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Theorem climdivf 36986
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 4484 . . 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 459 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  Z )
1412eldifad 3426 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
15 eldifsni 4098 . . . . . . 7  |-  ( ( G `  k )  e.  ( CC  \  { 0 } )  ->  ( G `  k )  =/=  0
)
1612, 15syl 17 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =/=  0 )
1714, 16reccld 10354 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
1  /  ( G `
 k ) )  e.  CC )
18 eqid 2402 . . . . . 6  |-  ( k  e.  Z  |->  ( 1  /  ( G `  k ) ) )  =  ( k  e.  Z  |->  ( 1  / 
( G `  k
) ) )
1918fvmpt2 5941 . . . . 5  |-  ( ( k  e.  Z  /\  ( 1  /  ( G `  k )
)  e.  CC )  ->  ( ( k  e.  Z  |->  ( 1  /  ( G `  k ) ) ) `
 k )  =  ( 1  /  ( G `  k )
) )
2013, 17, 19syl2anc 659 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |->  ( 1  /  ( G `  k )
) ) `  k
)  =  ( 1  /  ( G `  k ) ) )
21 fvex 5859 . . . . . . 7  |-  ( ZZ>= `  M )  e.  _V
225, 21eqeltri 2486 . . . . . 6  |-  Z  e. 
_V
2322mptex 6124 . . . . 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 36983 . . 3  |-  ( ph  ->  ( k  e.  Z  |->  ( 1  /  ( G `  k )
) )  ~~>  ( 1  /  B ) )
26 climdivf.11 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
2720, 17eqeltrd 2490 . . 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 10364 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  /  ( G `
 k ) )  =  ( ( F `
 k )  x.  ( 1  /  ( G `  k )
) ) )
3020eqcomd 2410 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
1  /  ( G `
 k ) )  =  ( ( k  e.  Z  |->  ( 1  /  ( G `  k ) ) ) `
 k ) )
3130oveq2d 6294 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  x.  ( 1  /  ( G `  k ) ) )  =  ( ( F `
 k )  x.  ( ( k  e.  Z  |->  ( 1  / 
( G `  k
) ) ) `  k ) ) )
3228, 29, 313eqtrd 2447 . . 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 36978 . 2  |-  ( ph  ->  H  ~~>  ( A  x.  ( 1  /  B
) ) )
34 climcl 13471 . . . 4  |-  ( F  ~~>  A  ->  A  e.  CC )
357, 34syl 17 . . 3  |-  ( ph  ->  A  e.  CC )
36 climcl 13471 . . . 4  |-  ( G  ~~>  B  ->  B  e.  CC )
3710, 36syl 17 . . 3  |-  ( ph  ->  B  e.  CC )
3835, 37, 11divrecd 10364 . 2  |-  ( ph  ->  ( A  /  B
)  =  ( A  x.  ( 1  /  B ) ) )
3933, 38breqtrrd 4421 1  |-  ( ph  ->  H  ~~>  ( A  /  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405   F/wnf 1637    e. wcel 1842   F/_wnfc 2550    =/= wne 2598   _Vcvv 3059    \ cdif 3411   {csn 3972   class class class wbr 4395    |-> cmpt 4453   ` cfv 5569  (class class class)co 6278   CCcc 9520   0cc0 9522   1c1 9523    x. cmul 9527    / cdiv 10247   ZZcz 10905   ZZ>=cuz 11127    ~~> cli 13456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-seq 12152  df-exp 12211  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460
This theorem is referenced by:  stirlinglem8  37231  fourierdlem103  37360  fourierdlem104  37361
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