Theorem cvgratnnlemseq 12271

Description: Lemma for cvgratnn 12276. (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 9937	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
2		1zzd 9650	. . . . . . 7 |- ( ph -> 1 e. ZZ )
3		cvgratnn.6	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
4	1, 2, 3	serf 10898	. . . . . 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 5835	. . . 4 |- ( ( ph /\ M < N ) -> ( seq 1 ( + , F ) ` M ) e. CC )
9		eqid 2238	. . . . . . 7 |- ( ZZ>= ` ( M + 1 ) ) = ( ZZ>= ` ( M + 1 ) )
10	6	nnzd 9746	. . . . . . . 8 |- ( ph -> M e. ZZ )
11	10	peano2zd 9750	. . . . . . 7 |- ( ph -> ( M + 1 ) e. ZZ )
12		fveq2 5690	. . . . . . . . 9 |- ( k = x -> ( F ` k ) = ( F ` x ) )
13	12	eleq1d 2307	. . . . . . . 8 |- ( k = x -> ( ( F ` k ) e. CC <-> ( F ` x ) e. CC ) )
14	3	ralrimiva 2623	. . . . . . . . 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 9298	. . . . . . . . 9 |- ( ph -> ( M + 1 ) e. NN )
17		eluznn 9979	. . . . . . . . 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 2934	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. CC )
20	9, 11, 19	serf 10898	. . . . . 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 9910	. . . . . . . 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 9678	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) )
28	10, 25, 27	syl2anc 415	. . . . . . 7 |- ( ph -> ( M < N <-> ( M + 1 ) <_ N ) )
29	28	biimpa 296	. . . . . 6 |- ( ( ph /\ M < N ) -> ( M + 1 ) <_ N )
30		eluz2 9906	. . . . . 6 |- ( N e. ( ZZ>= ` ( M + 1 ) ) <-> ( ( M + 1 ) e. ZZ /\ N e. ZZ /\ ( M + 1 ) <_ N ) )
31	22, 26, 29, 30	syl3anbrc 1212	. . . . 5 |- ( ( ph /\ M < N ) -> N e. ( ZZ>= ` ( M + 1 ) ) )
32	21, 31	ffvelcdmd 5835	. . . 4 |- ( ( ph /\ M < N ) -> ( seq ( M + 1 ) ( + , F ) ` N ) e. CC )
33	8, 32	pncan2d 8629	. . 3 |- ( ( ph /\ M < N ) -> ( ( ( seq 1 ( + , F ) ` M ) + ( seq ( M + 1 ) ( + , F ) ` N ) ) - ( seq 1 ( + , F ) ` M ) ) = ( seq ( M + 1 ) ( + , F ) ` N ) )
34		addcl 8294	. . . . . 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 8299	. . . . . 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 2331	. . . . . 6 |- ( ph -> M e. ( ZZ>= ` 1 ) )
39	38	adantr 276	. . . . 5 |- ( ( ph /\ M < N ) -> M e. ( ZZ>= ` 1 ) )
40	14	ad2antrr 492	. . . . . 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 2332	. . . . . 6 |- ( ( ( ph /\ M < N ) /\ x e. ( ZZ>= ` 1 ) ) -> x e. NN )
43	13, 40, 42	rspcdva 2934	. . . . 5 |- ( ( ( ph /\ M < N ) /\ x e. ( ZZ>= ` 1 ) ) -> ( F ` x ) e. CC )
44	35, 37, 31, 39, 43	seq3split 10903	. . . 4 |- ( ( ph /\ M < N ) -> ( seq 1 ( + , F ) ` N ) = ( ( seq 1 ( + , F ) ` M ) + ( seq ( M + 1 ) ( + , F ) ` N ) ) )
45	44	oveq1d 6090	. . 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 2239	. . . 4 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` i ) = ( F ` i ) )
47		fveq2 5690	. . . . . 6 |- ( k = i -> ( F ` k ) = ( F ` i ) )
48	47	eleq1d 2307	. . . . 5 |- ( k = i -> ( ( F ` k ) e. CC <-> ( F ` i ) e. CC ) )
49	14	ad2antrr 492	. . . . 5 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> A. k e. NN ( F ` k ) e. CC )
50	16	ad2antrr 492	. . . . . 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 9979	. . . . . 6 |- ( ( ( M + 1 ) e. NN /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> i e. NN )
53	50, 51, 52	syl2anc 415	. . . . 5 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> i e. NN )
54	48, 49, 53	rspcdva 2934	. . . 4 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` i ) e. CC )
55	46, 31, 54	fsum3ser 12142	. . 3 |- ( ( ph /\ M < N ) -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) = ( seq ( M + 1 ) ( + , F ) ` N ) )
56	33, 45, 55	3eqtr4d 2281	. 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 9296	. . . . . . . . 9 |- ( ph -> M e. RR )
59	58	ltp1d 9250	. . . . . . . 8 |- ( ph -> M < ( M + 1 ) )
60	59	adantr 276	. . . . . . 7 |- ( ( ph /\ M = N ) -> M < ( M + 1 ) )
61	57, 60	eqbrtrrd 4149	. . . . . 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 10425	. . . . . . 7 |- ( ( ( M + 1 ) e. ZZ /\ N e. ZZ ) -> ( N < ( M + 1 ) <-> ( ( M + 1 ) ... N ) = (/) ) )
65	62, 63, 64	syl2anc 415	. . . . . 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 12110	. . . 4 |- ( ( ph /\ M = N ) -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) = sum_ i e. (/) ( F ` i ) )
68		sum0 12133	. . . 4 |- sum_ i e. (/) ( F ` i ) = 0
69	67, 68	eqtrdi 2287	. . 3 |- ( ( ph /\ M = N ) -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) = 0 )
70	4, 6	ffvelcdmd 5835	. . . . 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 8615	. . 3 |- ( ( ph /\ M = N ) -> ( ( seq 1 ( + , F ) ` M ) - ( seq 1 ( + , F ) ` M ) ) = 0 )
73	57	fveq2d 5694	. . . 4 |- ( ( ph /\ M = N ) -> ( seq 1 ( + , F ) ` M ) = ( seq 1 ( + , F ) ` N ) )
74	73	oveq1d 6090	. . 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 2278	. 2 |- ( ( ph /\ M = N ) -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )
76		eluzle 9913	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
77	23, 76	syl 14	. . 3 |- ( ph -> M <_ N )
78		zleloe 9670	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( M < N \/ M = N ) ) )
79	10, 25, 78	syl2anc 415	. . 3 |- ( ph -> ( M <_ N <-> ( M < N \/ M = N ) ) )
80	77, 79	mpbid 147	. 2 |- ( ph -> ( M < N \/ M = N ) )
81	56, 75, 80	mpjaodan 810	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 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   (/)c0 3520   class class class wbr 4125   -->wf 5368   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   < clt 8350   <_ cle 8351   - cmin 8487   NNcn 9283   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   seqcseq 10862   abscabs 11741   sum_csu 12097

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-sumdc 12098

This theorem is referenced by:  cvgratnnlemrate  12275