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Theorem clim2divap 11567
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 2189 . . 3  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
2 clim2div.2 . . . . 5  |-  ( ph  ->  N  e.  Z )
3 eluzelz 9556 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
4 clim2div.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
53, 4eleq2s 2284 . . . . 5  |-  ( N  e.  Z  ->  N  e.  ZZ )
62, 5syl 14 . . . 4  |-  ( ph  ->  N  e.  ZZ )
76peano2zd 9397 . . 3  |-  ( ph  ->  ( N  +  1 )  e.  ZZ )
8 clim2div.4 . . 3  |-  ( ph  ->  seq M (  x.  ,  F )  ~~>  A )
9 eluzel2 9552 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
109, 4eleq2s 2284 . . . . . . 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 11565 . . . . 5  |-  ( ph  ->  seq M (  x.  ,  F ) : Z --> CC )
1413, 2ffvelcdmd 5668 . . . 4  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N )  e.  CC )
15 clim2divap.5 . . . 4  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N ) #  0 )
1614, 15recclapd 8757 . . 3  |-  ( ph  ->  ( 1  /  (  seq M (  x.  ,  F ) `  N
) )  e.  CC )
17 seqex 10466 . . . 4  |-  seq ( N  +  1 ) (  x.  ,  F
)  e.  _V
1817a1i 9 . . 3  |-  ( ph  ->  seq ( N  + 
1 ) (  x.  ,  F )  e. 
_V )
192, 4eleqtrdi 2282 . . . . . . 7  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
20 peano2uz 9602 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
2119, 20syl 14 . . . . . 6  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )
2221, 4eleqtrrdi 2283 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  Z )
234uztrn2 9564 . . . . 5  |-  ( ( ( N  +  1 )  e.  Z  /\  j  e.  ( ZZ>= `  ( N  +  1
) ) )  -> 
j  e.  Z )
2422, 23sylan 283 . . . 4  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  j  e.  Z )
2513ffvelcdmda 5667 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  x.  ,  F ) `  j
)  e.  CC )
2624, 25syldan 282 . . 3  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  x.  ,  F ) `  j
)  e.  CC )
27 mulcl 7957 . . . . . . . 8  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
2827adantl 277 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  x.  x
)  e.  CC )
29 mulass 7961 . . . . . . . 8  |-  ( ( k  e.  CC  /\  x  e.  CC  /\  y  e.  CC )  ->  (
( k  x.  x
)  x.  y )  =  ( k  x.  ( x  x.  y
) ) )
3029adantl 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 ) ) )
3219adantr 276 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  ( ZZ>= `  M )
)
334eleq2i 2256 . . . . . . . . 9  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
3433, 12sylan2br 288 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3534adantlr 477 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3628, 30, 31, 32, 35seq3split 10498 . . . . . 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 2195 . . . . 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 276 . . . . . 6  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  x.  ,  F ) `  N
)  e.  CC )
394uztrn2 9564 . . . . . . . . . 10  |-  ( ( ( N  +  1 )  e.  Z  /\  k  e.  ( ZZ>= `  ( N  +  1
) ) )  -> 
k  e.  Z )
4022, 39sylan 283 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  Z )
4140, 12syldan 282 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  k )  e.  CC )
421, 7, 41prodf 11565 . . . . . . 7  |-  ( ph  ->  seq ( N  + 
1 ) (  x.  ,  F ) : ( ZZ>= `  ( N  +  1 ) ) --> CC )
4342ffvelcdmda 5667 . . . . . 6  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq ( N  +  1
) (  x.  ,  F ) `  j
)  e.  CC )
4415adantr 276 . . . . . 6  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  x.  ,  F ) `  N
) #  0 )
4526, 38, 43, 44divmulapd 8788 . . . . 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 167 . . . 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 8770 . . . 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 2224 . . 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 11358 . 2  |-  ( ph  ->  seq ( N  + 
1 ) (  x.  ,  F )  ~~>  ( ( 1  /  (  seq M (  x.  ,  F ) `  N
) )  x.  A
) )
50 climcl 11309 . . . 4  |-  (  seq M (  x.  ,  F )  ~~>  A  ->  A  e.  CC )
518, 50syl 14 . . 3  |-  ( ph  ->  A  e.  CC )
5251, 14, 15divrecap2d 8770 . 2  |-  ( ph  ->  ( A  /  (  seq M (  x.  ,  F ) `  N
) )  =  ( ( 1  /  (  seq M (  x.  ,  F ) `  N
) )  x.  A
) )
5349, 52breqtrrd 4046 1  |-  ( ph  ->  seq ( N  + 
1 ) (  x.  ,  F )  ~~>  ( A  /  (  seq M
(  x.  ,  F
) `  N )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    = wceq 1364    e. wcel 2160   _Vcvv 2752   class class class wbr 4018   ` cfv 5231  (class class class)co 5891   CCcc 7828   0cc0 7830   1c1 7831    + caddc 7833    x. cmul 7835   # cap 8557    / cdiv 8648   ZZcz 9272   ZZ>=cuz 9547    seqcseq 10464    ~~> cli 11305
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947  ax-pre-mulext 7948  ax-arch 7949  ax-caucvg 7950
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-reap 8551  df-ap 8558  df-div 8649  df-inn 8939  df-2 8997  df-3 8998  df-4 8999  df-n0 9196  df-z 9273  df-uz 9548  df-rp 9673  df-fz 10028  df-seqfrec 10465  df-exp 10539  df-cj 10870  df-re 10871  df-im 10872  df-rsqrt 11026  df-abs 11027  df-clim 11306
This theorem is referenced by: (None)
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