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Theorem climneg 29951
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 4492 . . 3  |-  F/_ k
( k  e.  Z  |-> 
-u 1 )
3 climneg.2 . . 3  |-  F/_ k F
4 nfmpt1 4492 . . 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 5812 . . . . . . 7  |-  ( ZZ>= `  M )  e.  _V
85, 7eqeltri 2538 . . . . . 6  |-  Z  e. 
_V
98mptex 6060 . . . . 5  |-  ( k  e.  Z  |->  -u 1
)  e.  _V
109a1i 11 . . . 4  |-  ( ph  ->  ( k  e.  Z  |-> 
-u 1 )  e. 
_V )
11 ax-1cn 9454 . . . . . 6  |-  1  e.  CC
1211a1i 11 . . . . 5  |-  ( ph  ->  1  e.  CC )
1312negcld 9820 . . . 4  |-  ( ph  -> 
-u 1  e.  CC )
14 eqidd 2455 . . . . . 6  |-  ( j  e.  Z  ->  (
k  e.  Z  |->  -u
1 )  =  ( k  e.  Z  |->  -u
1 ) )
15 eqidd 2455 . . . . . 6  |-  ( ( j  e.  Z  /\  k  =  j )  -> 
-u 1  =  -u
1 )
16 id 22 . . . . . 6  |-  ( j  e.  Z  ->  j  e.  Z )
1711a1i 11 . . . . . . 7  |-  ( j  e.  Z  ->  1  e.  CC )
1817negcld 9820 . . . . . 6  |-  ( j  e.  Z  ->  -u 1  e.  CC )
1914, 15, 16, 18fvmptd 5891 . . . . 5  |-  ( j  e.  Z  ->  (
( k  e.  Z  |-> 
-u 1 ) `  j )  =  -u
1 )
2019adantl 466 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  (
( k  e.  Z  |-> 
-u 1 ) `  j )  =  -u
1 )
215, 6, 10, 13, 20climconst 13142 . . 3  |-  ( ph  ->  ( k  e.  Z  |-> 
-u 1 )  ~~>  -u 1
)
228mptex 6060 . . . 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 10539 . . . . . 6  |-  -u 1  e.  CC
26 eqid 2454 . . . . . . 7  |-  ( k  e.  Z  |->  -u 1
)  =  ( k  e.  Z  |->  -u 1
)
2726fvmpt2 5893 . . . . . 6  |-  ( ( k  e.  Z  /\  -u 1  e.  CC )  ->  ( ( k  e.  Z  |->  -u 1
) `  k )  =  -u 1 )
2825, 27mpan2 671 . . . . 5  |-  ( k  e.  Z  ->  (
( k  e.  Z  |-> 
-u 1 ) `  k )  =  -u
1 )
2928, 25syl6eqel 2550 . . . 4  |-  ( k  e.  Z  ->  (
( k  e.  Z  |-> 
-u 1 ) `  k )  e.  CC )
3029adantl 466 . . 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 461 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  Z )
3331negcld 9820 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  -u ( F `  k )  e.  CC )
34 eqid 2454 . . . . . 6  |-  ( k  e.  Z  |->  -u ( F `  k )
)  =  ( k  e.  Z  |->  -u ( F `  k )
)
3534fvmpt2 5893 . . . . 5  |-  ( ( k  e.  Z  /\  -u ( F `  k
)  e.  CC )  ->  ( ( k  e.  Z  |->  -u ( F `  k )
) `  k )  =  -u ( F `  k ) )
3632, 33, 35syl2anc 661 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |-> 
-u ( F `  k ) ) `  k )  =  -u ( F `  k ) )
3731mulm1d 9910 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( -u 1  x.  ( F `
 k ) )  =  -u ( F `  k ) )
3828eqcomd 2462 . . . . . 6  |-  ( k  e.  Z  ->  -u 1  =  ( ( k  e.  Z  |->  -u 1
) `  k )
)
3938adantl 466 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  -u 1  =  ( ( k  e.  Z  |->  -u 1
) `  k )
)
4039oveq1d 6218 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( -u 1  x.  ( F `
 k ) )  =  ( ( ( k  e.  Z  |->  -u
1 ) `  k
)  x.  ( F `
 k ) ) )
4136, 37, 403eqtr2d 2501 . . 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 29945 . 2  |-  ( ph  ->  ( k  e.  Z  |-> 
-u ( F `  k ) )  ~~>  ( -u
1  x.  A ) )
43 climcl 13098 . . . 4  |-  ( F  ~~>  A  ->  A  e.  CC )
4424, 43syl 16 . . 3  |-  ( ph  ->  A  e.  CC )
4544mulm1d 9910 . 2  |-  ( ph  ->  ( -u 1  x.  A )  =  -u A )
4642, 45breqtrd 4427 1  |-  ( ph  ->  ( k  e.  Z  |-> 
-u ( F `  k ) )  ~~>  -u A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   F/wnf 1590    e. wcel 1758   F/_wnfc 2602   _Vcvv 3078   class class class wbr 4403    |-> cmpt 4461   ` cfv 5529  (class class class)co 6203   CCcc 9394   1c1 9397    x. cmul 9401   -ucneg 9710   ZZcz 10760   ZZ>=cuz 10975    ~~> cli 13083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-seq 11927  df-exp 11986  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087
This theorem is referenced by: (None)
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