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Theorem eliccelico 28203
Description: Relate elementhood to a closed interval with elementhood to the same closed-below, open-above interval or to its upper bound. (Contributed by Thierry Arnoux, 3-Jul-2017.)
Assertion
Ref Expression
eliccelico  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  ( A [,) B
)  \/  C  =  B ) ) )

Proof of Theorem eliccelico
StepHypRef Expression
1 simpl1 1008 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  A  e.  RR* )
2 simpl2 1009 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  B  e.  RR* )
3 simprl 762 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  C  e.  ( A [,] B ) )
4 elicc1 11680 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B
) ) )
54biimpa 486 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  ( C  e. 
RR*  /\  A  <_  C  /\  C  <_  B
) )
65simp1d 1017 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  C  e.  RR* )
71, 2, 3, 6syl21anc 1263 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  C  e.  RR* )
85simp3d 1019 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  C  <_  B
)
91, 2, 3, 8syl21anc 1263 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  C  <_  B
)
101, 2jca 534 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  ( A  e. 
RR*  /\  B  e.  RR* ) )
11 simprr 764 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  -.  C  e.  ( A [,) B ) )
125simp2d 1018 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  A  <_  C
)
1310, 3, 12syl2anc 665 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  A  <_  C
)
14 elico1 11679 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B
) ) )
1514notbid 295 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  C  e.  ( A [,) B )  <->  -.  ( C  e.  RR*  /\  A  <_  C  /\  C  < 
B ) ) )
1615biimpa 486 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A [,) B ) )  ->  -.  ( C  e.  RR*  /\  A  <_  C  /\  C  < 
B ) )
17 df-3an 984 . . . . . . . . . 10  |-  ( ( C  e.  RR*  /\  A  <_  C  /\  C  < 
B )  <->  ( ( C  e.  RR*  /\  A  <_  C )  /\  C  <  B ) )
1817notbii 297 . . . . . . . . 9  |-  ( -.  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B )  <->  -.  (
( C  e.  RR*  /\  A  <_  C )  /\  C  <  B ) )
19 imnan 423 . . . . . . . . 9  |-  ( ( ( C  e.  RR*  /\  A  <_  C )  ->  -.  C  <  B
)  <->  -.  ( ( C  e.  RR*  /\  A  <_  C )  /\  C  <  B ) )
2018, 19bitr4i 255 . . . . . . . 8  |-  ( -.  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B )  <->  ( ( C  e.  RR*  /\  A  <_  C )  ->  -.  C  <  B ) )
2116, 20sylib 199 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A [,) B ) )  ->  ( ( C  e.  RR*  /\  A  <_  C )  ->  -.  C  <  B ) )
2221imp 430 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A [,) B
) )  /\  ( C  e.  RR*  /\  A  <_  C ) )  ->  -.  C  <  B )
2310, 11, 7, 13, 22syl22anc 1265 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  -.  C  <  B )
24 xeqlelt 28202 . . . . . 6  |-  ( ( C  e.  RR*  /\  B  e.  RR* )  ->  ( C  =  B  <->  ( C  <_  B  /\  -.  C  <  B ) ) )
2524biimpar 487 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B  e.  RR* )  /\  ( C  <_  B  /\  -.  C  <  B
) )  ->  C  =  B )
267, 2, 9, 23, 25syl22anc 1265 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  C  =  B )
2726ex 435 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B
) )  ->  C  =  B ) )
28 pm5.6 920 . . 3  |-  ( ( ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B
) )  ->  C  =  B )  <->  ( C  e.  ( A [,] B
)  ->  ( C  e.  ( A [,) B
)  \/  C  =  B ) ) )
2927, 28sylib 199 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( C  e.  ( A [,] B )  ->  ( C  e.  ( A [,) B )  \/  C  =  B ) ) )
30 icossicc 11721 . . . . 5  |-  ( A [,) B )  C_  ( A [,] B )
31 simpr 462 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  e.  ( A [,) B
) )  ->  C  e.  ( A [,) B
) )
3230, 31sseldi 3468 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  e.  ( A [,) B
) )  ->  C  e.  ( A [,] B
) )
33 simpr 462 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  C  =  B )
34 simpl2 1009 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  B  e.  RR* )
3533, 34eqeltrd 2517 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  C  e.  RR* )
36 simpl3 1010 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  A  <_  B )
3736, 33breqtrrd 4452 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  A  <_  C )
38 xrleid 11449 . . . . . . 7  |-  ( B  e.  RR*  ->  B  <_  B )
3934, 38syl 17 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  B  <_  B )
4033, 39eqbrtrd 4446 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  C  <_  B )
41 simpl1 1008 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  A  e.  RR* )
4241, 34, 4syl2anc 665 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B
) ) )
4335, 37, 40, 42mpbir3and 1188 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  C  e.  ( A [,] B
) )
4432, 43jaodan 792 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,) B )  \/  C  =  B ) )  ->  C  e.  ( A [,] B ) )
4544ex 435 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( C  e.  ( A [,) B )  \/  C  =  B )  ->  C  e.  ( A [,] B ) ) )
4629, 45impbid 193 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  ( A [,) B
)  \/  C  =  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   class class class wbr 4426  (class class class)co 6305   RR*cxr 9673    < clt 9674    <_ cle 9675   [,)cico 11637   [,]cicc 11638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-pre-lttri 9612  ax-pre-lttrn 9613
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-ico 11641  df-icc 11642
This theorem is referenced by:  xrge0adddir  28301  esumcvg  28754
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