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Theorem climneg 27736
Description: Complex limit of the negative of a sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Hypotheses
Ref Expression
climneg.1  |-  F/ k
ph
climneg.2  |-  F/_ k F
climneg.3  |-  Z  =  ( ZZ>= `  M )
climneg.4  |-  ( ph  ->  M  e.  ZZ )
climneg.5  |-  ( ph  ->  F  ~~>  A )
climneg.6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
Assertion
Ref Expression
climneg  |-  ( ph  ->  ( k  e.  Z  |-> 
-u ( F `  k ) )  ~~>  -u A
)
Distinct variable group:    k, Z
Allowed substitution hints:    ph( k)    A( k)    F( k)    M( k)

Proof of Theorem climneg
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 climneg.1 . . 3  |-  F/ k
ph
2 nfmpt1 4109 . . 3  |-  F/_ k
( k  e.  Z  |-> 
-u 1 )
3 climneg.2 . . 3  |-  F/_ k F
4 nfmpt1 4109 . . 3  |-  F/_ k
( k  e.  Z  |-> 
-u ( F `  k ) )
5 climneg.3 . . 3  |-  Z  =  ( ZZ>= `  M )
6 climneg.4 . . 3  |-  ( ph  ->  M  e.  ZZ )
7 fvex 5539 . . . . . . 7  |-  ( ZZ>= `  M )  e.  _V
85, 7eqeltri 2353 . . . . . 6  |-  Z  e. 
_V
98mptex 5746 . . . . 5  |-  ( k  e.  Z  |->  -u 1
)  e.  _V
109a1i 10 . . . 4  |-  ( ph  ->  ( k  e.  Z  |-> 
-u 1 )  e. 
_V )
11 ax-1cn 8795 . . . . . 6  |-  1  e.  CC
1211a1i 10 . . . . 5  |-  ( ph  ->  1  e.  CC )
1312negcld 9144 . . . 4  |-  ( ph  -> 
-u 1  e.  CC )
14 eqidd 2284 . . . . . 6  |-  ( j  e.  Z  ->  (
k  e.  Z  |->  -u
1 )  =  ( k  e.  Z  |->  -u
1 ) )
15 eqidd 2284 . . . . . 6  |-  ( ( j  e.  Z  /\  k  =  j )  -> 
-u 1  =  -u
1 )
16 id 19 . . . . . 6  |-  ( j  e.  Z  ->  j  e.  Z )
1711a1i 10 . . . . . . 7  |-  ( j  e.  Z  ->  1  e.  CC )
1817negcld 9144 . . . . . 6  |-  ( j  e.  Z  ->  -u 1  e.  CC )
1914, 15, 16, 18fvmptd 5606 . . . . 5  |-  ( j  e.  Z  ->  (
( k  e.  Z  |-> 
-u 1 ) `  j )  =  -u
1 )
2019adantl 452 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  (
( k  e.  Z  |-> 
-u 1 ) `  j )  =  -u
1 )
215, 6, 10, 13, 20climconst 12017 . . 3  |-  ( ph  ->  ( k  e.  Z  |-> 
-u 1 )  ~~>  -u 1
)
228mptex 5746 . . . 4  |-  ( k  e.  Z  |->  -u ( F `  k )
)  e.  _V
2322a1i 10 . . 3  |-  ( ph  ->  ( k  e.  Z  |-> 
-u ( F `  k ) )  e. 
_V )
24 climneg.5 . . 3  |-  ( ph  ->  F  ~~>  A )
25 neg1cn 9813 . . . . . . 7  |-  -u 1  e.  CC
2625jctr 526 . . . . . 6  |-  ( k  e.  Z  ->  (
k  e.  Z  /\  -u 1  e.  CC ) )
27 eqid 2283 . . . . . . 7  |-  ( k  e.  Z  |->  -u 1
)  =  ( k  e.  Z  |->  -u 1
)
2827fvmpt2 5608 . . . . . 6  |-  ( ( k  e.  Z  /\  -u 1  e.  CC )  ->  ( ( k  e.  Z  |->  -u 1
) `  k )  =  -u 1 )
2926, 28syl 15 . . . . 5  |-  ( k  e.  Z  ->  (
( k  e.  Z  |-> 
-u 1 ) `  k )  =  -u
1 )
3011a1i 10 . . . . . 6  |-  ( k  e.  Z  ->  1  e.  CC )
3130negcld 9144 . . . . 5  |-  ( k  e.  Z  ->  -u 1  e.  CC )
3229, 31eqeltrd 2357 . . . 4  |-  ( k  e.  Z  ->  (
( k  e.  Z  |-> 
-u 1 ) `  k )  e.  CC )
3332adantl 452 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |-> 
-u 1 ) `  k )  e.  CC )
34 climneg.6 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
35 simpr 447 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  Z )
3634negcld 9144 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  -u ( F `  k )  e.  CC )
3735, 36jca 518 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  (
k  e.  Z  /\  -u ( F `  k
)  e.  CC ) )
38 eqid 2283 . . . . . 6  |-  ( k  e.  Z  |->  -u ( F `  k )
)  =  ( k  e.  Z  |->  -u ( F `  k )
)
3938fvmpt2 5608 . . . . 5  |-  ( ( k  e.  Z  /\  -u ( F `  k
)  e.  CC )  ->  ( ( k  e.  Z  |->  -u ( F `  k )
) `  k )  =  -u ( F `  k ) )
4037, 39syl 15 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |-> 
-u ( F `  k ) ) `  k )  =  -u ( F `  k ) )
4134mulm1d 9231 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( -u 1  x.  ( F `
 k ) )  =  -u ( F `  k ) )
4241eqcomd 2288 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  -u ( F `  k )  =  ( -u 1  x.  ( F `  k
) ) )
4329eqcomd 2288 . . . . . 6  |-  ( k  e.  Z  ->  -u 1  =  ( ( k  e.  Z  |->  -u 1
) `  k )
)
4443adantl 452 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  -u 1  =  ( ( k  e.  Z  |->  -u 1
) `  k )
)
4544oveq1d 5873 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( -u 1  x.  ( F `
 k ) )  =  ( ( ( k  e.  Z  |->  -u
1 ) `  k
)  x.  ( F `
 k ) ) )
4640, 42, 453eqtrd 2319 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |-> 
-u ( F `  k ) ) `  k )  =  ( ( ( k  e.  Z  |->  -u 1 ) `  k )  x.  ( F `  k )
) )
471, 2, 3, 4, 5, 6, 21, 23, 24, 33, 34, 46climmulf 27730 . 2  |-  ( ph  ->  ( k  e.  Z  |-> 
-u ( F `  k ) )  ~~>  ( -u
1  x.  A ) )
48 climcl 11973 . . . 4  |-  ( F  ~~>  A  ->  A  e.  CC )
4924, 48syl 15 . . 3  |-  ( ph  ->  A  e.  CC )
5049mulm1d 9231 . 2  |-  ( ph  ->  ( -u 1  x.  A )  =  -u A )
5147, 50breqtrd 4047 1  |-  ( ph  ->  ( k  e.  Z  |-> 
-u ( F `  k ) )  ~~>  -u A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    x. cmul 8742   -ucneg 9038   ZZcz 10024   ZZ>=cuz 10230    ~~> cli 11958
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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