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Theorem climneg 27839
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 4125 . . 3  |-  F/_ k
( k  e.  Z  |-> 
-u 1 )
3 climneg.2 . . 3  |-  F/_ k F
4 nfmpt1 4125 . . 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 5555 . . . . . . 7  |-  ( ZZ>= `  M )  e.  _V
85, 7eqeltri 2366 . . . . . 6  |-  Z  e. 
_V
98mptex 5762 . . . . 5  |-  ( k  e.  Z  |->  -u 1
)  e.  _V
109a1i 10 . . . 4  |-  ( ph  ->  ( k  e.  Z  |-> 
-u 1 )  e. 
_V )
11 ax-1cn 8811 . . . . . 6  |-  1  e.  CC
1211a1i 10 . . . . 5  |-  ( ph  ->  1  e.  CC )
1312negcld 9160 . . . 4  |-  ( ph  -> 
-u 1  e.  CC )
14 eqidd 2297 . . . . . 6  |-  ( j  e.  Z  ->  (
k  e.  Z  |->  -u
1 )  =  ( k  e.  Z  |->  -u
1 ) )
15 eqidd 2297 . . . . . 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 9160 . . . . . 6  |-  ( j  e.  Z  ->  -u 1  e.  CC )
1914, 15, 16, 18fvmptd 5622 . . . . 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 12033 . . 3  |-  ( ph  ->  ( k  e.  Z  |-> 
-u 1 )  ~~>  -u 1
)
228mptex 5762 . . . 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 9829 . . . . . . 7  |-  -u 1  e.  CC
2625jctr 526 . . . . . 6  |-  ( k  e.  Z  ->  (
k  e.  Z  /\  -u 1  e.  CC ) )
27 eqid 2296 . . . . . . 7  |-  ( k  e.  Z  |->  -u 1
)  =  ( k  e.  Z  |->  -u 1
)
2827fvmpt2 5624 . . . . . 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 9160 . . . . 5  |-  ( k  e.  Z  ->  -u 1  e.  CC )
3229, 31eqeltrd 2370 . . . 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 9160 . . . . . 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 2296 . . . . . 6  |-  ( k  e.  Z  |->  -u ( F `  k )
)  =  ( k  e.  Z  |->  -u ( F `  k )
)
3938fvmpt2 5624 . . . . 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 9247 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( -u 1  x.  ( F `
 k ) )  =  -u ( F `  k ) )
4241eqcomd 2301 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  -u ( F `  k )  =  ( -u 1  x.  ( F `  k
) ) )
4329eqcomd 2301 . . . . . 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 5889 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( -u 1  x.  ( F `
 k ) )  =  ( ( ( k  e.  Z  |->  -u
1 ) `  k
)  x.  ( F `
 k ) ) )
4640, 42, 453eqtrd 2332 . . 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 27833 . 2  |-  ( ph  ->  ( k  e.  Z  |-> 
-u ( F `  k ) )  ~~>  ( -u
1  x.  A ) )
48 climcl 11989 . . . 4  |-  ( F  ~~>  A  ->  A  e.  CC )
4924, 48syl 15 . . 3  |-  ( ph  ->  A  e.  CC )
5049mulm1d 9247 . 2  |-  ( ph  ->  ( -u 1  x.  A )  =  -u A )
5147, 50breqtrd 4063 1  |-  ( ph  ->  ( k  e.  Z  |-> 
-u ( F `  k ) )  ~~>  -u A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   F/wnf 1534    = wceq 1632    e. wcel 1696   F/_wnfc 2419   _Vcvv 2801   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754    x. cmul 8758   -ucneg 9054   ZZcz 10040   ZZ>=cuz 10246    ~~> cli 11974
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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