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Theorem climneg 37309
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 4506 . . 3  |-  F/_ k
( k  e.  Z  |-> 
-u 1 )
3 climneg.2 . . 3  |-  F/_ k F
4 nfmpt1 4506 . . 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 5882 . . . . . . 7  |-  ( ZZ>= `  M )  e.  _V
85, 7eqeltri 2504 . . . . . 6  |-  Z  e. 
_V
98mptex 6142 . . . . 5  |-  ( k  e.  Z  |->  -u 1
)  e.  _V
109a1i 11 . . . 4  |-  ( ph  ->  ( k  e.  Z  |-> 
-u 1 )  e. 
_V )
11 1cnd 9648 . . . . 5  |-  ( ph  ->  1  e.  CC )
1211negcld 9962 . . . 4  |-  ( ph  -> 
-u 1  e.  CC )
13 eqidd 2421 . . . . . 6  |-  ( j  e.  Z  ->  (
k  e.  Z  |->  -u
1 )  =  ( k  e.  Z  |->  -u
1 ) )
14 eqidd 2421 . . . . . 6  |-  ( ( j  e.  Z  /\  k  =  j )  -> 
-u 1  =  -u
1 )
15 id 23 . . . . . 6  |-  ( j  e.  Z  ->  j  e.  Z )
16 1cnd 9648 . . . . . . 7  |-  ( j  e.  Z  ->  1  e.  CC )
1716negcld 9962 . . . . . 6  |-  ( j  e.  Z  ->  -u 1  e.  CC )
1813, 14, 15, 17fvmptd 5961 . . . . 5  |-  ( j  e.  Z  ->  (
( k  e.  Z  |-> 
-u 1 ) `  j )  =  -u
1 )
1918adantl 467 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  (
( k  e.  Z  |-> 
-u 1 ) `  j )  =  -u
1 )
205, 6, 10, 12, 19climconst 13574 . . 3  |-  ( ph  ->  ( k  e.  Z  |-> 
-u 1 )  ~~>  -u 1
)
218mptex 6142 . . . 4  |-  ( k  e.  Z  |->  -u ( F `  k )
)  e.  _V
2221a1i 11 . . 3  |-  ( ph  ->  ( k  e.  Z  |-> 
-u ( F `  k ) )  e. 
_V )
23 climneg.5 . . 3  |-  ( ph  ->  F  ~~>  A )
24 neg1cn 10702 . . . . . 6  |-  -u 1  e.  CC
25 eqid 2420 . . . . . . 7  |-  ( k  e.  Z  |->  -u 1
)  =  ( k  e.  Z  |->  -u 1
)
2625fvmpt2 5964 . . . . . 6  |-  ( ( k  e.  Z  /\  -u 1  e.  CC )  ->  ( ( k  e.  Z  |->  -u 1
) `  k )  =  -u 1 )
2724, 26mpan2 675 . . . . 5  |-  ( k  e.  Z  ->  (
( k  e.  Z  |-> 
-u 1 ) `  k )  =  -u
1 )
2827, 24syl6eqel 2516 . . . 4  |-  ( k  e.  Z  ->  (
( k  e.  Z  |-> 
-u 1 ) `  k )  e.  CC )
2928adantl 467 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |-> 
-u 1 ) `  k )  e.  CC )
30 climneg.6 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
31 simpr 462 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  Z )
3230negcld 9962 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  -u ( F `  k )  e.  CC )
33 eqid 2420 . . . . . 6  |-  ( k  e.  Z  |->  -u ( F `  k )
)  =  ( k  e.  Z  |->  -u ( F `  k )
)
3433fvmpt2 5964 . . . . 5  |-  ( ( k  e.  Z  /\  -u ( F `  k
)  e.  CC )  ->  ( ( k  e.  Z  |->  -u ( F `  k )
) `  k )  =  -u ( F `  k ) )
3531, 32, 34syl2anc 665 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |-> 
-u ( F `  k ) ) `  k )  =  -u ( F `  k ) )
3630mulm1d 10059 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( -u 1  x.  ( F `
 k ) )  =  -u ( F `  k ) )
3727eqcomd 2428 . . . . . 6  |-  ( k  e.  Z  ->  -u 1  =  ( ( k  e.  Z  |->  -u 1
) `  k )
)
3837adantl 467 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  -u 1  =  ( ( k  e.  Z  |->  -u 1
) `  k )
)
3938oveq1d 6311 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( -u 1  x.  ( F `
 k ) )  =  ( ( ( k  e.  Z  |->  -u
1 ) `  k
)  x.  ( F `
 k ) ) )
4035, 36, 393eqtr2d 2467 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |-> 
-u ( F `  k ) ) `  k )  =  ( ( ( k  e.  Z  |->  -u 1 ) `  k )  x.  ( F `  k )
) )
411, 2, 3, 4, 5, 6, 20, 22, 23, 29, 30, 40climmulf 37302 . 2  |-  ( ph  ->  ( k  e.  Z  |-> 
-u ( F `  k ) )  ~~>  ( -u
1  x.  A ) )
42 climcl 13530 . . . 4  |-  ( F  ~~>  A  ->  A  e.  CC )
4323, 42syl 17 . . 3  |-  ( ph  ->  A  e.  CC )
4443mulm1d 10059 . 2  |-  ( ph  ->  ( -u 1  x.  A )  =  -u A )
4541, 44breqtrd 4441 1  |-  ( ph  ->  ( k  e.  Z  |-> 
-u ( F `  k ) )  ~~>  -u A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   F/wnf 1663    e. wcel 1867   F/_wnfc 2568   _Vcvv 3078   class class class wbr 4417    |-> cmpt 4475   ` cfv 5592  (class class class)co 6296   CCcc 9526   1c1 9529    x. cmul 9533   -ucneg 9850   ZZcz 10926   ZZ>=cuz 11148    ~~> cli 13515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-sup 7953  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-seq 12200  df-exp 12259  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-clim 13519
This theorem is referenced by: (None)
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