Theorem clim2divap 11561

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 2187	. . 3 |- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) )
2		clim2div.2	. . . . 5 |- ( ph -> N e. Z )
3		eluzelz 9550	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4		clim2div.1	. . . . . 6 |- Z = ( ZZ>= ` M )
5	3, 4	eleq2s 2282	. . . . 5 |- ( N e. Z -> N e. ZZ )
6	2, 5	syl 14	. . . 4 |- ( ph -> N e. ZZ )
7	6	peano2zd 9391	. . 3 |- ( ph -> ( N + 1 ) e. ZZ )
8		clim2div.4	. . 3 |- ( ph -> seq M ( x. , F ) ~~> A )
9		eluzel2 9546	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
10	9, 4	eleq2s 2282	. . . . . . 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 11559	. . . . 5 |- ( ph -> seq M ( x. , F ) : Z --> CC )
14	13, 2	ffvelcdmd 5665	. . . 4 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
15		clim2divap.5	. . . 4 |- ( ph -> ( seq M ( x. , F ) ` N ) # 0 )
16	14, 15	recclapd 8751	. . 3 |- ( ph -> ( 1 / ( seq M ( x. , F ) ` N ) ) e. CC )
17		seqex 10460	. . . 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 2280	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
20		peano2uz 9596	. . . . . . 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 2281	. . . . 5 |- ( ph -> ( N + 1 ) e. Z )
23	4	uztrn2 9558	. . . . 5 |- ( ( ( N + 1 ) e. Z /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. Z )
24	22, 23	sylan 283	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. Z )
25	13	ffvelcdmda 5664	. . . 4 |- ( ( ph /\ j e. Z ) -> ( seq M ( x. , F ) ` j ) e. CC )
26	24, 25	syldan 282	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` j ) e. CC )
27		mulcl 7951	. . . . . . . 8 |- ( ( k e. CC /\ x e. CC ) -> ( k x. x ) e. CC )
28	27	adantl 277	. . . . . . 7 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( k e. CC /\ x e. CC ) ) -> ( k x. x ) e. CC )
29		mulass 7955	. . . . . . . 8 |- ( ( k e. CC /\ x e. CC /\ y e. CC ) -> ( ( k x. x ) x. y ) = ( k x. ( x x. y ) ) )
30	29	adantl 277	. . . . . . 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 110	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. ( ZZ>= ` ( N + 1 ) ) )
32	19	adantr 276	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> N e. ( ZZ>= ` M ) )
33	4	eleq2i 2254	. . . . . . . . 9 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
34	33, 12	sylan2br 288	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
35	34	adantlr 477	. . . . . . 7 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
36	28, 30, 31, 32, 35	seq3split 10492	. . . . . 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 2193	. . . . 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 276	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` N ) e. CC )
39	4	uztrn2 9558	. . . . . . . . . 10 |- ( ( ( N + 1 ) e. Z /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
40	22, 39	sylan 283	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
41	40, 12	syldan 282	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` k ) e. CC )
42	1, 7, 41	prodf 11559	. . . . . . 7 |- ( ph -> seq ( N + 1 ) ( x. , F ) : ( ZZ>= ` ( N + 1 ) ) --> CC )
43	42	ffvelcdmda 5664	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( x. , F ) ` j ) e. CC )
44	15	adantr 276	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( x. , F ) ` N ) # 0 )
45	26, 38, 43, 44	divmulapd 8782	. . . . 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 167	. . . 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 8764	. . . 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 2222	. . 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 11352	. 2 |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> ( ( 1 / ( seq M ( x. , F ) ` N ) ) x. A ) )
50		climcl 11303	. . . 4 |- ( seq M ( x. , F ) ~~> A -> A e. CC )
51	8, 50	syl 14	. . 3 |- ( ph -> A e. CC )
52	51, 14, 15	divrecap2d 8764	. 2 |- ( ph -> ( A / ( seq M ( x. , F ) ` N ) ) = ( ( 1 / ( seq M ( x. , F ) ` N ) ) x. A ) )
53	49, 52	breqtrrd 4043	1 |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> ( A / ( seq M ( x. , F ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 979   = wceq 1363   e. wcel 2158   _Vcvv 2749   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   CCcc 7822   0cc0 7824   1c1 7825   + caddc 7827   x. cmul 7829   # cap 8551   / cdiv 8642   ZZcz 9266   ZZ>=cuz 9541   seqcseq 10458   ~~> cli 11299

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-rp 9667  df-fz 10022  df-seqfrec 10459  df-exp 10533  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021  df-clim 11300

This theorem is referenced by: (None)