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Theorem clim1fr1 27727
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 10263 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1z 10053 . . . 4  |-  1  e.  ZZ
32a1i 10 . . 3  |-  ( ph  ->  1  e.  ZZ )
4 nnex 9752 . . . . . 6  |-  NN  e.  _V
54mptex 5746 . . . . 5  |-  ( n  e.  NN  |->  1 )  e.  _V
65a1i 10 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  1 )  e.  _V )
73zcnd 10118 . . . 4  |-  ( ph  ->  1  e.  CC )
8 eqidd 2284 . . . . . 6  |-  ( k  e.  NN  ->  (
n  e.  NN  |->  1 )  =  ( n  e.  NN  |->  1 ) )
9 eqidd 2284 . . . . . 6  |-  ( ( k  e.  NN  /\  n  =  k )  ->  1  =  1 )
10 id 19 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN )
11 ax-1cn 8795 . . . . . . 7  |-  1  e.  CC
1211a1i 10 . . . . . 6  |-  ( k  e.  NN  ->  1  e.  CC )
138, 9, 10, 12fvmptd 5606 . . . . 5  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  1 ) `  k
)  =  1 )
1413adantl 452 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  1 ) `  k )  =  1 )
151, 3, 6, 7, 14climconst 12017 . . 3  |-  ( ph  ->  ( n  e.  NN  |->  1 )  ~~>  1 )
16 clim1fr1.1 . . . . 5  |-  F  =  ( n  e.  NN  |->  ( ( ( A  x.  n )  +  B )  /  ( A  x.  n )
) )
174mptex 5746 . . . . 5  |-  ( n  e.  NN  |->  ( ( ( A  x.  n
)  +  B )  /  ( A  x.  n ) ) )  e.  _V
1816, 17eqeltri 2353 . . . 4  |-  F  e. 
_V
1918a1i 10 . . 3  |-  ( ph  ->  F  e.  _V )
20 clim1fr1.4 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
2120adantr 451 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  B  e.  CC )
22 clim1fr1.2 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
2322adantr 451 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  A  e.  CC )
24 nncn 9754 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  CC )
2524adantl 452 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  n  e.  CC )
26 clim1fr1.3 . . . . . . 7  |-  ( ph  ->  A  =/=  0 )
2726adantr 451 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  A  =/=  0 )
28 nnne0 9778 . . . . . . 7  |-  ( n  e.  NN  ->  n  =/=  0 )
2928adantl 452 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  n  =/=  0 )
3021, 23, 25, 27, 29divdiv1d 9567 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( B  /  A )  /  n )  =  ( B  /  ( A  x.  n )
) )
3130mpteq2dva 4106 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  ( ( B  /  A )  /  n
) )  =  ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) ) )
3220, 22, 26divcld 9536 . . . . 5  |-  ( ph  ->  ( B  /  A
)  e.  CC )
33 divcnv 12312 . . . . 5  |-  ( ( B  /  A )  e.  CC  ->  (
n  e.  NN  |->  ( ( B  /  A
)  /  n ) )  ~~>  0 )
3432, 33syl 15 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  ( ( B  /  A )  /  n
) )  ~~>  0 )
3531, 34eqbrtrrd 4045 . . 3  |-  ( ph  ->  ( n  e.  NN  |->  ( B  /  ( A  x.  n )
) )  ~~>  0 )
3611a1i 10 . . . . . . 7  |-  ( n  e.  NN  ->  1  e.  CC )
3736rgen 2608 . . . . . 6  |-  A. n  e.  NN  1  e.  CC
38 eqid 2283 . . . . . . 7  |-  ( n  e.  NN  |->  1 )  =  ( n  e.  NN  |->  1 )
3938fmpt 5681 . . . . . 6  |-  ( A. n  e.  NN  1  e.  CC  <->  ( n  e.  NN  |->  1 ) : NN --> CC )
4037, 39mpbi 199 . . . . 5  |-  ( n  e.  NN  |->  1 ) : NN --> CC
4140a1i 10 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  1 ) : NN --> CC )
4241ffvelrnda 5665 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  1 ) `  k )  e.  CC )
43 nfv 1605 . . . . . 6  |-  F/ n ph
4423, 25mulcld 8855 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( A  x.  n )  e.  CC )
4523, 25, 27, 29mulne0d 9420 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( A  x.  n )  =/=  0 )
4621, 44, 45divcld 9536 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( B  /  ( A  x.  n ) )  e.  CC )
4746ex 423 . . . . . 6  |-  ( ph  ->  ( n  e.  NN  ->  ( B  /  ( A  x.  n )
)  e.  CC ) )
4843, 47ralrimi 2624 . . . . 5  |-  ( ph  ->  A. n  e.  NN  ( B  /  ( A  x.  n )
)  e.  CC )
49 eqid 2283 . . . . . 6  |-  ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) )  =  ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) )
5049fmpt 5681 . . . . 5  |-  ( A. n  e.  NN  ( B  /  ( A  x.  n ) )  e.  CC  <->  ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) : NN --> CC )
5148, 50sylib 188 . . . 4  |-  ( ph  ->  ( n  e.  NN  |->  ( B  /  ( A  x.  n )
) ) : NN --> CC )
5251ffvelrnda 5665 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) ) `  k )  e.  CC )
5316a1i 10 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  F  =  ( n  e.  NN  |->  ( ( ( A  x.  n )  +  B )  /  ( A  x.  n )
) ) )
54 oveq2 5866 . . . . . . . 8  |-  ( n  =  k  ->  ( A  x.  n )  =  ( A  x.  k ) )
5554oveq1d 5873 . . . . . . 7  |-  ( n  =  k  ->  (
( A  x.  n
)  +  B )  =  ( ( A  x.  k )  +  B ) )
5655, 54oveq12d 5876 . . . . . 6  |-  ( n  =  k  ->  (
( ( A  x.  n )  +  B
)  /  ( A  x.  n ) )  =  ( ( ( A  x.  k )  +  B )  / 
( A  x.  k
) ) )
5756adantl 452 . . . . 5  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  -> 
( ( ( A  x.  n )  +  B )  /  ( A  x.  n )
)  =  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) ) )
58 simpr 447 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  NN )
5922adantr 451 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  A  e.  CC )
6058nncnd 9762 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  CC )
6159, 60mulcld 8855 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( A  x.  k )  e.  CC )
6220adantr 451 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  B  e.  CC )
6361, 62addcld 8854 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( A  x.  k )  +  B )  e.  CC )
6426adantr 451 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  A  =/=  0 )
6558nnne0d 9790 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  k  =/=  0 )
6659, 60, 64, 65mulne0d 9420 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( A  x.  k )  =/=  0 )
6763, 61, 66divcld 9536 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) )  e.  CC )
6853, 57, 58, 67fvmptd 5606 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  =  ( ( ( A  x.  k )  +  B )  /  ( A  x.  k )
) )
6961, 62, 61, 66divdird 9574 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) )  =  ( ( ( A  x.  k )  / 
( A  x.  k
) )  +  ( B  /  ( A  x.  k ) ) ) )
7061, 66dividd 9534 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( A  x.  k )  /  ( A  x.  k ) )  =  1 )
7170oveq1d 5873 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  /  ( A  x.  k ) )  +  ( B  / 
( A  x.  k
) ) )  =  ( 1  +  ( B  /  ( A  x.  k ) ) ) )
7269, 71eqtrd 2315 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( A  x.  k
)  +  B )  /  ( A  x.  k ) )  =  ( 1  +  ( B  /  ( A  x.  k ) ) ) )
7314eqcomd 2288 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  1  =  ( ( n  e.  NN  |->  1 ) `  k ) )
74 eqidd 2284 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) )  =  ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) )
75 simpr 447 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  ->  n  =  k )
7675oveq2d 5874 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  -> 
( A  x.  n
)  =  ( A  x.  k ) )
7776oveq2d 5874 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN )  /\  n  =  k )  -> 
( B  /  ( A  x.  n )
)  =  ( B  /  ( A  x.  k ) ) )
7862, 61, 66divcld 9536 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( B  /  ( A  x.  k ) )  e.  CC )
7974, 77, 58, 78fvmptd 5606 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( B  /  ( A  x.  n ) ) ) `  k )  =  ( B  / 
( A  x.  k
) ) )
8079eqcomd 2288 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( B  /  ( A  x.  k ) )  =  ( ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) `  k ) )
8173, 80oveq12d 5876 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( 1  +  ( B  / 
( A  x.  k
) ) )  =  ( ( ( n  e.  NN  |->  1 ) `
 k )  +  ( ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) `  k ) ) )
8268, 72, 813eqtrd 2319 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  =  ( ( ( n  e.  NN  |->  1 ) `
 k )  +  ( ( n  e.  NN  |->  ( B  / 
( A  x.  n
) ) ) `  k ) ) )
831, 3, 15, 19, 35, 42, 52, 82climadd 12105 . 2  |-  ( ph  ->  F  ~~>  ( 1  +  0 ) )
8411addid1i 8999 . 2  |-  ( 1  +  0 )  =  1
8583, 84syl6breq 4062 1  |-  ( ph  ->  F  ~~>  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    / cdiv 9423   NNcn 9746   ZZcz 10024    ~~> cli 11958
This theorem is referenced by:  wallispilem5  27818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963
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