Theorem cvgratnnlemseq 11808

Description: Lemma for cvgratnn 11813. (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 9683	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
2		1zzd 9398	. . . . . . 7 |- ( ph -> 1 e. ZZ )
3		cvgratnn.6	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
4	1, 2, 3	serf 10626	. . . . . 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 5715	. . . 4 |- ( ( ph /\ M < N ) -> ( seq 1 ( + , F ) ` M ) e. CC )
9		eqid 2204	. . . . . . 7 |- ( ZZ>= ` ( M + 1 ) ) = ( ZZ>= ` ( M + 1 ) )
10	6	nnzd 9493	. . . . . . . 8 |- ( ph -> M e. ZZ )
11	10	peano2zd 9497	. . . . . . 7 |- ( ph -> ( M + 1 ) e. ZZ )
12		fveq2 5575	. . . . . . . . 9 |- ( k = x -> ( F ` k ) = ( F ` x ) )
13	12	eleq1d 2273	. . . . . . . 8 |- ( k = x -> ( ( F ` k ) e. CC <-> ( F ` x ) e. CC ) )
14	3	ralrimiva 2578	. . . . . . . . 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 9050	. . . . . . . . 9 |- ( ph -> ( M + 1 ) e. NN )
17		eluznn 9720	. . . . . . . . 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 2881	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. CC )
20	9, 11, 19	serf 10626	. . . . . 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 9656	. . . . . . . 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 9426	. . . . . . . 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 9653	. . . . . 6 |- ( N e. ( ZZ>= ` ( M + 1 ) ) <-> ( ( M + 1 ) e. ZZ /\ N e. ZZ /\ ( M + 1 ) <_ N ) )
31	22, 26, 29, 30	syl3anbrc 1183	. . . . 5 |- ( ( ph /\ M < N ) -> N e. ( ZZ>= ` ( M + 1 ) ) )
32	21, 31	ffvelcdmd 5715	. . . 4 |- ( ( ph /\ M < N ) -> ( seq ( M + 1 ) ( + , F ) ` N ) e. CC )
33	8, 32	pncan2d 8384	. . 3 |- ( ( ph /\ M < N ) -> ( ( ( seq 1 ( + , F ) ` M ) + ( seq ( M + 1 ) ( + , F ) ` N ) ) - ( seq 1 ( + , F ) ` M ) ) = ( seq ( M + 1 ) ( + , F ) ` N ) )
34		addcl 8049	. . . . . 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 8054	. . . . . 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 2297	. . . . . 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 2298	. . . . . 6 |- ( ( ( ph /\ M < N ) /\ x e. ( ZZ>= ` 1 ) ) -> x e. NN )
43	13, 40, 42	rspcdva 2881	. . . . 5 |- ( ( ( ph /\ M < N ) /\ x e. ( ZZ>= ` 1 ) ) -> ( F ` x ) e. CC )
44	35, 37, 31, 39, 43	seq3split 10631	. . . 4 |- ( ( ph /\ M < N ) -> ( seq 1 ( + , F ) ` N ) = ( ( seq 1 ( + , F ) ` M ) + ( seq ( M + 1 ) ( + , F ) ` N ) ) )
45	44	oveq1d 5958	. . 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 2205	. . . 4 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` i ) = ( F ` i ) )
47		fveq2 5575	. . . . . 6 |- ( k = i -> ( F ` k ) = ( F ` i ) )
48	47	eleq1d 2273	. . . . 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 9720	. . . . . 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 2881	. . . 4 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` i ) e. CC )
55	46, 31, 54	fsum3ser 11679	. . 3 |- ( ( ph /\ M < N ) -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) = ( seq ( M + 1 ) ( + , F ) ` N ) )
56	33, 45, 55	3eqtr4d 2247	. 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 9048	. . . . . . . . 9 |- ( ph -> M e. RR )
59	58	ltp1d 9002	. . . . . . . 8 |- ( ph -> M < ( M + 1 ) )
60	59	adantr 276	. . . . . . 7 |- ( ( ph /\ M = N ) -> M < ( M + 1 ) )
61	57, 60	eqbrtrrd 4067	. . . . . 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 10163	. . . . . . 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 11648	. . . 4 |- ( ( ph /\ M = N ) -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) = sum_ i e. (/) ( F ` i ) )
68		sum0 11670	. . . 4 |- sum_ i e. (/) ( F ` i ) = 0
69	67, 68	eqtrdi 2253	. . 3 |- ( ( ph /\ M = N ) -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) = 0 )
70	4, 6	ffvelcdmd 5715	. . . . 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 8370	. . 3 |- ( ( ph /\ M = N ) -> ( ( seq 1 ( + , F ) ` M ) - ( seq 1 ( + , F ) ` M ) ) = 0 )
73	57	fveq2d 5579	. . . 4 |- ( ( ph /\ M = N ) -> ( seq 1 ( + , F ) ` M ) = ( seq 1 ( + , F ) ` N ) )
74	73	oveq1d 5958	. . 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 2244	. 2 |- ( ( ph /\ M = N ) -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )
76		eluzle 9659	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
77	23, 76	syl 14	. . 3 |- ( ph -> M <_ N )
78		zleloe 9418	. . . 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 799	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 709   /\ w3a 980   = wceq 1372   e. wcel 2175   A.wral 2483   (/)c0 3459   class class class wbr 4043   -->wf 5266   ` cfv 5270  (class class class)co 5943   CCcc 7922   RRcr 7923   0cc0 7924   1c1 7925   + caddc 7927   x. cmul 7929   < clt 8106   <_ cle 8107   - cmin 8242   NNcn 9035   ZZcz 9371   ZZ>=cuz 9647   ...cfz 10129   seqcseq 10590   abscabs 11279   sum_csu 11635

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-isom 5279  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-frec 6476  df-1o 6501  df-oadd 6505  df-er 6619  df-en 6827  df-dom 6828  df-fin 6829  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-z 9372  df-uz 9648  df-q 9740  df-rp 9775  df-fz 10130  df-fzo 10264  df-seqfrec 10591  df-exp 10682  df-ihash 10919  df-cj 11124  df-re 11125  df-im 11126  df-rsqrt 11280  df-abs 11281  df-clim 11561  df-sumdc 11636

This theorem is referenced by:  cvgratnnlemrate  11812