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Theorem climdivf 31109
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 4529 . . 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 461 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  Z )
1412eldifad 3481 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
15 eldifsni 4146 . . . . . . 7  |-  ( ( G `  k )  e.  ( CC  \  { 0 } )  ->  ( G `  k )  =/=  0
)
1612, 15syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =/=  0 )
1714, 16reccld 10302 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
1  /  ( G `
 k ) )  e.  CC )
18 eqid 2460 . . . . . 6  |-  ( k  e.  Z  |->  ( 1  /  ( G `  k ) ) )  =  ( k  e.  Z  |->  ( 1  / 
( G `  k
) ) )
1918fvmpt2 5948 . . . . 5  |-  ( ( k  e.  Z  /\  ( 1  /  ( G `  k )
)  e.  CC )  ->  ( ( k  e.  Z  |->  ( 1  /  ( G `  k ) ) ) `
 k )  =  ( 1  /  ( G `  k )
) )
2013, 17, 19syl2anc 661 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |->  ( 1  /  ( G `  k )
) ) `  k
)  =  ( 1  /  ( G `  k ) ) )
21 fvex 5867 . . . . . . 7  |-  ( ZZ>= `  M )  e.  _V
225, 21eqeltri 2544 . . . . . 6  |-  Z  e. 
_V
2322mptex 6122 . . . . 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 31106 . . 3  |-  ( ph  ->  ( k  e.  Z  |->  ( 1  /  ( G `  k )
) )  ~~>  ( 1  /  B ) )
26 climdivf.11 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
2720, 17eqeltrd 2548 . . 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 10312 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  /  ( G `
 k ) )  =  ( ( F `
 k )  x.  ( 1  /  ( G `  k )
) ) )
3020eqcomd 2468 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
1  /  ( G `
 k ) )  =  ( ( k  e.  Z  |->  ( 1  /  ( G `  k ) ) ) `
 k ) )
3130oveq2d 6291 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  x.  ( 1  /  ( G `  k ) ) )  =  ( ( F `
 k )  x.  ( ( k  e.  Z  |->  ( 1  / 
( G `  k
) ) ) `  k ) ) )
3228, 29, 313eqtrd 2505 . . 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 31101 . 2  |-  ( ph  ->  H  ~~>  ( A  x.  ( 1  /  B
) ) )
34 climcl 13271 . . . 4  |-  ( F  ~~>  A  ->  A  e.  CC )
357, 34syl 16 . . 3  |-  ( ph  ->  A  e.  CC )
36 climcl 13271 . . . 4  |-  ( G  ~~>  B  ->  B  e.  CC )
3710, 36syl 16 . . 3  |-  ( ph  ->  B  e.  CC )
3835, 37, 11divrecd 10312 . 2  |-  ( ph  ->  ( A  /  B
)  =  ( A  x.  ( 1  /  B ) ) )
3933, 38breqtrrd 4466 1  |-  ( ph  ->  H  ~~>  ( A  /  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374   F/wnf 1594    e. wcel 1762   F/_wnfc 2608    =/= wne 2655   _Vcvv 3106    \ cdif 3466   {csn 4020   class class class wbr 4440    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275   CCcc 9479   0cc0 9481   1c1 9482    x. cmul 9486    / cdiv 10195   ZZcz 10853   ZZ>=cuz 11071    ~~> cli 13256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260
This theorem is referenced by:  stirlinglem8  31336  fourierdlem103  31465  fourierdlem104  31466
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