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Theorem eliccelico 27411
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 999 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  A  e.  RR* )
2 simpl2 1000 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  B  e.  RR* )
3 simprl 755 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  C  e.  ( A [,] B ) )
4 elicc1 11585 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B
) ) )
54biimpa 484 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  ( C  e. 
RR*  /\  A  <_  C  /\  C  <_  B
) )
65simp1d 1008 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  C  e.  RR* )
71, 2, 3, 6syl21anc 1227 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  C  e.  RR* )
85simp3d 1010 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  C  <_  B
)
91, 2, 3, 8syl21anc 1227 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  C  <_  B
)
101, 2jca 532 . . . . . 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 756 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  -.  C  e.  ( A [,) B ) )
125simp2d 1009 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  A  <_  C
)
1310, 3, 12syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  A  <_  C
)
14 elico1 11584 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B
) ) )
1514notbid 294 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  C  e.  ( A [,) B )  <->  -.  ( C  e.  RR*  /\  A  <_  C  /\  C  < 
B ) ) )
1615biimpa 484 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A [,) B ) )  ->  -.  ( C  e.  RR*  /\  A  <_  C  /\  C  < 
B ) )
17 df-3an 975 . . . . . . . . . 10  |-  ( ( C  e.  RR*  /\  A  <_  C  /\  C  < 
B )  <->  ( ( C  e.  RR*  /\  A  <_  C )  /\  C  <  B ) )
1817notbii 296 . . . . . . . . 9  |-  ( -.  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B )  <->  -.  (
( C  e.  RR*  /\  A  <_  C )  /\  C  <  B ) )
19 imnan 422 . . . . . . . . 9  |-  ( ( ( C  e.  RR*  /\  A  <_  C )  ->  -.  C  <  B
)  <->  -.  ( ( C  e.  RR*  /\  A  <_  C )  /\  C  <  B ) )
2018, 19bitr4i 252 . . . . . . . 8  |-  ( -.  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B )  <->  ( ( C  e.  RR*  /\  A  <_  C )  ->  -.  C  <  B ) )
2116, 20sylib 196 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A [,) B ) )  ->  ( ( C  e.  RR*  /\  A  <_  C )  ->  -.  C  <  B ) )
2221imp 429 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A [,) B
) )  /\  ( C  e.  RR*  /\  A  <_  C ) )  ->  -.  C  <  B )
2310, 11, 7, 13, 22syl22anc 1229 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  -.  C  <  B )
24 xeqlelt 27410 . . . . . 6  |-  ( ( C  e.  RR*  /\  B  e.  RR* )  ->  ( C  =  B  <->  ( C  <_  B  /\  -.  C  <  B ) ) )
2524biimpar 485 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B  e.  RR* )  /\  ( C  <_  B  /\  -.  C  <  B
) )  ->  C  =  B )
267, 2, 9, 23, 25syl22anc 1229 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  C  =  B )
2726ex 434 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B
) )  ->  C  =  B ) )
28 pm5.6 910 . . 3  |-  ( ( ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B
) )  ->  C  =  B )  <->  ( C  e.  ( A [,] B
)  ->  ( C  e.  ( A [,) B
)  \/  C  =  B ) ) )
2927, 28sylib 196 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( C  e.  ( A [,] B )  ->  ( C  e.  ( A [,) B )  \/  C  =  B ) ) )
30 icossicc 11623 . . . . 5  |-  ( A [,) B )  C_  ( A [,] B )
31 simpr 461 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  e.  ( A [,) B
) )  ->  C  e.  ( A [,) B
) )
3230, 31sseldi 3507 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  e.  ( A [,) B
) )  ->  C  e.  ( A [,] B
) )
33 simpr 461 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  C  =  B )
34 simpl2 1000 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  B  e.  RR* )
3533, 34eqeltrd 2555 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  C  e.  RR* )
36 simpl3 1001 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  A  <_  B )
3736, 33breqtrrd 4479 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  A  <_  C )
38 xrleid 11368 . . . . . . 7  |-  ( B  e.  RR*  ->  B  <_  B )
3934, 38syl 16 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  B  <_  B )
4033, 39eqbrtrd 4473 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  C  <_  B )
41 simpl1 999 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  A  e.  RR* )
4241, 34, 4syl2anc 661 . . . . 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 1179 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  C  e.  ( A [,] B
) )
4432, 43jaodan 783 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,) B )  \/  C  =  B ) )  ->  C  e.  ( A [,] B ) )
4544ex 434 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( C  e.  ( A [,) B )  \/  C  =  B )  ->  C  e.  ( A [,] B ) ) )
4629, 45impbid 191 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 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4453  (class class class)co 6295   RR*cxr 9639    < clt 9640    <_ cle 9641   [,)cico 11543   [,]cicc 11544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-pre-lttri 9578  ax-pre-lttrn 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-ico 11547  df-icc 11548
This theorem is referenced by:  xrge0adddir  27506  esumcvg  27917
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