Theorem clim2divap 11503

Description: The limit of an infinite product with an initial segment removed. (Contributed by Scott Fenton, 20-Dec-2017.)

Hypotheses

Ref	Expression
clim2div.1	|- Z = ( ZZ>= ` M )
clim2div.2	|- ( ph -> N e. Z )
clim2div.3	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
clim2div.4	|- ( ph -> seq M ( x. , F ) ~~> A )
clim2divap.5	|- ( ph -> ( seq M ( x. , F ) ` N ) # 0 )

Assertion

Ref	Expression
clim2divap	|- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> ( A / ( seq M ( x. , F ) ` N ) ) )

Distinct variable groups:   k, F   ph, k   k, M   k, N   k, Z

Allowed substitution hint:   A( k)

Proof of Theorem clim2divap

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

Step	Hyp	Ref	Expression
1		eqid 2170	. . 3 |- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) )
2		clim2div.2	. . . . 5 |- ( ph -> N e. Z )
3		eluzelz 9496	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4		clim2div.1	. . . . . 6 |- Z = ( ZZ>= ` M )
5	3, 4	eleq2s 2265	. . . . 5 |- ( N e. Z -> N e. ZZ )
6	2, 5	syl 14	. . . 4 |- ( ph -> N e. ZZ )
7	6	peano2zd 9337	. . 3 |- ( ph -> ( N + 1 ) e. ZZ )
8		clim2div.4	. . 3 |- ( ph -> seq M ( x. , F ) ~~> A )
9		eluzel2 9492	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
10	9, 4	eleq2s 2265	. . . . . . 7 |- ( N e. Z -> M e. ZZ )
11	2, 10	syl 14	. . . . . 6 |- ( ph -> M e. ZZ )
12		clim2div.3	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
13	4, 11, 12	prodf 11501	. . . . 5 |- ( ph -> seq M ( x. , F ) : Z --> CC )
14	13, 2	ffvelrnd 5632	. . . 4 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
15		clim2divap.5	. . . 4 |- ( ph -> ( seq M ( x. , F ) ` N ) # 0 )
16	14, 15	recclapd 8698	. . 3 |- ( ph -> ( 1 / ( seq M ( x. , F ) ` N ) ) e. CC )
17		seqex 10403	. . . 4 |- seq ( N + 1 ) ( x. , F ) e. _V
18	17	a1i 9	. . 3 |- ( ph -> seq ( N + 1 ) ( x. , F ) e. _V )
19	2, 4	eleqtrdi 2263	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
20		peano2uz 9542	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
21	19, 20	syl 14	. . . . . 6 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )
22	21, 4	eleqtrrdi 2264	. . . . 5 |- ( ph -> ( N + 1 ) e. Z )
23	4	uztrn2 9504	. . . . 5 |- ( ( ( N + 1 ) e. Z /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. Z )
24	22, 23	sylan 281	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. Z )
25	13	ffvelrnda 5631	. . . 4 |- ( ( ph /\ j e. Z ) -> ( seq M ( x. , F ) ` j ) e. CC )
26	24, 25	syldan 280	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` j ) e. CC )
27		mulcl 7901	. . . . . . . 8 |- ( ( k e. CC /\ x e. CC ) -> ( k x. x ) e. CC )
28	27	adantl 275	. . . . . . 7 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( k e. CC /\ x e. CC ) ) -> ( k x. x ) e. CC )
29		mulass 7905	. . . . . . . 8 |- ( ( k e. CC /\ x e. CC /\ y e. CC ) -> ( ( k x. x ) x. y ) = ( k x. ( x x. y ) ) )
30	29	adantl 275	. . . . . . 7 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( k e. CC /\ x e. CC /\ y e. CC ) ) -> ( ( k x. x ) x. y ) = ( k x. ( x x. y ) ) )
31		simpr 109	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. ( ZZ>= ` ( N + 1 ) ) )
32	19	adantr 274	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> N e. ( ZZ>= ` M ) )
33	4	eleq2i 2237	. . . . . . . . 9 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
34	33, 12	sylan2br 286	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
35	34	adantlr 474	. . . . . . 7 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
36	28, 30, 31, 32, 35	seq3split 10435	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` j ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` j ) ) )
37	36	eqcomd 2176	. . . . 5 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` j ) ) = ( seq M ( x. , F ) ` j ) )
38	14	adantr 274	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` N ) e. CC )
39	4	uztrn2 9504	. . . . . . . . . 10 |- ( ( ( N + 1 ) e. Z /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
40	22, 39	sylan 281	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
41	40, 12	syldan 280	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` k ) e. CC )
42	1, 7, 41	prodf 11501	. . . . . . 7 |- ( ph -> seq ( N + 1 ) ( x. , F ) : ( ZZ>= ` ( N + 1 ) ) --> CC )
43	42	ffvelrnda 5631	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( x. , F ) ` j ) e. CC )
44	15	adantr 274	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` N ) # 0 )
45	26, 38, 43, 44	divmulapd 8729	. . . . 5 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( ( ( seq M ( x. , F ) ` j ) / ( seq M ( x. , F ) ` N ) ) = ( seq ( N + 1 ) ( x. , F ) ` j ) <-> ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` j ) ) = ( seq M ( x. , F ) ` j ) ) )
46	37, 45	mpbird 166	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( ( seq M ( x. , F ) ` j ) / ( seq M ( x. , F ) ` N ) ) = ( seq ( N + 1 ) ( x. , F ) ` j ) )
47	26, 38, 44	divrecap2d 8711	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( ( seq M ( x. , F ) ` j ) / ( seq M ( x. , F ) ` N ) ) = ( ( 1 / ( seq M ( x. , F ) ` N ) ) x. ( seq M ( x. , F ) ` j ) ) )
48	46, 47	eqtr3d 2205	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( x. , F ) ` j ) = ( ( 1 / ( seq M ( x. , F ) ` N ) ) x. ( seq M ( x. , F ) ` j ) ) )
49	1, 7, 8, 16, 18, 26, 48	climmulc2 11294	. 2 |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> ( ( 1 / ( seq M ( x. , F ) ` N ) ) x. A ) )
50		climcl 11245	. . . 4 |- ( seq M ( x. , F ) ~~> A -> A e. CC )
51	8, 50	syl 14	. . 3 |- ( ph -> A e. CC )
52	51, 14, 15	divrecap2d 8711	. 2 |- ( ph -> ( A / ( seq M ( x. , F ) ` N ) ) = ( ( 1 / ( seq M ( x. , F ) ` N ) ) x. A ) )
53	49, 52	breqtrrd 4017	1 |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> ( A / ( seq M ( x. , F ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   _Vcvv 2730   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   # cap 8500   / cdiv 8589   ZZcz 9212   ZZ>=cuz 9487   seqcseq 10401   ~~> cli 11241

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-fz 9966  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242

This theorem is referenced by: (None)