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Theorem climneg 27667
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 4290 . . 3  |-  F/_ k
( k  e.  Z  |-> 
-u 1 )
3 climneg.2 . . 3  |-  F/_ k F
4 nfmpt1 4290 . . 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 5734 . . . . . . 7  |-  ( ZZ>= `  M )  e.  _V
85, 7eqeltri 2505 . . . . . 6  |-  Z  e. 
_V
98mptex 5958 . . . . 5  |-  ( k  e.  Z  |->  -u 1
)  e.  _V
109a1i 11 . . . 4  |-  ( ph  ->  ( k  e.  Z  |-> 
-u 1 )  e. 
_V )
11 ax-1cn 9038 . . . . . 6  |-  1  e.  CC
1211a1i 11 . . . . 5  |-  ( ph  ->  1  e.  CC )
1312negcld 9388 . . . 4  |-  ( ph  -> 
-u 1  e.  CC )
14 eqidd 2436 . . . . . 6  |-  ( j  e.  Z  ->  (
k  e.  Z  |->  -u
1 )  =  ( k  e.  Z  |->  -u
1 ) )
15 eqidd 2436 . . . . . 6  |-  ( ( j  e.  Z  /\  k  =  j )  -> 
-u 1  =  -u
1 )
16 id 20 . . . . . 6  |-  ( j  e.  Z  ->  j  e.  Z )
1711a1i 11 . . . . . . 7  |-  ( j  e.  Z  ->  1  e.  CC )
1817negcld 9388 . . . . . 6  |-  ( j  e.  Z  ->  -u 1  e.  CC )
1914, 15, 16, 18fvmptd 5802 . . . . 5  |-  ( j  e.  Z  ->  (
( k  e.  Z  |-> 
-u 1 ) `  j )  =  -u
1 )
2019adantl 453 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  (
( k  e.  Z  |-> 
-u 1 ) `  j )  =  -u
1 )
215, 6, 10, 13, 20climconst 12327 . . 3  |-  ( ph  ->  ( k  e.  Z  |-> 
-u 1 )  ~~>  -u 1
)
228mptex 5958 . . . 4  |-  ( k  e.  Z  |->  -u ( F `  k )
)  e.  _V
2322a1i 11 . . 3  |-  ( ph  ->  ( k  e.  Z  |-> 
-u ( F `  k ) )  e. 
_V )
24 climneg.5 . . 3  |-  ( ph  ->  F  ~~>  A )
25 neg1cn 10057 . . . . . 6  |-  -u 1  e.  CC
26 eqid 2435 . . . . . . 7  |-  ( k  e.  Z  |->  -u 1
)  =  ( k  e.  Z  |->  -u 1
)
2726fvmpt2 5804 . . . . . 6  |-  ( ( k  e.  Z  /\  -u 1  e.  CC )  ->  ( ( k  e.  Z  |->  -u 1
) `  k )  =  -u 1 )
2825, 27mpan2 653 . . . . 5  |-  ( k  e.  Z  ->  (
( k  e.  Z  |-> 
-u 1 ) `  k )  =  -u
1 )
2928, 25syl6eqel 2523 . . . 4  |-  ( k  e.  Z  ->  (
( k  e.  Z  |-> 
-u 1 ) `  k )  e.  CC )
3029adantl 453 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |-> 
-u 1 ) `  k )  e.  CC )
31 climneg.6 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
32 simpr 448 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  Z )
3331negcld 9388 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  -u ( F `  k )  e.  CC )
34 eqid 2435 . . . . . 6  |-  ( k  e.  Z  |->  -u ( F `  k )
)  =  ( k  e.  Z  |->  -u ( F `  k )
)
3534fvmpt2 5804 . . . . 5  |-  ( ( k  e.  Z  /\  -u ( F `  k
)  e.  CC )  ->  ( ( k  e.  Z  |->  -u ( F `  k )
) `  k )  =  -u ( F `  k ) )
3632, 33, 35syl2anc 643 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |-> 
-u ( F `  k ) ) `  k )  =  -u ( F `  k ) )
3731mulm1d 9475 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( -u 1  x.  ( F `
 k ) )  =  -u ( F `  k ) )
3828eqcomd 2440 . . . . . 6  |-  ( k  e.  Z  ->  -u 1  =  ( ( k  e.  Z  |->  -u 1
) `  k )
)
3938adantl 453 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  -u 1  =  ( ( k  e.  Z  |->  -u 1
) `  k )
)
4039oveq1d 6088 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( -u 1  x.  ( F `
 k ) )  =  ( ( ( k  e.  Z  |->  -u
1 ) `  k
)  x.  ( F `
 k ) ) )
4136, 37, 403eqtr2d 2473 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |-> 
-u ( F `  k ) ) `  k )  =  ( ( ( k  e.  Z  |->  -u 1 ) `  k )  x.  ( F `  k )
) )
421, 2, 3, 4, 5, 6, 21, 23, 24, 30, 31, 41climmulf 27661 . 2  |-  ( ph  ->  ( k  e.  Z  |-> 
-u ( F `  k ) )  ~~>  ( -u
1  x.  A ) )
43 climcl 12283 . . . 4  |-  ( F  ~~>  A  ->  A  e.  CC )
4424, 43syl 16 . . 3  |-  ( ph  ->  A  e.  CC )
4544mulm1d 9475 . 2  |-  ( ph  ->  ( -u 1  x.  A )  =  -u A )
4642, 45breqtrd 4228 1  |-  ( ph  ->  ( k  e.  Z  |-> 
-u ( F `  k ) )  ~~>  -u A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2558   _Vcvv 2948   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   CCcc 8978   1c1 8981    x. cmul 8985   -ucneg 9282   ZZcz 10272   ZZ>=cuz 10478    ~~> cli 12268
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-rp 10603  df-seq 11314  df-exp 11373  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-clim 12272
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