Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  neglimc Structured version   Visualization version   Unicode version

Theorem neglimc 37728
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 22830 . . . 4  |-  ( F lim
CC  D )  C_  CC
2 neglimc.c . . . 4  |-  ( ph  ->  C  e.  ( F lim
CC  D ) )
31, 2sseldi 3430 . . 3  |-  ( ph  ->  C  e.  CC )
43negcld 9973 . 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 6046 . . . . . . . 8  |-  ( ph  ->  F : A --> CC )
86, 5, 2limcmptdm 37715 . . . . . . . 8  |-  ( ph  ->  A  C_  CC )
9 limcrcl 22829 . . . . . . . . . 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 1022 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
127, 8, 11ellimc3 22834 . . . . . . 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 2757 . . . 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 768 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  RR+ )  /\  w  e.  RR+ )  /\  v  e.  A )  ->  ph )
17163ad2ant1 1029 . . . . . . . 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 1033 . . . . . . . 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 1010 . . . . . . . . 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 1009 . . . . . . . . 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 1761 . . . . . . . . . . . . . . . 16  |-  F/ x
( ph  /\  v  e.  A )
23 neglimc.g . . . . . . . . . . . . . . . . . . 19  |-  G  =  ( x  e.  A  |-> 
-u B )
24 nfmpt1 4492 . . . . . . . . . . . . . . . . . . 19  |-  F/_ x
( x  e.  A  |-> 
-u B )
2523, 24nfcxfr 2590 . . . . . . . . . . . . . . . . . 18  |-  F/_ x G
26 nfcv 2592 . . . . . . . . . . . . . . . . . 18  |-  F/_ x
v
2725, 26nffv 5872 . . . . . . . . . . . . . . . . 17  |-  F/_ x
( G `  v
)
28 nfmpt1 4492 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ x
( x  e.  A  |->  B )
296, 28nfcxfr 2590 . . . . . . . . . . . . . . . . . . 19  |-  F/_ x F
3029, 26nffv 5872 . . . . . . . . . . . . . . . . . 18  |-  F/_ x
( F `  v
)
3130nfneg 9871 . . . . . . . . . . . . . . . . 17  |-  F/_ x -u ( F `  v
)
3227, 31nfeq 2603 . . . . . . . . . . . . . . . 16  |-  F/ x
( G `  v
)  =  -u ( F `  v )
3322, 32nfim 2003 . . . . . . . . . . . . . . 15  |-  F/ x
( ( ph  /\  v  e.  A )  ->  ( G `  v
)  =  -u ( F `  v )
)
34 eleq1 2517 . . . . . . . . . . . . . . . . 17  |-  ( x  =  v  ->  (
x  e.  A  <->  v  e.  A ) )
3534anbi2d 710 . . . . . . . . . . . . . . . 16  |-  ( x  =  v  ->  (
( ph  /\  x  e.  A )  <->  ( ph  /\  v  e.  A ) ) )
36 fveq2 5865 . . . . . . . . . . . . . . . . 17  |-  ( x  =  v  ->  ( G `  x )  =  ( G `  v ) )
37 fveq2 5865 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  v  ->  ( F `  x )  =  ( F `  v ) )
3837negeqd 9869 . . . . . . . . . . . . . . . . 17  |-  ( x  =  v  ->  -u ( F `  x )  =  -u ( F `  v ) )
3936, 38eqeq12d 2466 . . . . . . . . . . . . . . . 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 9973 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  A )  ->  -u B  e.  CC )
4323fvmpt2 5957 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  A  /\  -u B  e.  CC )  ->  ( G `  x )  =  -u B )
4441, 42, 43syl2anc 667 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  -u B )
456fvmpt2 5957 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  A  /\  B  e.  CC )  ->  ( F `  x
)  =  B )
4641, 5, 45syl2anc 667 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
4746negeqd 9869 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  A )  ->  -u ( F `  x )  =  -u B )
4844, 47eqtr4d 2488 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  -u ( F `  x ) )
4933, 40, 48chvar 2106 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  e.  A )  ->  ( G `  v )  =  -u ( F `  v ) )
5049oveq1d 6305 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  A )  ->  (
( G `  v
)  -  -u C
)  =  ( -u ( F `  v )  -  -u C ) )
517ffvelrnda 6022 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  e.  A )  ->  ( F `  v )  e.  CC )
523adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  e.  A )  ->  C  e.  CC )
5351, 52negsubdi3d 37507 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  A )  ->  -u (
( F `  v
)  -  C )  =  ( -u ( F `  v )  -  -u C ) )
5450, 53eqtr4d 2488 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  A )  ->  (
( G `  v
)  -  -u C
)  =  -u (
( F `  v
)  -  C ) )
5554fveq2d 5869 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  A )  ->  ( abs `  ( ( G `
 v )  -  -u C ) )  =  ( abs `  -u (
( F `  v
)  -  C ) ) )
5651, 52subcld 9986 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  A )  ->  (
( F `  v
)  -  C )  e.  CC )
5756absnegd 13511 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  A )  ->  ( abs `  -u ( ( F `
 v )  -  C ) )  =  ( abs `  (
( F `  v
)  -  C ) ) )
5855, 57eqtrd 2485 . . . . . . . . . 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 4423 . . . . . . . 8  |-  ( ( ( ph  /\  v  e.  A )  /\  ( abs `  ( ( F `
 v )  -  C ) )  < 
y )  ->  ( abs `  ( ( G `
 v )  -  -u C ) )  < 
y )
6217, 18, 21, 61syl21anc 1267 . . . . . . 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 1207 . . . . . 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 2796 . . . . 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 2862 . . . 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 2802 . 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 6046 . . 3  |-  ( ph  ->  G : A --> CC )
6968, 8, 11ellimc3 22834 . 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 933 1  |-  ( ph  -> 
-u C  e.  ( G lim CC  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738    C_ wss 3404   class class class wbr 4402    |-> cmpt 4461   dom cdm 4834   -->wf 5578   ` cfv 5582  (class class class)co 6290   CCcc 9537    < clt 9675    - cmin 9860   -ucneg 9861   RR+crp 11302   abscabs 13297   lim CC climc 22817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fi 7925  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-fz 11785  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-plusg 15203  df-mulr 15204  df-starv 15205  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-rest 15321  df-topn 15322  df-topgen 15342  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cnp 20244  df-xms 21335  df-ms 21336  df-limc 22821
This theorem is referenced by:  sublimc  37733  reclimc  37734
  Copyright terms: Public domain W3C validator