Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clim1fr1 Unicode version

Theorem clim1fr1 27388
Description: A class of sequences of fractions that converge to 1 (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Hypotheses
Ref Expression
clim1fr1.1  |-  F  =  ( n  e.  NN  |->  ( ( ( A  x.  n )  +  B )  /  ( A  x.  n )
) )
clim1fr1.2  |-  ( ph  ->  A  e.  CC )
clim1fr1.3  |-  ( ph  ->  A  =/=  0 )
clim1fr1.4  |-  ( ph  ->  B  e.  CC )
Assertion
Ref Expression
clim1fr1  |-  ( ph  ->  F  ~~>  1 )
Distinct variable groups:    ph, n    A, n    B, n
Allowed substitution hint:    F( n)

Proof of Theorem clim1fr1
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 nnuz 10446 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1z 10236 . . . 4  |-  1  e.  ZZ
32a1i 11 . . 3  |-  ( ph  ->  1  e.  ZZ )
4 nnex 9931 . . . . . 6  |-  NN  e.  _V
54mptex 5898 . . . . 5  |-  ( n  e.  NN  |->  1 )  e.  _V
65a1i 11 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  1 )  e.  _V )
73zcnd 10301 . . . 4  |-  ( ph  ->  1  e.  CC )
8 eqidd 2381 . . . . . 6  |-  ( k  e.  NN  ->  (
n  e.  NN  |->  1 )  =  ( n  e.  NN  |->  1 ) )
9 eqidd 2381 . . . . . 6  |-  ( ( k  e.  NN  /\  n  =  k )  ->  1  =  1 )
10 id 20 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN )
11 ax-1cn 8974 . . . . . . 7  |-  1  e.  CC
1211a1i 11 . . . . . 6  |-  ( k  e.  NN  ->  1  e.  CC )
138, 9, 10, 12fvmptd 5742 . . . . 5  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  1 ) `  k
)  =  1 )
1413adantl 453 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  1 ) `  k )  =  1 )
151, 3, 6, 7, 14climconst 12257 . . 3  |-  ( ph  ->  ( n  e.  NN  |->  1 )  ~~>  1 )
16 clim1fr1.1 . . . . 5  |-  F  =  ( n  e.  NN  |->  ( ( ( A  x.  n )  +  B )  /  ( A  x.  n )
) )
174mptex 5898 . . . . 5  |-  ( n  e.  NN  |->  ( ( ( A  x.  n
)  +  B )  /  ( A  x.  n ) ) )  e.  _V
1816, 17eqeltri 2450 . . . 4  |-  F  e. 
_V
1918a1i 11 . . 3  |-  ( ph  ->  F  e.  _V )
20 clim1fr1.4 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
2120adantr 452 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  B  e.  CC )
22 clim1fr1.2 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
2322adantr 452 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  A  e.  CC )
24 nncn 9933 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  CC )
2524adantl 453 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  n  e.  CC )
26 clim1fr1.3 . . . . . . 7  |-  ( ph  ->  A  =/=  0 )
2726adantr 452 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  A  =/=  0 )
28 nnne0 9957 . . . . . . 7  |-  ( n  e.  NN  ->  n  =/=  0 )
2928adantl 453 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  n  =/=  0 )
3021, 23, 25, 27, 29divdiv1d 9746 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( B  /  A )  /  n )  =  ( B  /  ( A  x.  n )
) )
3130mpteq2dva 4229 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  ( ( B  /  A )  /  n
) )  =  ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) ) )
3220, 22, 26divcld 9715 . . . . 5  |-  ( ph  ->  ( B  /  A
)  e.  CC )
33 divcnv 12553 . . . . 5  |-  ( ( B  /  A )  e.  CC  ->  (
n  e.  NN  |->  ( ( B  /  A
)  /  n ) )  ~~>  0 )
3432, 33syl 16 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  ( ( B  /  A )  /  n
) )  ~~>  0 )
3531, 34eqbrtrrd 4168 . . 3  |-  ( ph  ->  ( n  e.  NN  |->  ( B  /  ( A  x.  n )
) )  ~~>  0 )
36 eqid 2380 . . . . . 6  |-  ( n  e.  NN  |->  1 )  =  ( n  e.  NN  |->  1 )
3711a1i 11 . . . . . 6  |-  ( n  e.  NN  ->  1  e.  CC )
3836, 37fmpti 5824 . . . . 5  |-  ( n  e.  NN  |->  1 ) : NN --> CC
3938a1i 11 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  1 ) : NN --> CC )
4039ffvelrnda 5802 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  1 ) `  k )  e.  CC )
4123, 25mulcld 9034 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( A  x.  n )  e.  CC )
4223, 25, 27, 29mulne0d 9599 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( A  x.  n )  =/=  0 )
4321, 41, 42divcld 9715 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( B  /  ( A  x.  n ) )  e.  CC )
44 eqid 2380 . . . . 5  |-  ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) )  =  ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) )
4543, 44fmptd 5825 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  ( B  /  ( A  x.  n )
) ) : NN --> CC )
4645ffvelrnda 5802 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) ) `  k )  e.  CC )
4716a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  F  =  ( n  e.  NN  |->  ( ( ( A  x.  n )  +  B )  /  ( A  x.  n )
) ) )
48 oveq2 6021 . . . . . . . 8  |-  ( n  =  k  ->  ( A  x.  n )  =  ( A  x.  k ) )
4948oveq1d 6028 . . . . . . 7  |-  ( n  =  k  ->  (
( A  x.  n
)  +  B )  =  ( ( A  x.  k )  +  B ) )
5049, 48oveq12d 6031 . . . . . 6  |-  ( n  =  k  ->  (
( ( A  x.  n )  +  B
)  /  ( A  x.  n ) )  =  ( ( ( A  x.  k )  +  B )  / 
( A  x.  k
) ) )
5150adantl 453 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  -> 
( ( ( A  x.  n )  +  B )  /  ( A  x.  n )
)  =  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) ) )
52 simpr 448 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  NN )
5322adantr 452 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  A  e.  CC )
5452nncnd 9941 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  CC )
5553, 54mulcld 9034 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( A  x.  k )  e.  CC )
5620adantr 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  B  e.  CC )
5755, 56addcld 9033 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( A  x.  k )  +  B )  e.  CC )
5826adantr 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  A  =/=  0 )
5952nnne0d 9969 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  k  =/=  0 )
6053, 54, 58, 59mulne0d 9599 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( A  x.  k )  =/=  0 )
6157, 55, 60divcld 9715 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) )  e.  CC )
6247, 51, 52, 61fvmptd 5742 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  =  ( ( ( A  x.  k )  +  B )  /  ( A  x.  k )
) )
6355, 56, 55, 60divdird 9753 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) )  =  ( ( ( A  x.  k )  / 
( A  x.  k
) )  +  ( B  /  ( A  x.  k ) ) ) )
6455, 60dividd 9713 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( A  x.  k )  /  ( A  x.  k ) )  =  1 )
6564oveq1d 6028 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  /  ( A  x.  k ) )  +  ( B  / 
( A  x.  k
) ) )  =  ( 1  +  ( B  /  ( A  x.  k ) ) ) )
6663, 65eqtrd 2412 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) )  =  ( 1  +  ( B  /  ( A  x.  k ) ) ) )
6714eqcomd 2385 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  1  =  ( ( n  e.  NN  |->  1 ) `  k ) )
68 eqidd 2381 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) )  =  ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) )
69 simpr 448 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  ->  n  =  k )
7069oveq2d 6029 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  -> 
( A  x.  n
)  =  ( A  x.  k ) )
7170oveq2d 6029 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  -> 
( B  /  ( A  x.  n )
)  =  ( B  /  ( A  x.  k ) ) )
7256, 55, 60divcld 9715 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( B  /  ( A  x.  k ) )  e.  CC )
7368, 71, 52, 72fvmptd 5742 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) ) `  k )  =  ( B  / 
( A  x.  k
) ) )
7473eqcomd 2385 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( B  /  ( A  x.  k ) )  =  ( ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) `  k ) )
7567, 74oveq12d 6031 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( 1  +  ( B  / 
( A  x.  k
) ) )  =  ( ( ( n  e.  NN  |->  1 ) `
 k )  +  ( ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) `  k ) ) )
7662, 66, 753eqtrd 2416 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  =  ( ( ( n  e.  NN  |->  1 ) `
 k )  +  ( ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) `  k ) ) )
771, 3, 15, 19, 35, 40, 46, 76climadd 12345 . 2  |-  ( ph  ->  F  ~~>  ( 1  +  0 ) )
7811addid1i 9178 . 2  |-  ( 1  +  0 )  =  1
7977, 78syl6breq 4185 1  |-  ( ph  ->  F  ~~>  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543   _Vcvv 2892   class class class wbr 4146    e. cmpt 4200   -->wf 5383   ` cfv 5387  (class class class)co 6013   CCcc 8914   0cc0 8916   1c1 8917    + caddc 8919    x. cmul 8921    / cdiv 9602   NNcn 9925   ZZcz 10207    ~~> cli 12198
This theorem is referenced by:  wallispilem5  27479
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-fl 11122  df-seq 11244  df-exp 11303  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-rlim 12203
  Copyright terms: Public domain W3C validator