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Theorem eliccelico 26089
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 991 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  A  e.  RR* )
2 simpl2 992 . . . . . 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 11365 . . . . . . . 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 1000 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  C  e.  RR* )
71, 2, 3, 6syl21anc 1217 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  C  e.  RR* )
85simp3d 1002 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  C  <_  B
)
91, 2, 3, 8syl21anc 1217 . . . . 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 1001 . . . . . . 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 11364 . . . . . . . . . 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 967 . . . . . . . . . 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 1219 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  ( C  e.  ( A [,] B )  /\  -.  C  e.  ( A [,) B ) ) )  ->  -.  C  <  B )
24 xeqlelt 26088 . . . . . 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 1219 . . . 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 pm5.6 903 . . 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 26080 . . . . 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 3375 . . . 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 992 . . . . . 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 993 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  A  <_  B )
3736, 33breqtrrd 4339 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  A  <_  C )
38 xrleid 11148 . . . . . . 7  |-  ( B  e.  RR*  ->  B  <_  B )
3934, 38syl 16 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  B  <_  B )
4033, 39eqbrtrd 4333 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  /\  C  =  B )  ->  C  <_  B )
41 simpl1 991 . . . . . 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 1171 . . . 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 965    = wceq 1369    e. wcel 1756   class class class wbr 4313  (class class class)co 6112   RR*cxr 9438    < clt 9439    <_ cle 9440   [,)cico 11323   [,]cicc 11324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-pre-lttri 9377  ax-pre-lttrn 9378
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-po 4662  df-so 4663  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-ico 11327  df-icc 11328
This theorem is referenced by:  xrge0adddir  26177  esumcvg  26557
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