Theorem cvgratnnlemseq 12150

Description: Lemma for cvgratnn 12155. (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 9836	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
2		1zzd 9550	. . . . . . 7 |- ( ph -> 1 e. ZZ )
3		cvgratnn.6	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
4	1, 2, 3	serf 10791	. . . . . 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 5791	. . . 4 |- ( ( ph /\ M < N ) -> ( seq 1 ( + , F ) ` M ) e. CC )
9		eqid 2231	. . . . . . 7 |- ( ZZ>= ` ( M + 1 ) ) = ( ZZ>= ` ( M + 1 ) )
10	6	nnzd 9645	. . . . . . . 8 |- ( ph -> M e. ZZ )
11	10	peano2zd 9649	. . . . . . 7 |- ( ph -> ( M + 1 ) e. ZZ )
12		fveq2 5648	. . . . . . . . 9 |- ( k = x -> ( F ` k ) = ( F ` x ) )
13	12	eleq1d 2300	. . . . . . . 8 |- ( k = x -> ( ( F ` k ) e. CC <-> ( F ` x ) e. CC ) )
14	3	ralrimiva 2606	. . . . . . . . 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 9200	. . . . . . . . 9 |- ( ph -> ( M + 1 ) e. NN )
17		eluznn 9878	. . . . . . . . 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 2916	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. CC )
20	9, 11, 19	serf 10791	. . . . . 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 9809	. . . . . . . 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 9578	. . . . . . . 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 9805	. . . . . 6 |- ( N e. ( ZZ>= ` ( M + 1 ) ) <-> ( ( M + 1 ) e. ZZ /\ N e. ZZ /\ ( M + 1 ) <_ N ) )
31	22, 26, 29, 30	syl3anbrc 1208	. . . . 5 |- ( ( ph /\ M < N ) -> N e. ( ZZ>= ` ( M + 1 ) ) )
32	21, 31	ffvelcdmd 5791	. . . 4 |- ( ( ph /\ M < N ) -> ( seq ( M + 1 ) ( + , F ) ` N ) e. CC )
33	8, 32	pncan2d 8534	. . 3 |- ( ( ph /\ M < N ) -> ( ( ( seq 1 ( + , F ) ` M ) + ( seq ( M + 1 ) ( + , F ) ` N ) ) - ( seq 1 ( + , F ) ` M ) ) = ( seq ( M + 1 ) ( + , F ) ` N ) )
34		addcl 8200	. . . . . 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 8205	. . . . . 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 2324	. . . . . 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 2325	. . . . . 6 |- ( ( ( ph /\ M < N ) /\ x e. ( ZZ>= ` 1 ) ) -> x e. NN )
43	13, 40, 42	rspcdva 2916	. . . . 5 |- ( ( ( ph /\ M < N ) /\ x e. ( ZZ>= ` 1 ) ) -> ( F ` x ) e. CC )
44	35, 37, 31, 39, 43	seq3split 10796	. . . 4 |- ( ( ph /\ M < N ) -> ( seq 1 ( + , F ) ` N ) = ( ( seq 1 ( + , F ) ` M ) + ( seq ( M + 1 ) ( + , F ) ` N ) ) )
45	44	oveq1d 6043	. . 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 2232	. . . 4 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` i ) = ( F ` i ) )
47		fveq2 5648	. . . . . 6 |- ( k = i -> ( F ` k ) = ( F ` i ) )
48	47	eleq1d 2300	. . . . 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 9878	. . . . . 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 2916	. . . 4 |- ( ( ( ph /\ M < N ) /\ i e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` i ) e. CC )
55	46, 31, 54	fsum3ser 12021	. . 3 |- ( ( ph /\ M < N ) -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) = ( seq ( M + 1 ) ( + , F ) ` N ) )
56	33, 45, 55	3eqtr4d 2274	. 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 9198	. . . . . . . . 9 |- ( ph -> M e. RR )
59	58	ltp1d 9152	. . . . . . . 8 |- ( ph -> M < ( M + 1 ) )
60	59	adantr 276	. . . . . . 7 |- ( ( ph /\ M = N ) -> M < ( M + 1 ) )
61	57, 60	eqbrtrrd 4117	. . . . . 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 10322	. . . . . . 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 11989	. . . 4 |- ( ( ph /\ M = N ) -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) = sum_ i e. (/) ( F ` i ) )
68		sum0 12012	. . . 4 |- sum_ i e. (/) ( F ` i ) = 0
69	67, 68	eqtrdi 2280	. . 3 |- ( ( ph /\ M = N ) -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) = 0 )
70	4, 6	ffvelcdmd 5791	. . . . 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 8520	. . 3 |- ( ( ph /\ M = N ) -> ( ( seq 1 ( + , F ) ` M ) - ( seq 1 ( + , F ) ` M ) ) = 0 )
73	57	fveq2d 5652	. . . 4 |- ( ( ph /\ M = N ) -> ( seq 1 ( + , F ) ` M ) = ( seq 1 ( + , F ) ` N ) )
74	73	oveq1d 6043	. . 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 2271	. 2 |- ( ( ph /\ M = N ) -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )
76		eluzle 9812	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
77	23, 76	syl 14	. . 3 |- ( ph -> M <_ N )
78		zleloe 9570	. . . 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 806	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 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   (/)c0 3496   class class class wbr 4093   -->wf 5329   ` cfv 5333  (class class class)co 6028   CCcc 8073   RRcr 8074   0cc0 8075   1c1 8076   + caddc 8078   x. cmul 8080   < clt 8256   <_ cle 8257   - cmin 8392   NNcn 9185   ZZcz 9523   ZZ>=cuz 9799   ...cfz 10288   seqcseq 10755   abscabs 11620   sum_csu 11976

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-fz 10289  df-fzo 10423  df-seqfrec 10756  df-exp 10847  df-ihash 11084  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-clim 11902  df-sumdc 11977

This theorem is referenced by:  cvgratnnlemrate  12154