ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  clim2divap Unicode version

Theorem clim2divap 11496
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 2170 . . 3  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
2 clim2div.2 . . . . 5  |-  ( ph  ->  N  e.  Z )
3 eluzelz 9489 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
4 clim2div.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
53, 4eleq2s 2265 . . . . 5  |-  ( N  e.  Z  ->  N  e.  ZZ )
62, 5syl 14 . . . 4  |-  ( ph  ->  N  e.  ZZ )
76peano2zd 9330 . . 3  |-  ( ph  ->  ( N  +  1 )  e.  ZZ )
8 clim2div.4 . . 3  |-  ( ph  ->  seq M (  x.  ,  F )  ~~>  A )
9 eluzel2 9485 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
109, 4eleq2s 2265 . . . . . . 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 11494 . . . . 5  |-  ( ph  ->  seq M (  x.  ,  F ) : Z --> CC )
1413, 2ffvelrnd 5630 . . . 4  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N )  e.  CC )
15 clim2divap.5 . . . 4  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N ) #  0 )
1614, 15recclapd 8691 . . 3  |-  ( ph  ->  ( 1  /  (  seq M (  x.  ,  F ) `  N
) )  e.  CC )
17 seqex 10396 . . . 4  |-  seq ( N  +  1 ) (  x.  ,  F
)  e.  _V
1817a1i 9 . . 3  |-  ( ph  ->  seq ( N  + 
1 ) (  x.  ,  F )  e. 
_V )
192, 4eleqtrdi 2263 . . . . . . 7  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
20 peano2uz 9535 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
2119, 20syl 14 . . . . . 6  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )
2221, 4eleqtrrdi 2264 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  Z )
234uztrn2 9497 . . . . 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 5629 . . . 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 7894 . . . . . . . 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 7898 . . . . . . . 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 2237 . . . . . . . . 9  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
3433, 12sylan2br 286 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3534adantlr 474 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
3628, 30, 31, 32, 35seq3split 10428 . . . . . 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 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 ) )
3814adantr 274 . . . . . 6  |-  ( (
ph  /\  j  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  x.  ,  F ) `  N
)  e.  CC )
394uztrn2 9497 . . . . . . . . . 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 11494 . . . . . . 7  |-  ( ph  ->  seq ( N  + 
1 ) (  x.  ,  F ) : ( ZZ>= `  ( N  +  1 ) ) --> CC )
4342ffvelrnda 5629 . . . . . 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 8722 . . . . 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 8704 . . . 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 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
) ) )
491, 7, 8, 16, 18, 26, 48climmulc2 11287 . 2  |-  ( ph  ->  seq ( N  + 
1 ) (  x.  ,  F )  ~~>  ( ( 1  /  (  seq M (  x.  ,  F ) `  N
) )  x.  A
) )
50 climcl 11238 . . . 4  |-  (  seq M (  x.  ,  F )  ~~>  A  ->  A  e.  CC )
518, 50syl 14 . . 3  |-  ( ph  ->  A  e.  CC )
5251, 14, 15divrecap2d 8704 . 2  |-  ( ph  ->  ( A  /  (  seq M (  x.  ,  F ) `  N
) )  =  ( ( 1  /  (  seq M (  x.  ,  F ) `  N
) )  x.  A
) )
5349, 52breqtrrd 4015 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 973    = wceq 1348    e. wcel 2141   _Vcvv 2730   class class class wbr 3987   ` cfv 5196  (class class class)co 5851   CCcc 7765   0cc0 7767   1c1 7768    + caddc 7770    x. cmul 7772   # cap 8493    / cdiv 8582   ZZcz 9205   ZZ>=cuz 9480    seqcseq 10394    ~~> cli 11234
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 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-mulrcl 7866  ax-addcom 7867  ax-mulcom 7868  ax-addass 7869  ax-mulass 7870  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-1rid 7874  ax-0id 7875  ax-rnegex 7876  ax-precex 7877  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883  ax-pre-mulgt0 7884  ax-pre-mulext 7885  ax-arch 7886  ax-caucvg 7887
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 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-frec 6368  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-reap 8487  df-ap 8494  df-div 8583  df-inn 8872  df-2 8930  df-3 8931  df-4 8932  df-n0 9129  df-z 9206  df-uz 9481  df-rp 9604  df-fz 9959  df-seqfrec 10395  df-exp 10469  df-cj 10799  df-re 10800  df-im 10801  df-rsqrt 10955  df-abs 10956  df-clim 11235
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator