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Theorem neglimc 37825
Description: Limit of the negative function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
neglimc.f  |-  F  =  ( x  e.  A  |->  B )
neglimc.g  |-  G  =  ( x  e.  A  |-> 
-u B )
neglimc.b  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
neglimc.c  |-  ( ph  ->  C  e.  ( F lim
CC  D ) )
Assertion
Ref Expression
neglimc  |-  ( ph  -> 
-u C  e.  ( G lim CC  D ) )
Distinct variable groups:    x, A    ph, x
Allowed substitution hints:    B( x)    C( x)    D( x)    F( x)    G( x)

Proof of Theorem neglimc
Dummy variables  v  w  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limccl 22909 . . . 4  |-  ( F lim
CC  D )  C_  CC
2 neglimc.c . . . 4  |-  ( ph  ->  C  e.  ( F lim
CC  D ) )
31, 2sseldi 3416 . . 3  |-  ( ph  ->  C  e.  CC )
43negcld 9992 . 2  |-  ( ph  -> 
-u C  e.  CC )
5 neglimc.b . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
6 neglimc.f . . . . . . . . 9  |-  F  =  ( x  e.  A  |->  B )
75, 6fmptd 6061 . . . . . . . 8  |-  ( ph  ->  F : A --> CC )
86, 5, 2limcmptdm 37812 . . . . . . . 8  |-  ( ph  ->  A  C_  CC )
9 limcrcl 22908 . . . . . . . . . 10  |-  ( C  e.  ( F lim CC  D )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  D  e.  CC ) )
102, 9syl 17 . . . . . . . . 9  |-  ( ph  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  D  e.  CC ) )
1110simp3d 1044 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
127, 8, 11ellimc3 22913 . . . . . . 7  |-  ( ph  ->  ( C  e.  ( F lim CC  D )  <-> 
( C  e.  CC  /\ 
A. y  e.  RR+  E. w  e.  RR+  A. v  e.  A  ( (
v  =/=  D  /\  ( abs `  ( v  -  D ) )  <  w )  -> 
( abs `  (
( F `  v
)  -  C ) )  <  y ) ) ) )
132, 12mpbid 215 . . . . . 6  |-  ( ph  ->  ( C  e.  CC  /\ 
A. y  e.  RR+  E. w  e.  RR+  A. v  e.  A  ( (
v  =/=  D  /\  ( abs `  ( v  -  D ) )  <  w )  -> 
( abs `  (
( F `  v
)  -  C ) )  <  y ) ) )
1413simprd 470 . . . . 5  |-  ( ph  ->  A. y  e.  RR+  E. w  e.  RR+  A. v  e.  A  ( (
v  =/=  D  /\  ( abs `  ( v  -  D ) )  <  w )  -> 
( abs `  (
( F `  v
)  -  C ) )  <  y ) )
1514r19.21bi 2776 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. w  e.  RR+  A. v  e.  A  ( ( v  =/=  D  /\  ( abs `  ( v  -  D ) )  < 
w )  ->  ( abs `  ( ( F `
 v )  -  C ) )  < 
y ) )
16 simplll 776 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  w  e.  RR+ )  /\  v  e.  A )  ->  ph )
17163ad2ant1 1051 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  y  e.  RR+ )  /\  w  e.  RR+ )  /\  v  e.  A
)  /\  ( (
v  =/=  D  /\  ( abs `  ( v  -  D ) )  <  w )  -> 
( abs `  (
( F `  v
)  -  C ) )  <  y )  /\  ( v  =/= 
D  /\  ( abs `  ( v  -  D
) )  <  w
) )  ->  ph )
18 simp1r 1055 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  y  e.  RR+ )  /\  w  e.  RR+ )  /\  v  e.  A
)  /\  ( (
v  =/=  D  /\  ( abs `  ( v  -  D ) )  <  w )  -> 
( abs `  (
( F `  v
)  -  C ) )  <  y )  /\  ( v  =/= 
D  /\  ( abs `  ( v  -  D
) )  <  w
) )  ->  v  e.  A )
19 simp3 1032 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  y  e.  RR+ )  /\  w  e.  RR+ )  /\  v  e.  A
)  /\  ( (
v  =/=  D  /\  ( abs `  ( v  -  D ) )  <  w )  -> 
( abs `  (
( F `  v
)  -  C ) )  <  y )  /\  ( v  =/= 
D  /\  ( abs `  ( v  -  D
) )  <  w
) )  ->  (
v  =/=  D  /\  ( abs `  ( v  -  D ) )  <  w ) )
20 simp2 1031 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  y  e.  RR+ )  /\  w  e.  RR+ )  /\  v  e.  A
)  /\  ( (
v  =/=  D  /\  ( abs `  ( v  -  D ) )  <  w )  -> 
( abs `  (
( F `  v
)  -  C ) )  <  y )  /\  ( v  =/= 
D  /\  ( abs `  ( v  -  D
) )  <  w
) )  ->  (
( v  =/=  D  /\  ( abs `  (
v  -  D ) )  <  w )  ->  ( abs `  (
( F `  v
)  -  C ) )  <  y ) )
2119, 20mpd 15 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  y  e.  RR+ )  /\  w  e.  RR+ )  /\  v  e.  A
)  /\  ( (
v  =/=  D  /\  ( abs `  ( v  -  D ) )  <  w )  -> 
( abs `  (
( F `  v
)  -  C ) )  <  y )  /\  ( v  =/= 
D  /\  ( abs `  ( v  -  D
) )  <  w
) )  ->  ( abs `  ( ( F `
 v )  -  C ) )  < 
y )
22 nfv 1769 . . . . . . . . . . . . . . . 16  |-  F/ x
( ph  /\  v  e.  A )
23 neglimc.g . . . . . . . . . . . . . . . . . . 19  |-  G  =  ( x  e.  A  |-> 
-u B )
24 nfmpt1 4485 . . . . . . . . . . . . . . . . . . 19  |-  F/_ x
( x  e.  A  |-> 
-u B )
2523, 24nfcxfr 2610 . . . . . . . . . . . . . . . . . 18  |-  F/_ x G
26 nfcv 2612 . . . . . . . . . . . . . . . . . 18  |-  F/_ x
v
2725, 26nffv 5886 . . . . . . . . . . . . . . . . 17  |-  F/_ x
( G `  v
)
28 nfmpt1 4485 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ x
( x  e.  A  |->  B )
296, 28nfcxfr 2610 . . . . . . . . . . . . . . . . . . 19  |-  F/_ x F
3029, 26nffv 5886 . . . . . . . . . . . . . . . . . 18  |-  F/_ x
( F `  v
)
3130nfneg 9891 . . . . . . . . . . . . . . . . 17  |-  F/_ x -u ( F `  v
)
3227, 31nfeq 2623 . . . . . . . . . . . . . . . 16  |-  F/ x
( G `  v
)  =  -u ( F `  v )
3322, 32nfim 2023 . . . . . . . . . . . . . . 15  |-  F/ x
( ( ph  /\  v  e.  A )  ->  ( G `  v
)  =  -u ( F `  v )
)
34 eleq1 2537 . . . . . . . . . . . . . . . . 17  |-  ( x  =  v  ->  (
x  e.  A  <->  v  e.  A ) )
3534anbi2d 718 . . . . . . . . . . . . . . . 16  |-  ( x  =  v  ->  (
( ph  /\  x  e.  A )  <->  ( ph  /\  v  e.  A ) ) )
36 fveq2 5879 . . . . . . . . . . . . . . . . 17  |-  ( x  =  v  ->  ( G `  x )  =  ( G `  v ) )
37 fveq2 5879 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  v  ->  ( F `  x )  =  ( F `  v ) )
3837negeqd 9889 . . . . . . . . . . . . . . . . 17  |-  ( x  =  v  ->  -u ( F `  x )  =  -u ( F `  v ) )
3936, 38eqeq12d 2486 . . . . . . . . . . . . . . . 16  |-  ( x  =  v  ->  (
( G `  x
)  =  -u ( F `  x )  <->  ( G `  v )  =  -u ( F `  v ) ) )
4035, 39imbi12d 327 . . . . . . . . . . . . . . 15  |-  ( x  =  v  ->  (
( ( ph  /\  x  e.  A )  ->  ( G `  x
)  =  -u ( F `  x )
)  <->  ( ( ph  /\  v  e.  A )  ->  ( G `  v )  =  -u ( F `  v ) ) ) )
41 simpr 468 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
425negcld 9992 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  A )  ->  -u B  e.  CC )
4323fvmpt2 5972 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  A  /\  -u B  e.  CC )  ->  ( G `  x )  =  -u B )
4441, 42, 43syl2anc 673 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  -u B )
456fvmpt2 5972 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  A  /\  B  e.  CC )  ->  ( F `  x
)  =  B )
4641, 5, 45syl2anc 673 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
4746negeqd 9889 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  A )  ->  -u ( F `  x )  =  -u B )
4844, 47eqtr4d 2508 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  -u ( F `  x ) )
4933, 40, 48chvar 2119 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  e.  A )  ->  ( G `  v )  =  -u ( F `  v ) )
5049oveq1d 6323 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  A )  ->  (
( G `  v
)  -  -u C
)  =  ( -u ( F `  v )  -  -u C ) )
517ffvelrnda 6037 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  e.  A )  ->  ( F `  v )  e.  CC )
523adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  e.  A )  ->  C  e.  CC )
5351, 52negsubdi3d 37596 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  A )  ->  -u (
( F `  v
)  -  C )  =  ( -u ( F `  v )  -  -u C ) )
5450, 53eqtr4d 2508 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  A )  ->  (
( G `  v
)  -  -u C
)  =  -u (
( F `  v
)  -  C ) )
5554fveq2d 5883 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  A )  ->  ( abs `  ( ( G `
 v )  -  -u C ) )  =  ( abs `  -u (
( F `  v
)  -  C ) ) )
5651, 52subcld 10005 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  A )  ->  (
( F `  v
)  -  C )  e.  CC )
5756absnegd 13588 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  A )  ->  ( abs `  -u ( ( F `
 v )  -  C ) )  =  ( abs `  (
( F `  v
)  -  C ) ) )
5855, 57eqtrd 2505 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  A )  ->  ( abs `  ( ( G `
 v )  -  -u C ) )  =  ( abs `  (
( F `  v
)  -  C ) ) )
5958adantr 472 . . . . . . . . 9  |-  ( ( ( ph  /\  v  e.  A )  /\  ( abs `  ( ( F `
 v )  -  C ) )  < 
y )  ->  ( abs `  ( ( G `
 v )  -  -u C ) )  =  ( abs `  (
( F `  v
)  -  C ) ) )
60 simpr 468 . . . . . . . . 9  |-  ( ( ( ph  /\  v  e.  A )  /\  ( abs `  ( ( F `
 v )  -  C ) )  < 
y )  ->  ( abs `  ( ( F `
 v )  -  C ) )  < 
y )
6159, 60eqbrtrd 4416 . . . . . . . 8  |-  ( ( ( ph  /\  v  e.  A )  /\  ( abs `  ( ( F `
 v )  -  C ) )  < 
y )  ->  ( abs `  ( ( G `
 v )  -  -u C ) )  < 
y )
6217, 18, 21, 61syl21anc 1291 . . . . . . 7  |-  ( ( ( ( ( ph  /\  y  e.  RR+ )  /\  w  e.  RR+ )  /\  v  e.  A
)  /\  ( (
v  =/=  D  /\  ( abs `  ( v  -  D ) )  <  w )  -> 
( abs `  (
( F `  v
)  -  C ) )  <  y )  /\  ( v  =/= 
D  /\  ( abs `  ( v  -  D
) )  <  w
) )  ->  ( abs `  ( ( G `
 v )  -  -u C ) )  < 
y )
63623exp 1230 . . . . . 6  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  w  e.  RR+ )  /\  v  e.  A )  ->  ( ( ( v  =/=  D  /\  ( abs `  ( v  -  D ) )  < 
w )  ->  ( abs `  ( ( F `
 v )  -  C ) )  < 
y )  ->  (
( v  =/=  D  /\  ( abs `  (
v  -  D ) )  <  w )  ->  ( abs `  (
( G `  v
)  -  -u C
) )  <  y
) ) )
6463ralimdva 2805 . . . . 5  |-  ( ( ( ph  /\  y  e.  RR+ )  /\  w  e.  RR+ )  ->  ( A. v  e.  A  ( ( v  =/= 
D  /\  ( abs `  ( v  -  D
) )  <  w
)  ->  ( abs `  ( ( F `  v )  -  C
) )  <  y
)  ->  A. v  e.  A  ( (
v  =/=  D  /\  ( abs `  ( v  -  D ) )  <  w )  -> 
( abs `  (
( G `  v
)  -  -u C
) )  <  y
) ) )
6564reximdva 2858 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( E. w  e.  RR+  A. v  e.  A  ( (
v  =/=  D  /\  ( abs `  ( v  -  D ) )  <  w )  -> 
( abs `  (
( F `  v
)  -  C ) )  <  y )  ->  E. w  e.  RR+  A. v  e.  A  ( ( v  =/=  D  /\  ( abs `  (
v  -  D ) )  <  w )  ->  ( abs `  (
( G `  v
)  -  -u C
) )  <  y
) ) )
6615, 65mpd 15 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. w  e.  RR+  A. v  e.  A  ( ( v  =/=  D  /\  ( abs `  ( v  -  D ) )  < 
w )  ->  ( abs `  ( ( G `
 v )  -  -u C ) )  < 
y ) )
6766ralrimiva 2809 . 2  |-  ( ph  ->  A. y  e.  RR+  E. w  e.  RR+  A. v  e.  A  ( (
v  =/=  D  /\  ( abs `  ( v  -  D ) )  <  w )  -> 
( abs `  (
( G `  v
)  -  -u C
) )  <  y
) )
6842, 23fmptd 6061 . . 3  |-  ( ph  ->  G : A --> CC )
6968, 8, 11ellimc3 22913 . 2  |-  ( ph  ->  ( -u C  e.  ( G lim CC  D
)  <->  ( -u C  e.  CC  /\  A. y  e.  RR+  E. w  e.  RR+  A. v  e.  A  ( ( v  =/= 
D  /\  ( abs `  ( v  -  D
) )  <  w
)  ->  ( abs `  ( ( G `  v )  -  -u C
) )  <  y
) ) ) )
704, 67, 69mpbir2and 936 1  |-  ( ph  -> 
-u C  e.  ( G lim CC  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757    C_ wss 3390   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555    < clt 9693    - cmin 9880   -ucneg 9881   RR+crp 11325   abscabs 13374   lim CC climc 22896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-fz 11811  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cnp 20321  df-xms 21413  df-ms 21414  df-limc 22900
This theorem is referenced by:  sublimc  37830  reclimc  37831
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