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Theorem neglimc 37592
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 22822 . . . 4  |-  ( F lim
CC  D )  C_  CC
2 neglimc.c . . . 4  |-  ( ph  ->  C  e.  ( F lim
CC  D ) )
31, 2sseldi 3463 . . 3  |-  ( ph  ->  C  e.  CC )
43negcld 9975 . 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 6059 . . . . . . . 8  |-  ( ph  ->  F : A --> CC )
86, 5, 2limcmptdm 37579 . . . . . . . 8  |-  ( ph  ->  A  C_  CC )
9 limcrcl 22821 . . . . . . . . . 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 1020 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
127, 8, 11ellimc3 22826 . . . . . . 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 214 . . . . . 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 465 . . . . 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 2795 . . . 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 767 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  w  e.  RR+ )  /\  v  e.  A )  ->  ph )
17163ad2ant1 1027 . . . . . . . 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 1031 . . . . . . . 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 1008 . . . . . . . . 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 1007 . . . . . . . . 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 1752 . . . . . . . . . . . . . . . 16  |-  F/ x
( ph  /\  v  e.  A )
23 neglimc.g . . . . . . . . . . . . . . . . . . 19  |-  G  =  ( x  e.  A  |-> 
-u B )
24 nfmpt1 4511 . . . . . . . . . . . . . . . . . . 19  |-  F/_ x
( x  e.  A  |-> 
-u B )
2523, 24nfcxfr 2583 . . . . . . . . . . . . . . . . . 18  |-  F/_ x G
26 nfcv 2585 . . . . . . . . . . . . . . . . . 18  |-  F/_ x
v
2725, 26nffv 5886 . . . . . . . . . . . . . . . . 17  |-  F/_ x
( G `  v
)
28 nfmpt1 4511 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ x
( x  e.  A  |->  B )
296, 28nfcxfr 2583 . . . . . . . . . . . . . . . . . . 19  |-  F/_ x F
3029, 26nffv 5886 . . . . . . . . . . . . . . . . . 18  |-  F/_ x
( F `  v
)
3130nfneg 9873 . . . . . . . . . . . . . . . . 17  |-  F/_ x -u ( F `  v
)
3227, 31nfeq 2596 . . . . . . . . . . . . . . . 16  |-  F/ x
( G `  v
)  =  -u ( F `  v )
3322, 32nfim 1977 . . . . . . . . . . . . . . 15  |-  F/ x
( ( ph  /\  v  e.  A )  ->  ( G `  v
)  =  -u ( F `  v )
)
34 eleq1 2495 . . . . . . . . . . . . . . . . 17  |-  ( x  =  v  ->  (
x  e.  A  <->  v  e.  A ) )
3534anbi2d 709 . . . . . . . . . . . . . . . 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 9871 . . . . . . . . . . . . . . . . 17  |-  ( x  =  v  ->  -u ( F `  x )  =  -u ( F `  v ) )
3936, 38eqeq12d 2445 . . . . . . . . . . . . . . . 16  |-  ( x  =  v  ->  (
( G `  x
)  =  -u ( F `  x )  <->  ( G `  v )  =  -u ( F `  v ) ) )
4035, 39imbi12d 322 . . . . . . . . . . . . . . 15  |-  ( x  =  v  ->  (
( ( ph  /\  x  e.  A )  ->  ( G `  x
)  =  -u ( F `  x )
)  <->  ( ( ph  /\  v  e.  A )  ->  ( G `  v )  =  -u ( F `  v ) ) ) )
41 simpr 463 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
425negcld 9975 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  A )  ->  -u B  e.  CC )
4323fvmpt2 5971 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  A  /\  -u B  e.  CC )  ->  ( G `  x )  =  -u B )
4441, 42, 43syl2anc 666 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  -u B )
456fvmpt2 5971 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  A  /\  B  e.  CC )  ->  ( F `  x
)  =  B )
4641, 5, 45syl2anc 666 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
4746negeqd 9871 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  A )  ->  -u ( F `  x )  =  -u B )
4844, 47eqtr4d 2467 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  -u ( F `  x ) )
4933, 40, 48chvar 2068 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  e.  A )  ->  ( G `  v )  =  -u ( F `  v ) )
5049oveq1d 6318 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  A )  ->  (
( G `  v
)  -  -u C
)  =  ( -u ( F `  v )  -  -u C ) )
517ffvelrnda 6035 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  e.  A )  ->  ( F `  v )  e.  CC )
523adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  e.  A )  ->  C  e.  CC )
5351, 52negsubdi3d 37393 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  A )  ->  -u (
( F `  v
)  -  C )  =  ( -u ( F `  v )  -  -u C ) )
5450, 53eqtr4d 2467 . . . . . . . . . . . 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 9988 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  A )  ->  (
( F `  v
)  -  C )  e.  CC )
5756absnegd 13504 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  A )  ->  ( abs `  -u ( ( F `
 v )  -  C ) )  =  ( abs `  (
( F `  v
)  -  C ) ) )
5855, 57eqtrd 2464 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  A )  ->  ( abs `  ( ( G `
 v )  -  -u C ) )  =  ( abs `  (
( F `  v
)  -  C ) ) )
5958adantr 467 . . . . . . . . 9  |-  ( ( ( ph  /\  v  e.  A )  /\  ( abs `  ( ( F `
 v )  -  C ) )  < 
y )  ->  ( abs `  ( ( G `
 v )  -  -u C ) )  =  ( abs `  (
( F `  v
)  -  C ) ) )
60 simpr 463 . . . . . . . . 9  |-  ( ( ( ph  /\  v  e.  A )  /\  ( abs `  ( ( F `
 v )  -  C ) )  < 
y )  ->  ( abs `  ( ( F `
 v )  -  C ) )  < 
y )
6159, 60eqbrtrd 4442 . . . . . . . 8  |-  ( ( ( ph  /\  v  e.  A )  /\  ( abs `  ( ( F `
 v )  -  C ) )  < 
y )  ->  ( abs `  ( ( G `
 v )  -  -u C ) )  < 
y )
6217, 18, 21, 61syl21anc 1264 . . . . . . 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 1205 . . . . . 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 2834 . . . . 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 2901 . . . 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 2840 . 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 6059 . . 3  |-  ( ph  ->  G : A --> CC )
6968, 8, 11ellimc3 22826 . 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 931 1  |-  ( ph  -> 
-u C  e.  ( G lim CC  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   A.wral 2776   E.wrex 2777    C_ wss 3437   class class class wbr 4421    |-> cmpt 4480   dom cdm 4851   -->wf 5595   ` cfv 5599  (class class class)co 6303   CCcc 9539    < clt 9677    - cmin 9862   -ucneg 9863   RR+crp 11304   abscabs 13291   lim CC climc 22809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fi 7929  df-sup 7960  df-inf 7961  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-q 11267  df-rp 11305  df-xneg 11411  df-xadd 11412  df-xmul 11413  df-fz 11787  df-seq 12215  df-exp 12274  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-plusg 15196  df-mulr 15197  df-starv 15198  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-rest 15314  df-topn 15315  df-topgen 15335  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19913  df-bases 19914  df-topon 19915  df-topsp 19916  df-cnp 20236  df-xms 21327  df-ms 21328  df-limc 22813
This theorem is referenced by:  sublimc  37597  reclimc  37598
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