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Theorem idlimc 37706
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 463 . . . 4  |-  ( (
ph  /\  w  e.  RR+ )  ->  w  e.  RR+ )
3 simpr 463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
4 idlimc.f . . . . . . . . . . . . . . 15  |-  F  =  ( x  e.  A  |->  x )
54fvmpt2 5957 . . . . . . . . . . . . . 14  |-  ( ( x  e.  A  /\  x  e.  A )  ->  ( F `  x
)  =  x )
63, 3, 5syl2anc 667 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  x )
76oveq1d 6305 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
)  -  X )  =  ( x  -  X ) )
87fveq2d 5869 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  ( abs `  ( ( F `
 x )  -  X ) )  =  ( abs `  (
x  -  X ) ) )
98adantr 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  A )  /\  ( abs `  ( x  -  X ) )  < 
w )  ->  ( abs `  ( ( F `
 x )  -  X ) )  =  ( abs `  (
x  -  X ) ) )
10 simpr 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  A )  /\  ( abs `  ( x  -  X ) )  < 
w )  ->  ( abs `  ( x  -  X ) )  < 
w )
119, 10eqbrtrd 4423 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  /\  ( abs `  ( x  -  X ) )  < 
w )  ->  ( abs `  ( ( F `
 x )  -  X ) )  < 
w )
1211adantrl 722 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  A )  /\  (
x  =/=  X  /\  ( abs `  ( x  -  X ) )  <  w ) )  ->  ( abs `  (
( F `  x
)  -  X ) )  <  w )
1312ex 436 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  =/=  X  /\  ( abs `  (
x  -  X ) )  <  w )  ->  ( abs `  (
( F `  x
)  -  X ) )  <  w ) )
1413adantlr 721 . . . . . 6  |-  ( ( ( ph  /\  w  e.  RR+ )  /\  x  e.  A )  ->  (
( x  =/=  X  /\  ( abs `  (
x  -  X ) )  <  w )  ->  ( abs `  (
( F `  x
)  -  X ) )  <  w ) )
1514ralrimiva 2802 . . . . 5  |-  ( (
ph  /\  w  e.  RR+ )  ->  A. x  e.  A  ( (
x  =/=  X  /\  ( abs `  ( x  -  X ) )  <  w )  -> 
( abs `  (
( F `  x
)  -  X ) )  <  w ) )
16 nfcv 2592 . . . . . . . . 9  |-  F/_ z
x
17 nfcv 2592 . . . . . . . . 9  |-  F/_ z X
1816, 17nfne 2723 . . . . . . . 8  |-  F/ z  x  =/=  X
19 nfv 1761 . . . . . . . 8  |-  F/ z ( abs `  (
x  -  X ) )  <  w
2018, 19nfan 2011 . . . . . . 7  |-  F/ z ( x  =/=  X  /\  ( abs `  (
x  -  X ) )  <  w )
21 nfv 1761 . . . . . . 7  |-  F/ z ( abs `  (
( F `  x
)  -  X ) )  <  w
2220, 21nfim 2003 . . . . . 6  |-  F/ z ( ( x  =/= 
X  /\  ( abs `  ( x  -  X
) )  <  w
)  ->  ( abs `  ( ( F `  x )  -  X
) )  <  w
)
23 nfv 1761 . . . . . . 7  |-  F/ x
( z  =/=  X  /\  ( abs `  (
z  -  X ) )  <  w )
24 nfcv 2592 . . . . . . . . 9  |-  F/_ x abs
25 nfmpt1 4492 . . . . . . . . . . . 12  |-  F/_ x
( x  e.  A  |->  x )
264, 25nfcxfr 2590 . . . . . . . . . . 11  |-  F/_ x F
27 nfcv 2592 . . . . . . . . . . 11  |-  F/_ x
z
2826, 27nffv 5872 . . . . . . . . . 10  |-  F/_ x
( F `  z
)
29 nfcv 2592 . . . . . . . . . 10  |-  F/_ x  -
30 nfcv 2592 . . . . . . . . . 10  |-  F/_ x X
3128, 29, 30nfov 6316 . . . . . . . . 9  |-  F/_ x
( ( F `  z )  -  X
)
3224, 31nffv 5872 . . . . . . . 8  |-  F/_ x
( abs `  (
( F `  z
)  -  X ) )
33 nfcv 2592 . . . . . . . 8  |-  F/_ x  <
34 nfcv 2592 . . . . . . . 8  |-  F/_ x w
3532, 33, 34nfbr 4447 . . . . . . 7  |-  F/ x
( abs `  (
( F `  z
)  -  X ) )  <  w
3623, 35nfim 2003 . . . . . 6  |-  F/ x
( ( z  =/= 
X  /\  ( abs `  ( z  -  X
) )  <  w
)  ->  ( abs `  ( ( F `  z )  -  X
) )  <  w
)
37 neeq1 2686 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =/=  X  <->  z  =/=  X ) )
38 oveq1 6297 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x  -  X )  =  ( z  -  X ) )
3938fveq2d 5869 . . . . . . . . 9  |-  ( x  =  z  ->  ( abs `  ( x  -  X ) )  =  ( abs `  (
z  -  X ) ) )
4039breq1d 4412 . . . . . . . 8  |-  ( x  =  z  ->  (
( abs `  (
x  -  X ) )  <  w  <->  ( abs `  ( z  -  X
) )  <  w
) )
4137, 40anbi12d 717 . . . . . . 7  |-  ( x  =  z  ->  (
( x  =/=  X  /\  ( abs `  (
x  -  X ) )  <  w )  <-> 
( z  =/=  X  /\  ( abs `  (
z  -  X ) )  <  w ) ) )
42 fveq2 5865 . . . . . . . . . 10  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
4342oveq1d 6305 . . . . . . . . 9  |-  ( x  =  z  ->  (
( F `  x
)  -  X )  =  ( ( F `
 z )  -  X ) )
4443fveq2d 5869 . . . . . . . 8  |-  ( x  =  z  ->  ( abs `  ( ( F `
 x )  -  X ) )  =  ( abs `  (
( F `  z
)  -  X ) ) )
4544breq1d 4412 . . . . . . 7  |-  ( x  =  z  ->  (
( abs `  (
( F `  x
)  -  X ) )  <  w  <->  ( abs `  ( ( F `  z )  -  X
) )  <  w
) )
4641, 45imbi12d 322 . . . . . 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 3015 . . . . 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 200 . . . 4  |-  ( (
ph  /\  w  e.  RR+ )  ->  A. z  e.  A  ( (
z  =/=  X  /\  ( abs `  ( z  -  X ) )  <  w )  -> 
( abs `  (
( F `  z
)  -  X ) )  <  w ) )
49 breq2 4406 . . . . . . . 8  |-  ( y  =  w  ->  (
( abs `  (
z  -  X ) )  <  y  <->  ( abs `  ( z  -  X
) )  <  w
) )
5049anbi2d 710 . . . . . . 7  |-  ( y  =  w  ->  (
( z  =/=  X  /\  ( abs `  (
z  -  X ) )  <  y )  <-> 
( z  =/=  X  /\  ( abs `  (
z  -  X ) )  <  w ) ) )
5150imbi1d 319 . . . . . 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 2827 . . . . 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 3150 . . . 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 667 . . 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 2802 . 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 3432 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  CC )
5857, 4fmptd 6046 . . 3  |-  ( ph  ->  F : A --> CC )
5958, 56, 1ellimc3 22834 . 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 933 1  |-  ( ph  ->  X  e.  ( F lim
CC  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738    C_ wss 3404   class class class wbr 4402    |-> cmpt 4461   ` cfv 5582  (class class class)co 6290   CCcc 9537    < clt 9675    - cmin 9860   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:  fourierdlem53  38023  fourierdlem60  38030  fourierdlem61  38031  fourierdlem73  38043  fourierdlem74  38044  fourierdlem75  38045  fourierdlem76  38046
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