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Theorem eliccelico 28408
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 1017 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  A  e.  RR* )
2 simpl2 1018 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  B  e.  RR* )
3 simprl 769 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  C  e.  ( A [,] B ) )
4 elicc1 11709 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B
) ) )
54biimpa 491 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  ( C  e. 
RR*  /\  A  <_  C  /\  C  <_  B
) )
65simp1d 1026 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  C  e.  RR* )
71, 2, 3, 6syl21anc 1275 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  C  e.  RR* )
85simp3d 1028 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  C  <_  B
)
91, 2, 3, 8syl21anc 1275 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  C  <_  B
)
101, 2jca 539 . . . . . 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 771 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  -.  C  e.  ( A [,) B ) )
125simp2d 1027 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  A  <_  C
)
1310, 3, 12syl2anc 671 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  A  <_  C
)
14 elico1 11708 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B
) ) )
1514notbid 300 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  C  e.  ( A [,) B )  <->  -.  ( C  e.  RR*  /\  A  <_  C  /\  C  < 
B ) ) )
1615biimpa 491 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A [,) B ) )  ->  -.  ( C  e.  RR*  /\  A  <_  C  /\  C  < 
B ) )
17 df-3an 993 . . . . . . . . . 10  |-  ( ( C  e.  RR*  /\  A  <_  C  /\  C  < 
B )  <->  ( ( C  e.  RR*  /\  A  <_  C )  /\  C  <  B ) )
1817notbii 302 . . . . . . . . 9  |-  ( -.  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B )  <->  -.  (
( C  e.  RR*  /\  A  <_  C )  /\  C  <  B ) )
19 imnan 428 . . . . . . . . 9  |-  ( ( ( C  e.  RR*  /\  A  <_  C )  ->  -.  C  <  B
)  <->  -.  ( ( C  e.  RR*  /\  A  <_  C )  /\  C  <  B ) )
2018, 19bitr4i 260 . . . . . . . 8  |-  ( -.  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B )  <->  ( ( C  e.  RR*  /\  A  <_  C )  ->  -.  C  <  B ) )
2116, 20sylib 201 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A [,) B ) )  ->  ( ( C  e.  RR*  /\  A  <_  C )  ->  -.  C  <  B ) )
2221imp 435 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A [,) B
) )  /\  ( C  e.  RR*  /\  A  <_  C ) )  ->  -.  C  <  B )
2310, 11, 7, 13, 22syl22anc 1277 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  -.  C  <  B )
24 xeqlelt 28407 . . . . . 6  |-  ( ( C  e.  RR*  /\  B  e.  RR* )  ->  ( C  =  B  <->  ( C  <_  B  /\  -.  C  <  B ) ) )
2524biimpar 492 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B  e.  RR* )  /\  ( C  <_  B  /\  -.  C  <  B
) )  ->  C  =  B )
267, 2, 9, 23, 25syl22anc 1277 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  C  =  B )
2726ex 440 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B
) )  ->  C  =  B ) )
28 pm5.6 928 . . 3  |-  ( ( ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B
) )  ->  C  =  B )  <->  ( C  e.  ( A [,] B
)  ->  ( C  e.  ( A [,) B
)  \/  C  =  B ) ) )
2927, 28sylib 201 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( C  e.  ( A [,] B )  ->  ( C  e.  ( A [,) B )  \/  C  =  B ) ) )
30 icossicc 11750 . . . . 5  |-  ( A [,) B )  C_  ( A [,] B )
31 simpr 467 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  e.  ( A [,) B
) )  ->  C  e.  ( A [,) B
) )
3230, 31sseldi 3442 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  e.  ( A [,) B
) )  ->  C  e.  ( A [,] B
) )
33 simpr 467 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  C  =  B )
34 simpl2 1018 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  B  e.  RR* )
3533, 34eqeltrd 2540 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  C  e.  RR* )
36 simpl3 1019 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  A  <_  B )
3736, 33breqtrrd 4443 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  A  <_  C )
38 xrleid 11478 . . . . . . 7  |-  ( B  e.  RR*  ->  B  <_  B )
3934, 38syl 17 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  B  <_  B )
4033, 39eqbrtrd 4437 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  C  <_  B )
41 simpl1 1017 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  A  e.  RR* )
4241, 34, 4syl2anc 671 . . . . 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 1197 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  C  e.  ( A [,] B
) )
4432, 43jaodan 799 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,) B )  \/  C  =  B ) )  ->  C  e.  ( A [,] B ) )
4544ex 440 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( C  e.  ( A [,) B )  \/  C  =  B )  ->  C  e.  ( A [,] B ) ) )
4629, 45impbid 195 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 189    \/ wo 374    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898   class class class wbr 4416  (class class class)co 6315   RR*cxr 9700    < clt 9701    <_ cle 9702   [,)cico 11666   [,]cicc 11667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-pre-lttri 9639  ax-pre-lttrn 9640
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  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 6318  df-oprab 6319  df-mpt2 6320  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-ico 11670  df-icc 11671
This theorem is referenced by:  xrge0adddir  28504  esumcvg  28956
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