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Theorem clim2divap 11316
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.
StepHypRef Expression
1 eqid 2139 . . 3  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
2 clim2div.2 . . . . 5  |-  ( ph  ->  N  e.  Z )
3 eluzelz 9342 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
4 clim2div.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
53, 4eleq2s 2234 . . . . 5  |-  ( N  e.  Z  ->  N  e.  ZZ )
62, 5syl 14 . . . 4  |-  ( ph  ->  N  e.  ZZ )
76peano2zd 9183 . . 3  |-  ( ph  ->  ( N  +  1 )  e.  ZZ )
8 clim2div.4 . . 3  |-  ( ph  ->  seq M (  x.  ,  F )  ~~>  A )
9 eluzel2 9338 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
109, 4eleq2s 2234 . . . . . . 7  |-  ( N  e.  Z  ->  M  e.  ZZ )
112, 10syl 14 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
12 clim2div.3 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
134, 11, 12prodf 11314 . . . . 5  |-  ( ph  ->  seq M (  x.  ,  F ) : Z --> CC )
1413, 2ffvelrnd 5556 . . . 4  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N )  e.  CC )
15 clim2divap.5 . . . 4  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N ) #  0 )
1614, 15recclapd 8548 . . 3  |-  ( ph  ->  ( 1  /  (  seq M (  x.  ,  F ) `  N
) )  e.  CC )
17 seqex 10227 . . . 4  |-  seq ( N  +  1 ) (  x.  ,  F
)  e.  _V
1817a1i 9 . . 3  |-  ( ph  ->  seq ( N  + 
1 ) (  x.  ,  F )  e. 
_V )
192, 4eleqtrdi 2232 . . . . . . 7  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
20 peano2uz 9385 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
2119, 20syl 14 . . . . . 6  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )
2221, 4eleqtrrdi 2233 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  Z )
234uztrn2 9350 . . . . 5  |-  ( ( ( N  +  1 )  e.  Z  /\  j  e.  ( ZZ>= `  ( N  +  1
) ) )  -> 
j  e.  Z )
2422, 23sylan 281 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  j  e.  Z )
2513ffvelrnda 5555 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  x.  ,  F ) `  j
)  e.  CC )
2624, 25syldan 280 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  x.  ,  F ) `  j
)  e.  CC )
27 mulcl 7754 . . . . . . . 8  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
2827adantl 275 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  x.  x
)  e.  CC )
29 mulass 7758 . . . . . . . 8  |-  ( ( k  e.  CC  /\  x  e.  CC  /\  y  e.  CC )  ->  (
( k  x.  x
)  x.  y )  =  ( k  x.  ( x  x.  y
) ) )
3029adantl 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 ) ) )
3219adantr 274 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  ( ZZ>= `  M )
)
334eleq2i 2206 . . . . . . . . 9  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
3433, 12sylan2br 286 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3534adantlr 468 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3628, 30, 31, 32, 35seq3split 10259 . . . . . 6  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  x.  ,  F ) `  j
)  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  j
) ) )
3736eqcomd 2145 . . . . 5  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( (  seq M (  x.  ,  F ) `  N
)  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  j
) )  =  (  seq M (  x.  ,  F ) `  j ) )
3814adantr 274 . . . . . 6  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  x.  ,  F ) `  N
)  e.  CC )
394uztrn2 9350 . . . . . . . . . 10  |-  ( ( ( N  +  1 )  e.  Z  /\  k  e.  ( ZZ>= `  ( N  +  1
) ) )  -> 
k  e.  Z )
4022, 39sylan 281 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  Z )
4140, 12syldan 280 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  k )  e.  CC )
421, 7, 41prodf 11314 . . . . . . 7  |-  ( ph  ->  seq ( N  + 
1 ) (  x.  ,  F ) : ( ZZ>= `  ( N  +  1 ) ) --> CC )
4342ffvelrnda 5555 . . . . . 6  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq ( N  +  1
) (  x.  ,  F ) `  j
)  e.  CC )
4415adantr 274 . . . . . 6  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  x.  ,  F ) `  N
) #  0 )
4526, 38, 43, 44divmulapd 8579 . . . . 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 ) ) )
4637, 45mpbird 166 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( (  seq M (  x.  ,  F ) `  j
)  /  (  seq M (  x.  ,  F ) `  N
) )  =  (  seq ( N  + 
1 ) (  x.  ,  F ) `  j ) )
4726, 38, 44divrecap2d 8561 . . . 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
) ) )
4846, 47eqtr3d 2174 . . 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
) ) )
491, 7, 8, 16, 18, 26, 48climmulc2 11107 . 2  |-  ( ph  ->  seq ( N  + 
1 ) (  x.  ,  F )  ~~>  ( ( 1  /  (  seq M (  x.  ,  F ) `  N
) )  x.  A
) )
50 climcl 11058 . . . 4  |-  (  seq M (  x.  ,  F )  ~~>  A  ->  A  e.  CC )
518, 50syl 14 . . 3  |-  ( ph  ->  A  e.  CC )
5251, 14, 15divrecap2d 8561 . 2  |-  ( ph  ->  ( A  /  (  seq M (  x.  ,  F ) `  N
) )  =  ( ( 1  /  (  seq M (  x.  ,  F ) `  N
) )  x.  A
) )
5349, 52breqtrrd 3956 1  |-  ( ph  ->  seq ( N  + 
1 ) (  x.  ,  F )  ~~>  ( A  /  (  seq M
(  x.  ,  F
) `  N )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    = wceq 1331    e. wcel 1480   _Vcvv 2686   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7625   0cc0 7627   1c1 7628    + caddc 7630    x. cmul 7632   # cap 8350    / cdiv 8439   ZZcz 9061   ZZ>=cuz 9333    seqcseq 10225    ~~> cli 11054
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-n0 8985  df-z 9062  df-uz 9334  df-rp 9449  df-fz 9798  df-seqfrec 10226  df-exp 10300  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-clim 11055
This theorem is referenced by: (None)
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