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

Theorem idlimc 37281
Description: Limit of the identity function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
idlimc.a  |-  ( ph  ->  A  C_  CC )
idlimc.f  |-  F  =  ( x  e.  A  |->  x )
idlimc.x  |-  ( ph  ->  X  e.  CC )
Assertion
Ref Expression
idlimc  |-  ( ph  ->  X  e.  ( F lim
CC  X ) )
Distinct variable groups:    x, A    x, X    ph, x
Allowed substitution hint:    F( x)

Proof of Theorem idlimc
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idlimc.x . 2  |-  ( ph  ->  X  e.  CC )
2 simpr 462 . . . 4  |-  ( (
ph  /\  w  e.  RR+ )  ->  w  e.  RR+ )
3 simpr 462 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
4 idlimc.f . . . . . . . . . . . . . . 15  |-  F  =  ( x  e.  A  |->  x )
54fvmpt2 5973 . . . . . . . . . . . . . 14  |-  ( ( x  e.  A  /\  x  e.  A )  ->  ( F `  x
)  =  x )
63, 3, 5syl2anc 665 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  x )
76oveq1d 6320 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
)  -  X )  =  ( x  -  X ) )
87fveq2d 5885 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  ( abs `  ( ( F `
 x )  -  X ) )  =  ( abs `  (
x  -  X ) ) )
98adantr 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  A )  /\  ( abs `  ( x  -  X ) )  < 
w )  ->  ( abs `  ( ( F `
 x )  -  X ) )  =  ( abs `  (
x  -  X ) ) )
10 simpr 462 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  A )  /\  ( abs `  ( x  -  X ) )  < 
w )  ->  ( abs `  ( x  -  X ) )  < 
w )
119, 10eqbrtrd 4446 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  ( abs `  ( x  -  X ) )  < 
w )  ->  ( abs `  ( ( F `
 x )  -  X ) )  < 
w )
1211adantrl 720 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  (
x  =/=  X  /\  ( abs `  ( x  -  X ) )  <  w ) )  ->  ( abs `  (
( F `  x
)  -  X ) )  <  w )
1312ex 435 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  =/=  X  /\  ( abs `  (
x  -  X ) )  <  w )  ->  ( abs `  (
( F `  x
)  -  X ) )  <  w ) )
1413adantlr 719 . . . . . 6  |-  ( ( ( ph  /\  w  e.  RR+ )  /\  x  e.  A )  ->  (
( x  =/=  X  /\  ( abs `  (
x  -  X ) )  <  w )  ->  ( abs `  (
( F `  x
)  -  X ) )  <  w ) )
1514ralrimiva 2846 . . . . 5  |-  ( (
ph  /\  w  e.  RR+ )  ->  A. x  e.  A  ( (
x  =/=  X  /\  ( abs `  ( x  -  X ) )  <  w )  -> 
( abs `  (
( F `  x
)  -  X ) )  <  w ) )
16 nfcv 2591 . . . . . . . . 9  |-  F/_ z
x
17 nfcv 2591 . . . . . . . . 9  |-  F/_ z X
1816, 17nfne 2763 . . . . . . . 8  |-  F/ z  x  =/=  X
19 nfv 1754 . . . . . . . 8  |-  F/ z ( abs `  (
x  -  X ) )  <  w
2018, 19nfan 1986 . . . . . . 7  |-  F/ z ( x  =/=  X  /\  ( abs `  (
x  -  X ) )  <  w )
21 nfv 1754 . . . . . . 7  |-  F/ z ( abs `  (
( F `  x
)  -  X ) )  <  w
2220, 21nfim 1978 . . . . . 6  |-  F/ z ( ( x  =/= 
X  /\  ( abs `  ( x  -  X
) )  <  w
)  ->  ( abs `  ( ( F `  x )  -  X
) )  <  w
)
23 nfv 1754 . . . . . . 7  |-  F/ x
( z  =/=  X  /\  ( abs `  (
z  -  X ) )  <  w )
24 nfcv 2591 . . . . . . . . 9  |-  F/_ x abs
25 nfmpt1 4515 . . . . . . . . . . . 12  |-  F/_ x
( x  e.  A  |->  x )
264, 25nfcxfr 2589 . . . . . . . . . . 11  |-  F/_ x F
27 nfcv 2591 . . . . . . . . . . 11  |-  F/_ x
z
2826, 27nffv 5888 . . . . . . . . . 10  |-  F/_ x
( F `  z
)
29 nfcv 2591 . . . . . . . . . 10  |-  F/_ x  -
30 nfcv 2591 . . . . . . . . . 10  |-  F/_ x X
3128, 29, 30nfov 6331 . . . . . . . . 9  |-  F/_ x
( ( F `  z )  -  X
)
3224, 31nffv 5888 . . . . . . . 8  |-  F/_ x
( abs `  (
( F `  z
)  -  X ) )
33 nfcv 2591 . . . . . . . 8  |-  F/_ x  <
34 nfcv 2591 . . . . . . . 8  |-  F/_ x w
3532, 33, 34nfbr 4470 . . . . . . 7  |-  F/ x
( abs `  (
( F `  z
)  -  X ) )  <  w
3623, 35nfim 1978 . . . . . 6  |-  F/ x
( ( z  =/= 
X  /\  ( abs `  ( z  -  X
) )  <  w
)  ->  ( abs `  ( ( F `  z )  -  X
) )  <  w
)
37 neeq1 2712 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =/=  X  <->  z  =/=  X ) )
38 oveq1 6312 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x  -  X )  =  ( z  -  X ) )
3938fveq2d 5885 . . . . . . . . 9  |-  ( x  =  z  ->  ( abs `  ( x  -  X ) )  =  ( abs `  (
z  -  X ) ) )
4039breq1d 4436 . . . . . . . 8  |-  ( x  =  z  ->  (
( abs `  (
x  -  X ) )  <  w  <->  ( abs `  ( z  -  X
) )  <  w
) )
4137, 40anbi12d 715 . . . . . . 7  |-  ( x  =  z  ->  (
( x  =/=  X  /\  ( abs `  (
x  -  X ) )  <  w )  <-> 
( z  =/=  X  /\  ( abs `  (
z  -  X ) )  <  w ) ) )
42 fveq2 5881 . . . . . . . . . 10  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
4342oveq1d 6320 . . . . . . . . 9  |-  ( x  =  z  ->  (
( F `  x
)  -  X )  =  ( ( F `
 z )  -  X ) )
4443fveq2d 5885 . . . . . . . 8  |-  ( x  =  z  ->  ( abs `  ( ( F `
 x )  -  X ) )  =  ( abs `  (
( F `  z
)  -  X ) ) )
4544breq1d 4436 . . . . . . 7  |-  ( x  =  z  ->  (
( abs `  (
( F `  x
)  -  X ) )  <  w  <->  ( abs `  ( ( F `  z )  -  X
) )  <  w
) )
4641, 45imbi12d 321 . . . . . 6  |-  ( x  =  z  ->  (
( ( x  =/= 
X  /\  ( abs `  ( x  -  X
) )  <  w
)  ->  ( abs `  ( ( F `  x )  -  X
) )  <  w
)  <->  ( ( z  =/=  X  /\  ( abs `  ( z  -  X ) )  < 
w )  ->  ( abs `  ( ( F `
 z )  -  X ) )  < 
w ) ) )
4722, 36, 46cbvral 3058 . . . . 5  |-  ( A. x  e.  A  (
( x  =/=  X  /\  ( abs `  (
x  -  X ) )  <  w )  ->  ( abs `  (
( F `  x
)  -  X ) )  <  w )  <->  A. z  e.  A  ( ( z  =/= 
X  /\  ( abs `  ( z  -  X
) )  <  w
)  ->  ( abs `  ( ( F `  z )  -  X
) )  <  w
) )
4815, 47sylib 199 . . . 4  |-  ( (
ph  /\  w  e.  RR+ )  ->  A. z  e.  A  ( (
z  =/=  X  /\  ( abs `  ( z  -  X ) )  <  w )  -> 
( abs `  (
( F `  z
)  -  X ) )  <  w ) )
49 breq2 4430 . . . . . . . 8  |-  ( y  =  w  ->  (
( abs `  (
z  -  X ) )  <  y  <->  ( abs `  ( z  -  X
) )  <  w
) )
5049anbi2d 708 . . . . . . 7  |-  ( y  =  w  ->  (
( z  =/=  X  /\  ( abs `  (
z  -  X ) )  <  y )  <-> 
( z  =/=  X  /\  ( abs `  (
z  -  X ) )  <  w ) ) )
5150imbi1d 318 . . . . . 6  |-  ( y  =  w  ->  (
( ( z  =/= 
X  /\  ( abs `  ( z  -  X
) )  <  y
)  ->  ( abs `  ( ( F `  z )  -  X
) )  <  w
)  <->  ( ( z  =/=  X  /\  ( abs `  ( z  -  X ) )  < 
w )  ->  ( abs `  ( ( F `
 z )  -  X ) )  < 
w ) ) )
5251ralbidv 2871 . . . . 5  |-  ( y  =  w  ->  ( A. z  e.  A  ( ( z  =/= 
X  /\  ( abs `  ( z  -  X
) )  <  y
)  ->  ( abs `  ( ( F `  z )  -  X
) )  <  w
)  <->  A. z  e.  A  ( ( z  =/= 
X  /\  ( abs `  ( z  -  X
) )  <  w
)  ->  ( abs `  ( ( F `  z )  -  X
) )  <  w
) ) )
5352rspcev 3188 . . . 4  |-  ( ( w  e.  RR+  /\  A. z  e.  A  (
( z  =/=  X  /\  ( abs `  (
z  -  X ) )  <  w )  ->  ( abs `  (
( F `  z
)  -  X ) )  <  w ) )  ->  E. y  e.  RR+  A. z  e.  A  ( ( z  =/=  X  /\  ( abs `  ( z  -  X ) )  < 
y )  ->  ( abs `  ( ( F `
 z )  -  X ) )  < 
w ) )
542, 48, 53syl2anc 665 . . 3  |-  ( (
ph  /\  w  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  A  ( ( z  =/=  X  /\  ( abs `  ( z  -  X ) )  < 
y )  ->  ( abs `  ( ( F `
 z )  -  X ) )  < 
w ) )
5554ralrimiva 2846 . 2  |-  ( ph  ->  A. w  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( (
z  =/=  X  /\  ( abs `  ( z  -  X ) )  <  y )  -> 
( abs `  (
( F `  z
)  -  X ) )  <  w ) )
56 idlimc.a . . . . 5  |-  ( ph  ->  A  C_  CC )
5756sselda 3470 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  CC )
5857, 4fmptd 6061 . . 3  |-  ( ph  ->  F : A --> CC )
5958, 56, 1ellimc3 22711 . 2  |-  ( ph  ->  ( X  e.  ( F lim CC  X )  <-> 
( X  e.  CC  /\ 
A. w  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( (
z  =/=  X  /\  ( abs `  ( z  -  X ) )  <  y )  -> 
( abs `  (
( F `  z
)  -  X ) )  <  w ) ) ) )
601, 55, 59mpbir2and 930 1  |-  ( ph  ->  X  e.  ( F lim
CC  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   E.wrex 2783    C_ wss 3442   class class class wbr 4426    |-> cmpt 4484   ` cfv 5601  (class class class)co 6305   CCcc 9536    < clt 9674    - cmin 9859   RR+crp 11302   abscabs 13276   lim CC climc 22694
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fi 7931  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  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 11783  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-plusg 15165  df-mulr 15166  df-starv 15167  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-rest 15280  df-topn 15281  df-topgen 15301  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cnp 20175  df-xms 21266  df-ms 21267  df-limc 22698
This theorem is referenced by:  fourierdlem53  37594  fourierdlem60  37601  fourierdlem61  37602  fourierdlem73  37614  fourierdlem74  37615  fourierdlem75  37616  fourierdlem76  37617
  Copyright terms: Public domain W3C validator