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Theorem idlimc 31491
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 . . 3  |-  ( ph  ->  X  e.  CC )
2 simpr 461 . . . . 5  |-  ( (
ph  /\  w  e.  RR+ )  ->  w  e.  RR+ )
3 simpr 461 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
4 idlimc.f . . . . . . . . . . . . . . . 16  |-  F  =  ( x  e.  A  |->  x )
54fvmpt2 5964 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  A  /\  x  e.  A )  ->  ( F `  x
)  =  x )
63, 3, 5syl2anc 661 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  x )
76oveq1d 6310 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
)  -  X )  =  ( x  -  X ) )
87fveq2d 5876 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  ( abs `  ( ( F `
 x )  -  X ) )  =  ( abs `  (
x  -  X ) ) )
98adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  A )  /\  ( abs `  ( x  -  X ) )  < 
w )  ->  ( abs `  ( ( F `
 x )  -  X ) )  =  ( abs `  (
x  -  X ) ) )
10 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  A )  /\  ( abs `  ( x  -  X ) )  < 
w )  ->  ( abs `  ( x  -  X ) )  < 
w )
119, 10eqbrtrd 4473 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  A )  /\  ( abs `  ( x  -  X ) )  < 
w )  ->  ( abs `  ( ( F `
 x )  -  X ) )  < 
w )
1211adantrl 715 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  (
x  =/=  X  /\  ( abs `  ( x  -  X ) )  <  w ) )  ->  ( abs `  (
( F `  x
)  -  X ) )  <  w )
1312ex 434 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  =/=  X  /\  ( abs `  (
x  -  X ) )  <  w )  ->  ( abs `  (
( F `  x
)  -  X ) )  <  w ) )
1413adantlr 714 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  RR+ )  /\  x  e.  A )  ->  (
( x  =/=  X  /\  ( abs `  (
x  -  X ) )  <  w )  ->  ( abs `  (
( F `  x
)  -  X ) )  <  w ) )
1514ralrimiva 2881 . . . . . 6  |-  ( (
ph  /\  w  e.  RR+ )  ->  A. x  e.  A  ( (
x  =/=  X  /\  ( abs `  ( x  -  X ) )  <  w )  -> 
( abs `  (
( F `  x
)  -  X ) )  <  w ) )
16 nfcv 2629 . . . . . . . . . 10  |-  F/_ z
x
17 nfcv 2629 . . . . . . . . . 10  |-  F/_ z X
1816, 17nfne 2798 . . . . . . . . 9  |-  F/ z  x  =/=  X
19 nfv 1683 . . . . . . . . 9  |-  F/ z ( abs `  (
x  -  X ) )  <  w
2018, 19nfan 1875 . . . . . . . 8  |-  F/ z ( x  =/=  X  /\  ( abs `  (
x  -  X ) )  <  w )
21 nfv 1683 . . . . . . . 8  |-  F/ z ( abs `  (
( F `  x
)  -  X ) )  <  w
2220, 21nfim 1867 . . . . . . 7  |-  F/ z ( ( x  =/= 
X  /\  ( abs `  ( x  -  X
) )  <  w
)  ->  ( abs `  ( ( F `  x )  -  X
) )  <  w
)
23 nfv 1683 . . . . . . . 8  |-  F/ x
( z  =/=  X  /\  ( abs `  (
z  -  X ) )  <  w )
24 nfcv 2629 . . . . . . . . . 10  |-  F/_ x abs
25 nfmpt1 4542 . . . . . . . . . . . . 13  |-  F/_ x
( x  e.  A  |->  x )
264, 25nfcxfr 2627 . . . . . . . . . . . 12  |-  F/_ x F
27 nfcv 2629 . . . . . . . . . . . 12  |-  F/_ x
z
2826, 27nffv 5879 . . . . . . . . . . 11  |-  F/_ x
( F `  z
)
29 nfcv 2629 . . . . . . . . . . 11  |-  F/_ x  -
30 nfcv 2629 . . . . . . . . . . 11  |-  F/_ x X
3128, 29, 30nfov 6318 . . . . . . . . . 10  |-  F/_ x
( ( F `  z )  -  X
)
3224, 31nffv 5879 . . . . . . . . 9  |-  F/_ x
( abs `  (
( F `  z
)  -  X ) )
33 nfcv 2629 . . . . . . . . 9  |-  F/_ x  <
34 nfcv 2629 . . . . . . . . 9  |-  F/_ x w
3532, 33, 34nfbr 4497 . . . . . . . 8  |-  F/ x
( abs `  (
( F `  z
)  -  X ) )  <  w
3623, 35nfim 1867 . . . . . . 7  |-  F/ x
( ( z  =/= 
X  /\  ( abs `  ( z  -  X
) )  <  w
)  ->  ( abs `  ( ( F `  z )  -  X
) )  <  w
)
37 neeq1 2748 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  =/=  X  <->  z  =/=  X ) )
38 oveq1 6302 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
x  -  X )  =  ( z  -  X ) )
3938fveq2d 5876 . . . . . . . . . 10  |-  ( x  =  z  ->  ( abs `  ( x  -  X ) )  =  ( abs `  (
z  -  X ) ) )
4039breq1d 4463 . . . . . . . . 9  |-  ( x  =  z  ->  (
( abs `  (
x  -  X ) )  <  w  <->  ( abs `  ( z  -  X
) )  <  w
) )
4137, 40anbi12d 710 . . . . . . . 8  |-  ( x  =  z  ->  (
( x  =/=  X  /\  ( abs `  (
x  -  X ) )  <  w )  <-> 
( z  =/=  X  /\  ( abs `  (
z  -  X ) )  <  w ) ) )
42 fveq2 5872 . . . . . . . . . . 11  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
4342oveq1d 6310 . . . . . . . . . 10  |-  ( x  =  z  ->  (
( F `  x
)  -  X )  =  ( ( F `
 z )  -  X ) )
4443fveq2d 5876 . . . . . . . . 9  |-  ( x  =  z  ->  ( abs `  ( ( F `
 x )  -  X ) )  =  ( abs `  (
( F `  z
)  -  X ) ) )
4544breq1d 4463 . . . . . . . 8  |-  ( x  =  z  ->  (
( abs `  (
( F `  x
)  -  X ) )  <  w  <->  ( abs `  ( ( F `  z )  -  X
) )  <  w
) )
4641, 45imbi12d 320 . . . . . . 7  |-  ( 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 3089 . . . . . 6  |-  ( 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 196 . . . . 5  |-  ( (
ph  /\  w  e.  RR+ )  ->  A. z  e.  A  ( (
z  =/=  X  /\  ( abs `  ( z  -  X ) )  <  w )  -> 
( abs `  (
( F `  z
)  -  X ) )  <  w ) )
49 breq2 4457 . . . . . . . . . 10  |-  ( y  =  w  ->  (
( abs `  (
z  -  X ) )  <  y  <->  ( abs `  ( z  -  X
) )  <  w
) )
5049anbi2d 703 . . . . . . . . 9  |-  ( y  =  w  ->  (
( z  =/=  X  /\  ( abs `  (
z  -  X ) )  <  y )  <-> 
( z  =/=  X  /\  ( abs `  (
z  -  X ) )  <  w ) ) )
5150imbi1d 317 . . . . . . . 8  |-  ( y  =  w  ->  (
( ( z  =/= 
X  /\  ( abs `  ( z  -  X
) )  <  y
)  ->  ( abs `  ( ( F `  z )  -  X
) )  <  w
)  <->  ( ( z  =/=  X  /\  ( abs `  ( z  -  X ) )  < 
w )  ->  ( abs `  ( ( F `
 z )  -  X ) )  < 
w ) ) )
5251adantr 465 . . . . . . 7  |-  ( ( y  =  w  /\  z  e.  A )  ->  ( ( ( z  =/=  X  /\  ( abs `  ( z  -  X ) )  < 
y )  ->  ( abs `  ( ( F `
 z )  -  X ) )  < 
w )  <->  ( (
z  =/=  X  /\  ( abs `  ( z  -  X ) )  <  w )  -> 
( abs `  (
( F `  z
)  -  X ) )  <  w ) ) )
5352ralbidva 2903 . . . . . 6  |-  ( 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
) ) )
5453rspcev 3219 . . . . 5  |-  ( ( 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 ) )
552, 48, 54syl2anc 661 . . . 4  |-  ( (
ph  /\  w  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  A  ( ( z  =/=  X  /\  ( abs `  ( z  -  X ) )  < 
y )  ->  ( abs `  ( ( F `
 z )  -  X ) )  < 
w ) )
5655ralrimiva 2881 . . 3  |-  ( ph  ->  A. w  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( (
z  =/=  X  /\  ( abs `  ( z  -  X ) )  <  y )  -> 
( abs `  (
( F `  z
)  -  X ) )  <  w ) )
571, 56jca 532 . 2  |-  ( ph  ->  ( 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 ) ) )
58 idlimc.a . . . . 5  |-  ( ph  ->  A  C_  CC )
5958sselda 3509 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  CC )
6059, 4fmptd 6056 . . 3  |-  ( ph  ->  F : A --> CC )
6160, 58, 1ellimc3 22151 . 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 ) ) ) )
6257, 61mpbird 232 1  |-  ( ph  ->  X  e.  ( F lim
CC  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818    C_ wss 3481   class class class wbr 4453    |-> cmpt 4511   ` cfv 5594  (class class class)co 6295   CCcc 9502    < clt 9640    - cmin 9817   RR+crp 11232   abscabs 13047   lim CC climc 22134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fi 7883  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-fz 11685  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-plusg 14585  df-mulr 14586  df-starv 14587  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-rest 14695  df-topn 14696  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cnp 19597  df-xms 20691  df-ms 20692  df-limc 22138
This theorem is referenced by:  fourierdlem53  31783  fourierdlem60  31790  fourierdlem61  31791  fourierdlem73  31803  fourierdlem74  31804  fourierdlem75  31805  fourierdlem76  31806
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