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Theorem climdivf 27738
Description: Limit of the ratio of two converging sequences. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Hypotheses
Ref Expression
climdivf.1  |-  F/ k
ph
climdivf.2  |-  F/_ k F
climdivf.3  |-  F/_ k G
climdivf.4  |-  F/_ k H
climdivf.5  |-  Z  =  ( ZZ>= `  M )
climdivf.6  |-  ( ph  ->  M  e.  ZZ )
climdivf.7  |-  ( ph  ->  F  ~~>  A )
climdivf.8  |-  ( ph  ->  H  e.  X )
climdivf.9  |-  ( ph  ->  G  ~~>  B )
climdivf.10  |-  ( ph  ->  B  =/=  0 )
climdivf.11  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
climdivf.12  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  ( CC  \  {
0 } ) )
climdivf.13  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k )  / 
( G `  k
) ) )
Assertion
Ref Expression
climdivf  |-  ( ph  ->  H  ~~>  ( A  /  B ) )
Distinct variable group:    k, Z
Allowed substitution hints:    ph( k)    A( k)    B( k)    F( k)    G( k)    H( k)    M( k)    X( k)

Proof of Theorem climdivf
StepHypRef Expression
1 climdivf.1 . . 3  |-  F/ k
ph
2 climdivf.2 . . 3  |-  F/_ k F
3 nfmpt1 4109 . . 3  |-  F/_ k
( k  e.  Z  |->  ( 1  /  ( G `  k )
) )
4 climdivf.4 . . 3  |-  F/_ k H
5 climdivf.5 . . 3  |-  Z  =  ( ZZ>= `  M )
6 climdivf.6 . . 3  |-  ( ph  ->  M  e.  ZZ )
7 climdivf.7 . . 3  |-  ( ph  ->  F  ~~>  A )
8 climdivf.8 . . 3  |-  ( ph  ->  H  e.  X )
9 climdivf.3 . . . 4  |-  F/_ k G
10 climdivf.9 . . . 4  |-  ( ph  ->  G  ~~>  B )
11 climdivf.10 . . . 4  |-  ( ph  ->  B  =/=  0 )
12 climdivf.12 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  ( CC  \  {
0 } ) )
13 simpr 447 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  Z )
1412eldifad 3164 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
15 eldifsni 3750 . . . . . . . 8  |-  ( ( G `  k )  e.  ( CC  \  { 0 } )  ->  ( G `  k )  =/=  0
)
1612, 15syl 15 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =/=  0 )
1714, 16reccld 9529 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  (
1  /  ( G `
 k ) )  e.  CC )
1813, 17jca 518 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
k  e.  Z  /\  ( 1  /  ( G `  k )
)  e.  CC ) )
19 eqid 2283 . . . . . 6  |-  ( k  e.  Z  |->  ( 1  /  ( G `  k ) ) )  =  ( k  e.  Z  |->  ( 1  / 
( G `  k
) ) )
2019fvmpt2 5608 . . . . 5  |-  ( ( k  e.  Z  /\  ( 1  /  ( G `  k )
)  e.  CC )  ->  ( ( k  e.  Z  |->  ( 1  /  ( G `  k ) ) ) `
 k )  =  ( 1  /  ( G `  k )
) )
2118, 20syl 15 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |->  ( 1  /  ( G `  k )
) ) `  k
)  =  ( 1  /  ( G `  k ) ) )
22 uzssz 10247 . . . . . . . 8  |-  ( ZZ>= `  M )  C_  ZZ
23 zex 10033 . . . . . . . . 9  |-  ZZ  e.  _V
2423ssex 4158 . . . . . . . 8  |-  ( (
ZZ>= `  M )  C_  ZZ  ->  ( ZZ>= `  M
)  e.  _V )
2522, 24ax-mp 8 . . . . . . 7  |-  ( ZZ>= `  M )  e.  _V
265, 25eqeltri 2353 . . . . . 6  |-  Z  e. 
_V
2726mptex 5746 . . . . 5  |-  ( k  e.  Z  |->  ( 1  /  ( G `  k ) ) )  e.  _V
2827a1i 10 . . . 4  |-  ( ph  ->  ( k  e.  Z  |->  ( 1  /  ( G `  k )
) )  e.  _V )
291, 9, 3, 5, 6, 10, 11, 12, 21, 28climrecf 27735 . . 3  |-  ( ph  ->  ( k  e.  Z  |->  ( 1  /  ( G `  k )
) )  ~~>  ( 1  /  B ) )
30 climdivf.11 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
3121, 17eqeltrd 2357 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |->  ( 1  /  ( G `  k )
) ) `  k
)  e.  CC )
32 climdivf.13 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k )  / 
( G `  k
) ) )
3330, 14, 16divrecd 9539 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  /  ( G `
 k ) )  =  ( ( F `
 k )  x.  ( 1  /  ( G `  k )
) ) )
3421eqcomd 2288 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
1  /  ( G `
 k ) )  =  ( ( k  e.  Z  |->  ( 1  /  ( G `  k ) ) ) `
 k ) )
3534oveq2d 5874 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  x.  ( 1  /  ( G `  k ) ) )  =  ( ( F `
 k )  x.  ( ( k  e.  Z  |->  ( 1  / 
( G `  k
) ) ) `  k ) ) )
3632, 33, 353eqtrd 2319 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k )  x.  ( ( k  e.  Z  |->  ( 1  / 
( G `  k
) ) ) `  k ) ) )
371, 2, 3, 4, 5, 6, 7, 8, 29, 30, 31, 36climmulf 27730 . 2  |-  ( ph  ->  H  ~~>  ( A  x.  ( 1  /  B
) ) )
38 climcl 11973 . . . 4  |-  ( F  ~~>  A  ->  A  e.  CC )
397, 38syl 15 . . 3  |-  ( ph  ->  A  e.  CC )
40 climcl 11973 . . . 4  |-  ( G  ~~>  B  ->  B  e.  CC )
4110, 40syl 15 . . 3  |-  ( ph  ->  B  e.  CC )
4239, 41, 11divrecd 9539 . 2  |-  ( ph  ->  ( A  /  B
)  =  ( A  x.  ( 1  /  B ) ) )
4337, 42breqtrrd 4049 1  |-  ( ph  ->  H  ~~>  ( A  /  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406    =/= wne 2446   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742    / cdiv 9423   ZZcz 10024   ZZ>=cuz 10230    ~~> cli 11958
This theorem is referenced by:  stirlinglem8  27830
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-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-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962
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