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Theorem cncfiooicc 31943
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 can be complex valued. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
cncfiooicc.x  |-  F/ x ph
cncfiooicc.g  |-  G  =  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )
cncfiooicc.a  |-  ( ph  ->  A  e.  RR )
cncfiooicc.b  |-  ( ph  ->  B  e.  RR )
cncfiooicc.f  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> CC ) )
cncfiooicc.l  |-  ( ph  ->  L  e.  ( F lim
CC  B ) )
cncfiooicc.r  |-  ( ph  ->  R  e.  ( F lim
CC  A ) )
Assertion
Ref Expression
cncfiooicc  |-  ( ph  ->  G  e.  ( ( A [,] B )
-cn-> CC ) )
Distinct variable groups:    x, A    x, B    x, F    x, L    x, R    ph, x
Allowed substitution hint:    G( x)

Proof of Theorem cncfiooicc
StepHypRef Expression
1 nfv 1708 . . 3  |-  F/ x
( ph  /\  A  < 
B )
2 cncfiooicc.g . . 3  |-  G  =  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )
3 cncfiooicc.a . . . 4  |-  ( ph  ->  A  e.  RR )
43adantr 465 . . 3  |-  ( (
ph  /\  A  <  B )  ->  A  e.  RR )
5 cncfiooicc.b . . . 4  |-  ( ph  ->  B  e.  RR )
65adantr 465 . . 3  |-  ( (
ph  /\  A  <  B )  ->  B  e.  RR )
7 simpr 461 . . 3  |-  ( (
ph  /\  A  <  B )  ->  A  <  B )
8 cncfiooicc.f . . . 4  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> CC ) )
98adantr 465 . . 3  |-  ( (
ph  /\  A  <  B )  ->  F  e.  ( ( A (,) B ) -cn-> CC ) )
10 cncfiooicc.l . . . 4  |-  ( ph  ->  L  e.  ( F lim
CC  B ) )
1110adantr 465 . . 3  |-  ( (
ph  /\  A  <  B )  ->  L  e.  ( F lim CC  B ) )
12 cncfiooicc.r . . . 4  |-  ( ph  ->  R  e.  ( F lim
CC  A ) )
1312adantr 465 . . 3  |-  ( (
ph  /\  A  <  B )  ->  R  e.  ( F lim CC  A ) )
141, 2, 4, 6, 7, 9, 11, 13cncfiooicclem1 31942 . 2  |-  ( (
ph  /\  A  <  B )  ->  G  e.  ( ( A [,] B ) -cn-> CC ) )
15 limccl 22496 . . . . . . . . . 10  |-  ( F lim
CC  A )  C_  CC
1615, 12sseldi 3497 . . . . . . . . 9  |-  ( ph  ->  R  e.  CC )
1716snssd 4177 . . . . . . . 8  |-  ( ph  ->  { R }  C_  CC )
18 ssid 3518 . . . . . . . . 9  |-  CC  C_  CC
1918a1i 11 . . . . . . . 8  |-  ( ph  ->  CC  C_  CC )
20 cncfss 21620 . . . . . . . 8  |-  ( ( { R }  C_  CC  /\  CC  C_  CC )  ->  ( { A } -cn-> { R } ) 
C_  ( { A } -cn-> CC ) )
2117, 19, 20syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( { A } -cn->
{ R } ) 
C_  ( { A } -cn-> CC ) )
2221adantr 465 . . . . . 6  |-  ( (
ph  /\  A  =  B )  ->  ( { A } -cn-> { R } )  C_  ( { A } -cn-> CC ) )
233rexrd 9660 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  RR* )
24 iccid 11599 . . . . . . . . . . . 12  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
2523, 24syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( A [,] A
)  =  { A } )
26 oveq2 6304 . . . . . . . . . . 11  |-  ( A  =  B  ->  ( A [,] A )  =  ( A [,] B
) )
2725, 26sylan9req 2519 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  B )  ->  { A }  =  ( A [,] B ) )
2827eqcomd 2465 . . . . . . . . 9  |-  ( (
ph  /\  A  =  B )  ->  ( A [,] B )  =  { A } )
29 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =  B )  /\  x  e.  ( A [,] B
) )  ->  x  e.  ( A [,] B
) )
3028adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =  B )  /\  x  e.  ( A [,] B
) )  ->  ( A [,] B )  =  { A } )
3129, 30eleqtrd 2547 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =  B )  /\  x  e.  ( A [,] B
) )  ->  x  e.  { A } )
32 elsni 4057 . . . . . . . . . . 11  |-  ( x  e.  { A }  ->  x  =  A )
3331, 32syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =  B )  /\  x  e.  ( A [,] B
) )  ->  x  =  A )
3433iftrued 3952 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =  B )  /\  x  e.  ( A [,] B
) )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  R )
3528, 34mpteq12dva 4534 . . . . . . . 8  |-  ( (
ph  /\  A  =  B )  ->  (
x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) ) )  =  ( x  e.  { A }  |->  R ) )
362, 35syl5eq 2510 . . . . . . 7  |-  ( (
ph  /\  A  =  B )  ->  G  =  ( x  e. 
{ A }  |->  R ) )
373recnd 9639 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
3837adantr 465 . . . . . . . 8  |-  ( (
ph  /\  A  =  B )  ->  A  e.  CC )
3916adantr 465 . . . . . . . 8  |-  ( (
ph  /\  A  =  B )  ->  R  e.  CC )
40 cncfdmsn 31939 . . . . . . . 8  |-  ( ( A  e.  CC  /\  R  e.  CC )  ->  ( x  e.  { A }  |->  R )  e.  ( { A } -cn-> { R } ) )
4138, 39, 40syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  A  =  B )  ->  (
x  e.  { A }  |->  R )  e.  ( { A } -cn->
{ R } ) )
4236, 41eqeltrd 2545 . . . . . 6  |-  ( (
ph  /\  A  =  B )  ->  G  e.  ( { A } -cn->
{ R } ) )
4322, 42sseldd 3500 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  G  e.  ( { A } -cn->
CC ) )
4427oveq1d 6311 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  ( { A } -cn-> CC )  =  ( ( A [,] B ) -cn-> CC ) )
4543, 44eleqtrd 2547 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  G  e.  ( ( A [,] B ) -cn-> CC ) )
4645adantlr 714 . . 3  |-  ( ( ( ph  /\  -.  A  <  B )  /\  A  =  B )  ->  G  e.  ( ( A [,] B )
-cn-> CC ) )
47 simpll 753 . . . 4  |-  ( ( ( ph  /\  -.  A  <  B )  /\  -.  A  =  B
)  ->  ph )
48 eqcom 2466 . . . . . . . . 9  |-  ( B  =  A  <->  A  =  B )
4948biimpi 194 . . . . . . . 8  |-  ( B  =  A  ->  A  =  B )
5049con3i 135 . . . . . . 7  |-  ( -.  A  =  B  ->  -.  B  =  A
)
5150adantl 466 . . . . . 6  |-  ( ( ( ph  /\  -.  A  <  B )  /\  -.  A  =  B
)  ->  -.  B  =  A )
52 simplr 755 . . . . . 6  |-  ( ( ( ph  /\  -.  A  <  B )  /\  -.  A  =  B
)  ->  -.  A  <  B )
53 pm4.56 495 . . . . . . 7  |-  ( ( -.  B  =  A  /\  -.  A  < 
B )  <->  -.  ( B  =  A  \/  A  <  B ) )
5453biimpi 194 . . . . . 6  |-  ( ( -.  B  =  A  /\  -.  A  < 
B )  ->  -.  ( B  =  A  \/  A  <  B ) )
5551, 52, 54syl2anc 661 . . . . 5  |-  ( ( ( ph  /\  -.  A  <  B )  /\  -.  A  =  B
)  ->  -.  ( B  =  A  \/  A  <  B ) )
5647, 5syl 16 . . . . . 6  |-  ( ( ( ph  /\  -.  A  <  B )  /\  -.  A  =  B
)  ->  B  e.  RR )
5747, 3syl 16 . . . . . 6  |-  ( ( ( ph  /\  -.  A  <  B )  /\  -.  A  =  B
)  ->  A  e.  RR )
5856, 57lttrid 9740 . . . . 5  |-  ( ( ( ph  /\  -.  A  <  B )  /\  -.  A  =  B
)  ->  ( B  <  A  <->  -.  ( B  =  A  \/  A  <  B ) ) )
5955, 58mpbird 232 . . . 4  |-  ( ( ( ph  /\  -.  A  <  B )  /\  -.  A  =  B
)  ->  B  <  A )
60 0ss 3823 . . . . . . . 8  |-  (/)  C_  CC
61 eqid 2457 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
6261cnfldtop 21508 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  e.  Top
63 rest0 19888 . . . . . . . . . . 11  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  (/) )  =  { (/)
} )
6462, 63ax-mp 5 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  (/) )  =  { (/)
}
6564eqcomi 2470 . . . . . . . . 9  |-  { (/) }  =  ( ( TopOpen ` fld )t  (/) )
6661, 65, 65cncfcn 21630 . . . . . . . 8  |-  ( (
(/)  C_  CC  /\  (/)  C_  CC )  ->  ( (/) -cn-> (/) )  =  ( { (/) }  Cn  {
(/) } ) )
6760, 60, 66mp2an 672 . . . . . . 7  |-  ( (/) -cn-> (/) )  =  ( {
(/) }  Cn  { (/) } )
68 cncfss 21620 . . . . . . . 8  |-  ( (
(/)  C_  CC  /\  CC  C_  CC )  ->  ( (/)
-cn->
(/) )  C_  ( (/)
-cn-> CC ) )
6960, 18, 68mp2an 672 . . . . . . 7  |-  ( (/) -cn-> (/) )  C_  ( (/) -cn-> CC )
7067, 69eqsstr3i 3530 . . . . . 6  |-  ( {
(/) }  Cn  { (/) } )  C_  ( (/) -cn-> CC )
712a1i 11 . . . . . . . 8  |-  ( (
ph  /\  B  <  A )  ->  G  =  ( x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) ) ) )
72 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  B  <  A )  ->  B  <  A )
7323adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  B  <  A )  ->  A  e.  RR* )
745rexrd 9660 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  RR* )
7574adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  B  <  A )  ->  B  e.  RR* )
76 icc0 11602 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,] B
)  =  (/)  <->  B  <  A ) )
7773, 75, 76syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  B  <  A )  ->  ( ( A [,] B )  =  (/) 
<->  B  <  A ) )
7872, 77mpbird 232 . . . . . . . . 9  |-  ( (
ph  /\  B  <  A )  ->  ( A [,] B )  =  (/) )
7978mpteq1d 4538 . . . . . . . 8  |-  ( (
ph  /\  B  <  A )  ->  ( x  e.  ( A [,] B
)  |->  if ( x  =  A ,  R ,  if ( x  =  B ,  L , 
( F `  x
) ) ) )  =  ( x  e.  (/)  |->  if ( x  =  A ,  R ,  if ( x  =  B ,  L , 
( F `  x
) ) ) ) )
80 mpt0 5714 . . . . . . . . 9  |-  ( x  e.  (/)  |->  if ( x  =  A ,  R ,  if ( x  =  B ,  L , 
( F `  x
) ) ) )  =  (/)
8180a1i 11 . . . . . . . 8  |-  ( (
ph  /\  B  <  A )  ->  ( x  e.  (/)  |->  if ( x  =  A ,  R ,  if ( x  =  B ,  L , 
( F `  x
) ) ) )  =  (/) )
8271, 79, 813eqtrd 2502 . . . . . . 7  |-  ( (
ph  /\  B  <  A )  ->  G  =  (/) )
83 0cnf 31925 . . . . . . 7  |-  (/)  e.  ( { (/) }  Cn  { (/)
} )
8482, 83syl6eqel 2553 . . . . . 6  |-  ( (
ph  /\  B  <  A )  ->  G  e.  ( { (/) }  Cn  { (/)
} ) )
8570, 84sseldi 3497 . . . . 5  |-  ( (
ph  /\  B  <  A )  ->  G  e.  ( (/) -cn-> CC ) )
8678eqcomd 2465 . . . . . 6  |-  ( (
ph  /\  B  <  A )  ->  (/)  =  ( A [,] B ) )
8786oveq1d 6311 . . . . 5  |-  ( (
ph  /\  B  <  A )  ->  ( (/) -cn-> CC )  =  ( ( A [,] B ) -cn-> CC ) )
8885, 87eleqtrd 2547 . . . 4  |-  ( (
ph  /\  B  <  A )  ->  G  e.  ( ( A [,] B ) -cn-> CC ) )
8947, 59, 88syl2anc 661 . . 3  |-  ( ( ( ph  /\  -.  A  <  B )  /\  -.  A  =  B
)  ->  G  e.  ( ( A [,] B ) -cn-> CC ) )
9046, 89pm2.61dan 791 . 2  |-  ( (
ph  /\  -.  A  <  B )  ->  G  e.  ( ( A [,] B ) -cn-> CC ) )
9114, 90pm2.61dan 791 1  |-  ( ph  ->  G  e.  ( ( A [,] B )
-cn-> CC ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1395   F/wnf 1617    e. wcel 1819    C_ wss 3471   (/)c0 3793   ifcif 3944   {csn 4032   class class class wbr 4456    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   RR*cxr 9644    < clt 9645   (,)cioo 11554   [,]cicc 11557   ↾t crest 14929   TopOpenctopn 14930  ℂfldccnfld 18638   Topctop 19612    Cn ccn 19943   -cn->ccncf 21597   lim CC climc 22483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fi 7889  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-seq 12111  df-exp 12170  df-cj 13035  df-re 13036  df-im 13037  df-sqrt 13171  df-abs 13172  df-struct 14737  df-ndx 14738  df-slot 14739  df-base 14740  df-plusg 14816  df-mulr 14817  df-starv 14818  df-tset 14822  df-ple 14823  df-ds 14825  df-unif 14826  df-rest 14931  df-topn 14932  df-topgen 14952  df-psmet 18629  df-xmet 18630  df-met 18631  df-bl 18632  df-mopn 18633  df-cnfld 18639  df-top 19617  df-bases 19619  df-topon 19620  df-topsp 19621  df-cld 19738  df-ntr 19739  df-cls 19740  df-cn 19946  df-cnp 19947  df-xms 21040  df-ms 21041  df-cncf 21599  df-limc 22487
This theorem is referenced by:  cncfiooiccre  31944  cncfioobd  31946  itgioocnicc  32022  iblcncfioo  32023  fourierdlem73  32208  fourierdlem81  32216  fourierdlem82  32217
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