Theorem clim2div 14621

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 )
clim2div.5	|- ( ph -> ( seq M ( x. , F ) ` N ) =/= 0 )

Assertion

Ref	Expression
clim2div	|- ( 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 clim2div

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) )
2		clim2div.2	. . . . 5 |- ( ph -> N e. Z )
3		eluzelz 11697	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4		clim2div.1	. . . . . 6 |- Z = ( ZZ>= ` M )
5	3, 4	eleq2s 2719	. . . . 5 |- ( N e. Z -> N e. ZZ )
6	2, 5	syl 17	. . . 4 |- ( ph -> N e. ZZ )
7	6	peano2zd 11485	. . 3 |- ( ph -> ( N + 1 ) e. ZZ )
8		clim2div.4	. . 3 |- ( ph -> seq M ( x. , F ) ~~> A )
9		eluzel2 11692	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
10	9, 4	eleq2s 2719	. . . . . . 7 |- ( N e. Z -> M e. ZZ )
11	2, 10	syl 17	. . . . . 6 |- ( ph -> M e. ZZ )
12		clim2div.3	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
13	4, 11, 12	prodf 14619	. . . . 5 |- ( ph -> seq M ( x. , F ) : Z --> CC )
14	13, 2	ffvelrnd 6360	. . . 4 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
15		clim2div.5	. . . 4 |- ( ph -> ( seq M ( x. , F ) ` N ) =/= 0 )
16	14, 15	reccld 10794	. . 3 |- ( ph -> ( 1 / ( seq M ( x. , F ) ` N ) ) e. CC )
17		seqex 12803	. . . 4 |- seq ( N + 1 ) ( x. , F ) e. _V
18	17	a1i 11	. . 3 |- ( ph -> seq ( N + 1 ) ( x. , F ) e. _V )
19	2, 4	syl6eleq 2711	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
20		peano2uz 11741	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
21	19, 20	syl 17	. . . . . 6 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )
22	21, 4	syl6eleqr 2712	. . . . 5 |- ( ph -> ( N + 1 ) e. Z )
23	4	uztrn2 11705	. . . . 5 |- ( ( ( N + 1 ) e. Z /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. Z )
24	22, 23	sylan 488	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. Z )
25	13	ffvelrnda 6359	. . . 4 |- ( ( ph /\ j e. Z ) -> ( seq M ( x. , F ) ` j ) e. CC )
26	24, 25	syldan 487	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` j ) e. CC )
27		mulcl 10020	. . . . . . . 8 |- ( ( k e. CC /\ x e. CC ) -> ( k x. x ) e. CC )
28	27	adantl 482	. . . . . . 7 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( k e. CC /\ x e. CC ) ) -> ( k x. x ) e. CC )
29		mulass 10024	. . . . . . . 8 |- ( ( k e. CC /\ x e. CC /\ y e. CC ) -> ( ( k x. x ) x. y ) = ( k x. ( x x. y ) ) )
30	29	adantl 482	. . . . . . 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 477	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. ( ZZ>= ` ( N + 1 ) ) )
32	19	adantr 481	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> N e. ( ZZ>= ` M ) )
33		elfzuz 12338	. . . . . . . . . 10 |- ( k e. ( M ... j ) -> k e. ( ZZ>= ` M ) )
34	33, 4	syl6eleqr 2712	. . . . . . . . 9 |- ( k e. ( M ... j ) -> k e. Z )
35	34, 12	sylan2 491	. . . . . . . 8 |- ( ( ph /\ k e. ( M ... j ) ) -> ( F ` k ) e. CC )
36	35	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ k e. ( M ... j ) ) -> ( F ` k ) e. CC )
37	28, 30, 31, 32, 36	seqsplit 12834	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` j ) = ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` j ) ) )
38	37	eqcomd 2628	. . . . 5 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( ( seq M ( x. , F ) ` N ) x. ( seq ( N + 1 ) ( x. , F ) ` j ) ) = ( seq M ( x. , F ) ` j ) )
39	14	adantr 481	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` N ) e. CC )
40	4	uztrn2 11705	. . . . . . . . . 10 |- ( ( ( N + 1 ) e. Z /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
41	22, 40	sylan 488	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
42	41, 12	syldan 487	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` k ) e. CC )
43	1, 7, 42	prodf 14619	. . . . . . 7 |- ( ph -> seq ( N + 1 ) ( x. , F ) : ( ZZ>= ` ( N + 1 ) ) --> CC )
44	43	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( x. , F ) ` j ) e. CC )
45	15	adantr 481	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` N ) =/= 0 )
46	26, 39, 44, 45	divmuld 10823	. . . . 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 ) ) )
47	38, 46	mpbird 247	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( ( seq M ( x. , F ) ` j ) / ( seq M ( x. , F ) ` N ) ) = ( seq ( N + 1 ) ( x. , F ) ` j ) )
48	26, 39, 45	divrec2d 10805	. . . 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 ) ) )
49	47, 48	eqtr3d 2658	. . 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 ) ) )
50	1, 7, 8, 16, 18, 26, 49	climmulc2 14367	. 2 |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> ( ( 1 / ( seq M ( x. , F ) ` N ) ) x. A ) )
51		climcl 14230	. . . 4 |- ( seq M ( x. , F ) ~~> A -> A e. CC )
52	8, 51	syl 17	. . 3 |- ( ph -> A e. CC )
53	52, 14, 15	divrec2d 10805	. 2 |- ( ph -> ( A / ( seq M ( x. , F ) ` N ) ) = ( ( 1 / ( seq M ( x. , F ) ` N ) ) x. A ) )
54	50, 53	breqtrrd 4681	1 |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> ( A / ( seq M ( x. , F ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   / cdiv 10684   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219

This theorem is referenced by:  ntrivcvgtail  14632