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

Theorem cncfiooiccre 31652
Description: A continuous function  F on an open interval  ( A (,) B ) can be extended to a continuous function  G on the corresponding close interval, if it has a finite right limit  R in  A and a finite left limit  L in  B.  F is assumed to be real valued. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
cncfiooiccre.x  |-  F/ x ph
cncfiooiccre.g  |-  G  =  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )
cncfiooiccre.a  |-  ( ph  ->  A  e.  RR )
cncfiooiccre.b  |-  ( ph  ->  B  e.  RR )
cncfiooiccre.altb  |-  ( ph  ->  A  <  B )
cncfiooiccre.f  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> RR ) )
cncfiooiccre.l  |-  ( ph  ->  L  e.  ( F lim
CC  B ) )
cncfiooiccre.r  |-  ( ph  ->  R  e.  ( F lim
CC  A ) )
Assertion
Ref Expression
cncfiooiccre  |-  ( ph  ->  G  e.  ( ( A [,] B )
-cn-> RR ) )
Distinct variable groups:    x, A    x, B    x, F    x, L    x, R    ph, x
Allowed substitution hint:    G( x)

Proof of Theorem cncfiooiccre
StepHypRef Expression
1 iftrue 3932 . . . . . . 7  |-  ( x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  R )
21adantl 466 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  R )
3 cncfiooiccre.f . . . . . . . . 9  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> RR ) )
4 cncff 21375 . . . . . . . . 9  |-  ( F  e.  ( ( A (,) B ) -cn-> RR )  ->  F :
( A (,) B
) --> RR )
53, 4syl 16 . . . . . . . 8  |-  ( ph  ->  F : ( A (,) B ) --> RR )
6 ioosscn 31481 . . . . . . . . 9  |-  ( A (,) B )  C_  CC
76a1i 11 . . . . . . . 8  |-  ( ph  ->  ( A (,) B
)  C_  CC )
8 eqid 2443 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
9 cncfiooiccre.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR )
109rexrd 9646 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR* )
11 cncfiooiccre.a . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
12 cncfiooiccre.altb . . . . . . . . 9  |-  ( ph  ->  A  <  B )
138, 10, 11, 12lptioo1cn 31606 . . . . . . . 8  |-  ( ph  ->  A  e.  ( (
limPt `  ( TopOpen ` fld ) ) `  ( A (,) B ) ) )
14 cncfiooiccre.r . . . . . . . 8  |-  ( ph  ->  R  e.  ( F lim
CC  A ) )
155, 7, 13, 14limcrecl 31589 . . . . . . 7  |-  ( ph  ->  R  e.  RR )
1615adantr 465 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  R  e.  RR )
172, 16eqeltrd 2531 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  RR )
1817adantlr 714 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  RR )
19 iffalse 3935 . . . . . . . . 9  |-  ( -.  x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
20 iftrue 3932 . . . . . . . . 9  |-  ( x  =  B  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  L )
2119, 20sylan9eq 2504 . . . . . . . 8  |-  ( ( -.  x  =  A  /\  x  =  B )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  L )
2221adantll 713 . . . . . . 7  |-  ( ( ( ph  /\  -.  x  =  A )  /\  x  =  B
)  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  L )
2311rexrd 9646 . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR* )
248, 23, 9, 12lptioo2cn 31605 . . . . . . . . 9  |-  ( ph  ->  B  e.  ( (
limPt `  ( TopOpen ` fld ) ) `  ( A (,) B ) ) )
25 cncfiooiccre.l . . . . . . . . 9  |-  ( ph  ->  L  e.  ( F lim
CC  B ) )
265, 7, 24, 25limcrecl 31589 . . . . . . . 8  |-  ( ph  ->  L  e.  RR )
2726ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  -.  x  =  A )  /\  x  =  B
)  ->  L  e.  RR )
2822, 27eqeltrd 2531 . . . . . 6  |-  ( ( ( ph  /\  -.  x  =  A )  /\  x  =  B
)  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  e.  RR )
2928adantllr 718 . . . . 5  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  RR )
30 iffalse 3935 . . . . . . . 8  |-  ( -.  x  =  B  ->  if ( x  =  B ,  L ,  ( F `  x ) )  =  ( F `
 x ) )
3119, 30sylan9eq 2504 . . . . . . 7  |-  ( ( -.  x  =  A  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  ( F `  x ) )
3231adantll 713 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  ( F `  x ) )
335ad3antrrr 729 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  F : ( A (,) B ) --> RR )
3423ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  A  e.  RR* )
3510ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  B  e.  RR* )
3611adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR )
379adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
38 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  ( A [,] B ) )
39 eliccre 31494 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  x  e.  ( A [,] B
) )  ->  x  e.  RR )
4036, 37, 38, 39syl3anc 1229 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  RR )
4140ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  e.  RR )
4211ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  e.  RR )
4340adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  e.  RR )
4423ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  e.  RR* )
4510ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  B  e.  RR* )
4638adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  e.  ( A [,] B ) )
47 iccgelb 11592 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e.  ( A [,] B
) )  ->  A  <_  x )
4844, 45, 46, 47syl3anc 1229 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  <_  x )
49 neqne 31388 . . . . . . . . . . 11  |-  ( -.  x  =  A  ->  x  =/=  A )
5049adantl 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  =/=  A )
5142, 43, 48, 50leneltd 31448 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  <  x )
5251adantr 465 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  A  <  x )
5340adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  e.  RR )
549ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  B  e.  RR )
5523ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  A  e.  RR* )
5610ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  B  e.  RR* )
5738adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  e.  ( A [,] B ) )
58 iccleub 11591 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e.  ( A [,] B
) )  ->  x  <_  B )
5955, 56, 57, 58syl3anc 1229 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  <_  B )
60 neqne 31388 . . . . . . . . . . . 12  |-  ( -.  x  =  B  ->  x  =/=  B )
6160necomd 2714 . . . . . . . . . . 11  |-  ( -.  x  =  B  ->  B  =/=  x )
6261adantl 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  B  =/=  x )
6353, 54, 59, 62leneltd 31448 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  <  B )
6463adantlr 714 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  <  B )
6534, 35, 41, 52, 64eliood 31485 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  e.  ( A (,) B
) )
6633, 65ffvelrnd 6017 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ( F `  x )  e.  RR )
6732, 66eqeltrd 2531 . . . . 5  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  RR )
6829, 67pm2.61dan 791 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  RR )
6918, 68pm2.61dan 791 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  e.  RR )
70 cncfiooiccre.g . . 3  |-  G  =  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )
7169, 70fmptd 6040 . 2  |-  ( ph  ->  G : ( A [,] B ) --> RR )
72 ax-resscn 9552 . . 3  |-  RR  C_  CC
73 cncfiooiccre.x . . . 4  |-  F/ x ph
74 ssid 3508 . . . . . 6  |-  CC  C_  CC
75 cncfss 21381 . . . . . 6  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( A (,) B
) -cn-> RR )  C_  (
( A (,) B
) -cn-> CC ) )
7672, 74, 75mp2an 672 . . . . 5  |-  ( ( A (,) B )
-cn-> RR )  C_  (
( A (,) B
) -cn-> CC )
7776, 3sseldi 3487 . . . 4  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> CC ) )
7873, 70, 11, 9, 77, 25, 14cncfiooicc 31651 . . 3  |-  ( ph  ->  G  e.  ( ( A [,] B )
-cn-> CC ) )
79 cncffvrn 21380 . . 3  |-  ( ( RR  C_  CC  /\  G  e.  ( ( A [,] B ) -cn-> CC ) )  ->  ( G  e.  ( ( A [,] B ) -cn-> RR )  <-> 
G : ( A [,] B ) --> RR ) )
8072, 78, 79sylancr 663 . 2  |-  ( ph  ->  ( G  e.  ( ( A [,] B
) -cn-> RR )  <->  G :
( A [,] B
) --> RR ) )
8171, 80mpbird 232 1  |-  ( ph  ->  G  e.  ( ( A [,] B )
-cn-> RR ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383   F/wnf 1603    e. wcel 1804    =/= wne 2638    C_ wss 3461   ifcif 3926   class class class wbr 4437    |-> cmpt 4495   -->wf 5574   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   RR*cxr 9630    < clt 9631    <_ cle 9632   (,)cioo 11540   [,]cicc 11543   TopOpenctopn 14801  ℂfldccnfld 18399   -cn->ccncf 21358   lim CC climc 22244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fi 7873  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-q 11194  df-rp 11232  df-xneg 11329  df-xadd 11330  df-xmul 11331  df-ioo 11544  df-ioc 11545  df-ico 11546  df-icc 11547  df-fz 11684  df-seq 12090  df-exp 12149  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-plusg 14692  df-mulr 14693  df-starv 14694  df-tset 14698  df-ple 14699  df-ds 14701  df-unif 14702  df-rest 14802  df-topn 14803  df-topgen 14823  df-psmet 18390  df-xmet 18391  df-met 18392  df-bl 18393  df-mopn 18394  df-cnfld 18400  df-top 19377  df-bases 19379  df-topon 19380  df-topsp 19381  df-cld 19498  df-ntr 19499  df-cls 19500  df-nei 19577  df-lp 19615  df-cn 19706  df-cnp 19707  df-xms 20801  df-ms 20802  df-cncf 21360  df-limc 22248
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator