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Theorem climdivf 27841
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 4125 . . 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 3177 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
15 eldifsni 3763 . . . . . . . 8  |-  ( ( G `  k )  e.  ( CC  \  { 0 } )  ->  ( G `  k )  =/=  0
)
1612, 15syl 15 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =/=  0 )
1714, 16reccld 9545 . . . . . 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 2296 . . . . . 6  |-  ( k  e.  Z  |->  ( 1  /  ( G `  k ) ) )  =  ( k  e.  Z  |->  ( 1  / 
( G `  k
) ) )
2019fvmpt2 5624 . . . . 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 10263 . . . . . . . 8  |-  ( ZZ>= `  M )  C_  ZZ
23 zex 10049 . . . . . . . . 9  |-  ZZ  e.  _V
2423ssex 4174 . . . . . . . 8  |-  ( (
ZZ>= `  M )  C_  ZZ  ->  ( ZZ>= `  M
)  e.  _V )
2522, 24ax-mp 8 . . . . . . 7  |-  ( ZZ>= `  M )  e.  _V
265, 25eqeltri 2366 . . . . . 6  |-  Z  e. 
_V
2726mptex 5762 . . . . 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 27838 . . 3  |-  ( ph  ->  ( k  e.  Z  |->  ( 1  /  ( G `  k )
) )  ~~>  ( 1  /  B ) )
30 climdivf.11 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
3121, 17eqeltrd 2370 . . 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 9555 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  /  ( G `
 k ) )  =  ( ( F `
 k )  x.  ( 1  /  ( G `  k )
) ) )
3421eqcomd 2301 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
1  /  ( G `
 k ) )  =  ( ( k  e.  Z  |->  ( 1  /  ( G `  k ) ) ) `
 k ) )
3534oveq2d 5890 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  x.  ( 1  /  ( G `  k ) ) )  =  ( ( F `
 k )  x.  ( ( k  e.  Z  |->  ( 1  / 
( G `  k
) ) ) `  k ) ) )
3632, 33, 353eqtrd 2332 . . 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 27833 . 2  |-  ( ph  ->  H  ~~>  ( A  x.  ( 1  /  B
) ) )
38 climcl 11989 . . . 4  |-  ( F  ~~>  A  ->  A  e.  CC )
397, 38syl 15 . . 3  |-  ( ph  ->  A  e.  CC )
40 climcl 11989 . . . 4  |-  ( G  ~~>  B  ->  B  e.  CC )
4110, 40syl 15 . . 3  |-  ( ph  ->  B  e.  CC )
4239, 41, 11divrecd 9555 . 2  |-  ( ph  ->  ( A  /  B
)  =  ( A  x.  ( 1  /  B ) ) )
4337, 42breqtrrd 4065 1  |-  ( ph  ->  H  ~~>  ( A  /  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   F/wnf 1534    = wceq 1632    e. wcel 1696   F/_wnfc 2419    =/= wne 2459   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {csn 3653   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    x. cmul 8758    / cdiv 9439   ZZcz 10040   ZZ>=cuz 10246    ~~> cli 11974
This theorem is referenced by:  stirlinglem8  27933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978
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