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Theorem climdivf 27714
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 4298 . . 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 448 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  Z )
1412eldifad 3332 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
15 eldifsni 3928 . . . . . . 7  |-  ( ( G `  k )  e.  ( CC  \  { 0 } )  ->  ( G `  k )  =/=  0
)
1612, 15syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =/=  0 )
1714, 16reccld 9783 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
1  /  ( G `
 k ) )  e.  CC )
18 eqid 2436 . . . . . 6  |-  ( k  e.  Z  |->  ( 1  /  ( G `  k ) ) )  =  ( k  e.  Z  |->  ( 1  / 
( G `  k
) ) )
1918fvmpt2 5812 . . . . 5  |-  ( ( k  e.  Z  /\  ( 1  /  ( G `  k )
)  e.  CC )  ->  ( ( k  e.  Z  |->  ( 1  /  ( G `  k ) ) ) `
 k )  =  ( 1  /  ( G `  k )
) )
2013, 17, 19syl2anc 643 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |->  ( 1  /  ( G `  k )
) ) `  k
)  =  ( 1  /  ( G `  k ) ) )
21 fvex 5742 . . . . . . 7  |-  ( ZZ>= `  M )  e.  _V
225, 21eqeltri 2506 . . . . . 6  |-  Z  e. 
_V
2322mptex 5966 . . . . 5  |-  ( k  e.  Z  |->  ( 1  /  ( G `  k ) ) )  e.  _V
2423a1i 11 . . . 4  |-  ( ph  ->  ( k  e.  Z  |->  ( 1  /  ( G `  k )
) )  e.  _V )
251, 9, 3, 5, 6, 10, 11, 12, 20, 24climrecf 27711 . . 3  |-  ( ph  ->  ( k  e.  Z  |->  ( 1  /  ( G `  k )
) )  ~~>  ( 1  /  B ) )
26 climdivf.11 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
2720, 17eqeltrd 2510 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |->  ( 1  /  ( G `  k )
) ) `  k
)  e.  CC )
28 climdivf.13 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k )  / 
( G `  k
) ) )
2926, 14, 16divrecd 9793 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  /  ( G `
 k ) )  =  ( ( F `
 k )  x.  ( 1  /  ( G `  k )
) ) )
3020eqcomd 2441 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
1  /  ( G `
 k ) )  =  ( ( k  e.  Z  |->  ( 1  /  ( G `  k ) ) ) `
 k ) )
3130oveq2d 6097 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  x.  ( 1  /  ( G `  k ) ) )  =  ( ( F `
 k )  x.  ( ( k  e.  Z  |->  ( 1  / 
( G `  k
) ) ) `  k ) ) )
3228, 29, 313eqtrd 2472 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k )  x.  ( ( k  e.  Z  |->  ( 1  / 
( G `  k
) ) ) `  k ) ) )
331, 2, 3, 4, 5, 6, 7, 8, 25, 26, 27, 32climmulf 27706 . 2  |-  ( ph  ->  H  ~~>  ( A  x.  ( 1  /  B
) ) )
34 climcl 12293 . . . 4  |-  ( F  ~~>  A  ->  A  e.  CC )
357, 34syl 16 . . 3  |-  ( ph  ->  A  e.  CC )
36 climcl 12293 . . . 4  |-  ( G  ~~>  B  ->  B  e.  CC )
3710, 36syl 16 . . 3  |-  ( ph  ->  B  e.  CC )
3835, 37, 11divrecd 9793 . 2  |-  ( ph  ->  ( A  /  B
)  =  ( A  x.  ( 1  /  B ) ) )
3933, 38breqtrrd 4238 1  |-  ( ph  ->  H  ~~>  ( A  /  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2559    =/= wne 2599   _Vcvv 2956    \ cdif 3317   {csn 3814   class class class wbr 4212    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    x. cmul 8995    / cdiv 9677   ZZcz 10282   ZZ>=cuz 10488    ~~> cli 12278
This theorem is referenced by:  stirlinglem8  27806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282
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