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Theorem cncfiooiccre 31864
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 3863 . . . . . . 7  |-  ( x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  R )
21adantl 464 . . . . . 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 21482 . . . . . . . . 9  |-  ( F  e.  ( ( A (,) B ) -cn-> RR )  ->  F :
( A (,) B
) --> RR )
53, 4syl 16 . . . . . . . 8  |-  ( ph  ->  F : ( A (,) B ) --> RR )
6 ioosscn 31693 . . . . . . . . 9  |-  ( A (,) B )  C_  CC
76a1i 11 . . . . . . . 8  |-  ( ph  ->  ( A (,) B
)  C_  CC )
8 eqid 2382 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
9 cncfiooiccre.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR )
109rexrd 9554 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR* )
11 cncfiooiccre.a . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
12 cncfiooiccre.altb . . . . . . . . 9  |-  ( ph  ->  A  <  B )
138, 10, 11, 12lptioo1cn 31818 . . . . . . . 8  |-  ( ph  ->  A  e.  ( (
limPt `  ( TopOpen ` fld ) ) `  ( A (,) B ) ) )
14 cncfiooiccre.r . . . . . . . 8  |-  ( ph  ->  R  e.  ( F lim
CC  A ) )
155, 7, 13, 14limcrecl 31801 . . . . . . 7  |-  ( ph  ->  R  e.  RR )
1615adantr 463 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  R  e.  RR )
172, 16eqeltrd 2470 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  RR )
1817adantlr 712 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  RR )
19 iffalse 3866 . . . . . . . . 9  |-  ( -.  x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
20 iftrue 3863 . . . . . . . . 9  |-  ( x  =  B  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  L )
2119, 20sylan9eq 2443 . . . . . . . 8  |-  ( ( -.  x  =  A  /\  x  =  B )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  L )
2221adantll 711 . . . . . . 7  |-  ( ( ( ph  /\  -.  x  =  A )  /\  x  =  B
)  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  L )
2311rexrd 9554 . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR* )
248, 23, 9, 12lptioo2cn 31817 . . . . . . . . 9  |-  ( ph  ->  B  e.  ( (
limPt `  ( TopOpen ` fld ) ) `  ( A (,) B ) ) )
25 cncfiooiccre.l . . . . . . . . 9  |-  ( ph  ->  L  e.  ( F lim
CC  B ) )
265, 7, 24, 25limcrecl 31801 . . . . . . . 8  |-  ( ph  ->  L  e.  RR )
2726ad2antrr 723 . . . . . . 7  |-  ( ( ( ph  /\  -.  x  =  A )  /\  x  =  B
)  ->  L  e.  RR )
2822, 27eqeltrd 2470 . . . . . 6  |-  ( ( ( ph  /\  -.  x  =  A )  /\  x  =  B
)  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  e.  RR )
2928adantllr 716 . . . . 5  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  RR )
30 iffalse 3866 . . . . . . . 8  |-  ( -.  x  =  B  ->  if ( x  =  B ,  L ,  ( F `  x ) )  =  ( F `
 x ) )
3119, 30sylan9eq 2443 . . . . . . 7  |-  ( ( -.  x  =  A  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  ( F `  x ) )
3231adantll 711 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  ( F `  x ) )
335ad3antrrr 727 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  F : ( A (,) B ) --> RR )
3423ad3antrrr 727 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  A  e.  RR* )
3510ad3antrrr 727 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  B  e.  RR* )
3611adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR )
379adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
38 simpr 459 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  ( A [,] B ) )
39 eliccre 31706 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  x  e.  ( A [,] B
) )  ->  x  e.  RR )
4036, 37, 38, 39syl3anc 1226 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  RR )
4140ad2antrr 723 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  e.  RR )
4211ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  e.  RR )
4340adantr 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  e.  RR )
4423ad2antrr 723 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  e.  RR* )
4510ad2antrr 723 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  B  e.  RR* )
4638adantr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  e.  ( A [,] B ) )
47 iccgelb 11502 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e.  ( A [,] B
) )  ->  A  <_  x )
4844, 45, 46, 47syl3anc 1226 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  <_  x )
49 neqne 31601 . . . . . . . . . . 11  |-  ( -.  x  =  A  ->  x  =/=  A )
5049adantl 464 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  =/=  A )
5142, 43, 48, 50leneltd 31660 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  <  x )
5251adantr 463 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  A  <  x )
5340adantr 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  e.  RR )
549ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  B  e.  RR )
5523ad2antrr 723 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  A  e.  RR* )
5610ad2antrr 723 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  B  e.  RR* )
5738adantr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  e.  ( A [,] B ) )
58 iccleub 11501 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e.  ( A [,] B
) )  ->  x  <_  B )
5955, 56, 57, 58syl3anc 1226 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  <_  B )
60 neqne 31601 . . . . . . . . . . . 12  |-  ( -.  x  =  B  ->  x  =/=  B )
6160necomd 2653 . . . . . . . . . . 11  |-  ( -.  x  =  B  ->  B  =/=  x )
6261adantl 464 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  B  =/=  x )
6353, 54, 59, 62leneltd 31660 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  <  B )
6463adantlr 712 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  <  B )
6534, 35, 41, 52, 64eliood 31697 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  e.  ( A (,) B
) )
6633, 65ffvelrnd 5934 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  ( F `  x )  e.  RR )
6732, 66eqeltrd 2470 . . . . 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 789 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  e.  RR )
6918, 68pm2.61dan 789 . . 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 5957 . 2  |-  ( ph  ->  G : ( A [,] B ) --> RR )
72 ax-resscn 9460 . . 3  |-  RR  C_  CC
73 cncfiooiccre.x . . . 4  |-  F/ x ph
74 ssid 3436 . . . . . 6  |-  CC  C_  CC
75 cncfss 21488 . . . . . 6  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( A (,) B
) -cn-> RR )  C_  (
( A (,) B
) -cn-> CC ) )
7672, 74, 75mp2an 670 . . . . 5  |-  ( ( A (,) B )
-cn-> RR )  C_  (
( A (,) B
) -cn-> CC )
7776, 3sseldi 3415 . . . 4  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> CC ) )
7873, 70, 11, 9, 77, 25, 14cncfiooicc 31863 . . 3  |-  ( ph  ->  G  e.  ( ( A [,] B )
-cn-> CC ) )
79 cncffvrn 21487 . . 3  |-  ( ( RR  C_  CC  /\  G  e.  ( ( A [,] B ) -cn-> CC ) )  ->  ( G  e.  ( ( A [,] B ) -cn-> RR )  <-> 
G : ( A [,] B ) --> RR ) )
8072, 78, 79sylancr 661 . 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 367    = wceq 1399   F/wnf 1624    e. wcel 1826    =/= wne 2577    C_ wss 3389   ifcif 3857   class class class wbr 4367    |-> cmpt 4425   -->wf 5492   ` cfv 5496  (class class class)co 6196   CCcc 9401   RRcr 9402   RR*cxr 9538    < clt 9539    <_ cle 9540   (,)cioo 11450   [,]cicc 11453   TopOpenctopn 14829  ℂfldccnfld 18533   -cn->ccncf 21465   lim CC climc 22351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fi 7786  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ioo 11454  df-ioc 11455  df-ico 11456  df-icc 11457  df-fz 11594  df-seq 12011  df-exp 12070  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-plusg 14715  df-mulr 14716  df-starv 14717  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-rest 14830  df-topn 14831  df-topgen 14851  df-psmet 18524  df-xmet 18525  df-met 18526  df-bl 18527  df-mopn 18528  df-cnfld 18534  df-top 19484  df-bases 19486  df-topon 19487  df-topsp 19488  df-cld 19605  df-ntr 19606  df-cls 19607  df-nei 19685  df-lp 19723  df-cn 19814  df-cnp 19815  df-xms 20908  df-ms 20909  df-cncf 21467  df-limc 22355
This theorem is referenced by: (None)
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