40	14	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ M < N ) /\ x e. ( ZZ>= ` 1 ) ) -> A. k e. NN ( F ` k ) e. CC )
41		simpr 110	. . . . . . 7 |- ( ( ( ph /\ M < N ) /\ x e. ( ZZ>= ` 1 ) ) -> x e. ( ZZ>= ` 1 ) )
42	41, 1	eleqtrrdi 2301	. . . . . 6 |- ( ( ( ph /\ M < N ) /\ x e. ( ZZ>= ` 1 ) ) -> x e. NN )
43	13, 40, 42	rspcdva 2889	. . . . 5 |- ( ( ( ph /\ M < N ) /\ x e. ( ZZ>= ` 1 ) ) -> ( F ` x ) e. CC )
44	35, 37, 31, 39, 43	seq3split 10670	. . . 4 |- ( ( ph /\ M < N ) -> ( seq 1 ( + , F ) ` N ) = ( ( seq 1 ( + , F ) ` M ) + ( seq ( M + 1 ) ( + , F ) ` N ) ) )
45	44	oveq1d 5982	. . 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 2208	. . . 4 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` i ) = ( F ` i ) )
47		fveq2 5599	. . . . . 6 |- ( k = i -> ( F ` k ) = ( F ` i ) )
48	47	eleq1d 2276	. . . . 5 |- ( k = i -> ( ( F ` k ) e. CC <-> ( F ` i ) e. CC ) )
49	14	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> A. k e. NN ( F ` k ) e. CC )
50	16	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> ( M + 1 ) e. NN )
51		simpr 110	. . . . . 6 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> i e. ( ZZ>= ` ( M + 1 ) ) )
52		eluznn 9756	. . . . . 6 |- ( ( ( M + 1 ) e. NN /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> i e. NN )
53	50, 51, 52	syl2anc 411	. . . . 5 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> i e. NN )
54	48, 49, 53	rspcdva 2889	. . . 4 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` i ) e. CC )
55	46, 31, 54	fsum3ser 11823	. . 3 |- ( ( ph /\ M < N ) -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) = ( seq ( M + 1 ) ( + , F ) ` N ) )
56	33, 45, 55	3eqtr4d 2250	. 2 |- ( ( ph /\ M < N ) -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )
57		simpr 110	. . . . . . 7 |- ( ( ph /\ M = N ) -> M = N )
58	6	nnred 9084	. . . . . . . . 9 |- ( ph -> M e. RR )
59	58	ltp1d 9038	. . . . . . . 8 |- ( ph -> M < ( M + 1 ) )
60	59	adantr 276	. . . . . . 7 |- ( ( ph /\ M = N ) -> M < ( M + 1 ) )
61	57, 60	eqbrtrrd 4083	. . . . . 6 |- ( ( ph /\ M = N ) -> N < ( M + 1 ) )
62	11	adantr 276	. . . . . . 7 |- ( ( ph /\ M = N ) -> ( M + 1 ) e. ZZ )
63	25	adantr 276	. . . . . . 7 |- ( ( ph /\ M = N ) -> N e. ZZ )
64		fzn 10199	. . . . . . 7 |- ( ( ( M + 1 ) e. ZZ /\ N e. ZZ ) -> ( N < ( M + 1 ) <-> ( ( M + 1 ) ... N ) = (/) ) )
65	62, 63, 64	syl2anc 411	. . . . . 6 |- ( ( ph /\ M = N ) -> ( N < ( M + 1 ) <-> ( ( M + 1 ) ... N ) = (/) ) )
66	61, 65	mpbid 147	. . . . 5 |- ( ( ph /\ M = N ) -> ( ( M + 1 ) ... N ) = (/) )
67	66	sumeq1d 11792	. . . 4 |- ( ( ph /\ M = N ) -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) = sum_ i e. (/) ( F ` i ) )
68		sum0 11814	. . . 4 |- sum_ i e. (/) ( F ` i ) = 0
69	67, 68	eqtrdi 2256	. . 3 |- ( ( ph /\ M = N ) -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) = 0 )
70	4, 6	ffvelcdmd 5739	. . . . 5 |- ( ph -> ( seq 1 ( + , F ) ` M ) e. CC )
71	70	adantr 276	. . . 4 |- ( ( ph /\ M = N ) -> ( seq 1 ( + , F ) ` M ) e. CC )
72	71	subidd 8406	. . 3 |- ( ( ph /\ M = N ) -> ( ( seq 1 ( + , F ) ` M ) - ( seq 1 ( + , F ) ` M ) ) = 0 )
73	57	fveq2d 5603	. . . 4 |- ( ( ph /\ M = N ) -> ( seq 1 ( + , F ) ` M ) = ( seq 1 ( + , F ) ` N ) )
74	73	oveq1d 5982	. . 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 2247	. 2 |- ( ( ph /\ M = N ) -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )
76		eluzle 9695	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
77	23, 76	syl 14	. . 3 |- ( ph -> M <_ N )
78		zleloe 9454	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( M < N \/ M = N ) ) )
79	10, 25, 78	syl2anc 411	. . 3 |- ( ph -> ( M <_ N <-> ( M < N \/ M = N ) ) )
80	77, 79	mpbid 147	. 2 |- ( ph -> ( M < N \/ M = N ) )
81	56, 75, 80	mpjaodan 800	1 |- ( ph -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   /\ w3a 981   = wceq 1373   e. wcel 2178   A.wral 2486   (/)c0 3468   class class class wbr 4059   -->wf 5286   ` cfv 5290  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   1c1 7961   + caddc 7963   x. cmul 7965   < clt 8142   <_ cle 8143   - cmin 8278   NNcn 9071   ZZcz 9407   ZZ>=cuz 9683   ...cfz 10165   seqcseq 10629   abscabs 11423   sum_csu 11779

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-frec 6500  df-1o 6525  df-oadd 6529  df-er 6643  df-en 6851  df-dom 6852  df-fin 6853  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-seqfrec 10630  df-exp 10721  df-ihash 10958  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-clim 11705  df-sumdc 11780

This theorem is referenced by:  cvgratnnlemrate  11956