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Theorem elicoelioo 28372
Description: Relate elementhood to a closed-below, open-above interval with elementhood to the same open interval or to its lower bound. (Contributed by Thierry Arnoux, 6-Jul-2017.)
Assertion
Ref Expression
elicoelioo  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( C  e.  ( A [,) B )  <->  ( C  =  A  \/  C  e.  ( A (,) B
) ) ) )

Proof of Theorem elicoelioo
StepHypRef Expression
1 simpl1 1012 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  A  e.  RR* )
2 simpl2 1013 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  B  e.  RR* )
3 simprl 765 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  C  e.  ( A [,) B ) )
4 elico1 11686 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B
) ) )
54biimpa 487 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,) B ) )  ->  ( C  e. 
RR*  /\  A  <_  C  /\  C  <  B
) )
65simp1d 1021 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,) B ) )  ->  C  e.  RR* )
71, 2, 3, 6syl21anc 1268 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  C  e.  RR* )
85simp2d 1022 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,) B ) )  ->  A  <_  C
)
91, 2, 3, 8syl21anc 1268 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  A  <_  C
)
101, 2jca 535 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  ( A  e. 
RR*  /\  B  e.  RR* ) )
11 simprr 767 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  -.  C  e.  ( A (,) B ) )
125simp3d 1023 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,) B ) )  ->  C  <  B
)
1310, 3, 12syl2anc 667 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  C  <  B
)
14 elioo1 11683 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <  B ) ) )
1514notbid 296 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  C  e.  ( A (,) B )  <->  -.  ( C  e.  RR*  /\  A  <  C  /\  C  < 
B ) ) )
1615biimpa 487 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A (,) B ) )  ->  -.  ( C  e.  RR*  /\  A  <  C  /\  C  < 
B ) )
17 3anan32 998 . . . . . . . . . . . 12  |-  ( ( C  e.  RR*  /\  A  <  C  /\  C  < 
B )  <->  ( ( C  e.  RR*  /\  C  <  B )  /\  A  <  C ) )
1817notbii 298 . . . . . . . . . . 11  |-  ( -.  ( C  e.  RR*  /\  A  <  C  /\  C  <  B )  <->  -.  (
( C  e.  RR*  /\  C  <  B )  /\  A  <  C
) )
19 imnan 424 . . . . . . . . . . 11  |-  ( ( ( C  e.  RR*  /\  C  <  B )  ->  -.  A  <  C )  <->  -.  ( ( C  e.  RR*  /\  C  <  B )  /\  A  <  C ) )
2018, 19bitr4i 256 . . . . . . . . . 10  |-  ( -.  ( C  e.  RR*  /\  A  <  C  /\  C  <  B )  <->  ( ( C  e.  RR*  /\  C  <  B )  ->  -.  A  <  C ) )
2116, 20sylib 200 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A (,) B ) )  ->  ( ( C  e.  RR*  /\  C  <  B )  ->  -.  A  <  C ) )
2221imp 431 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A (,) B
) )  /\  ( C  e.  RR*  /\  C  <  B ) )  ->  -.  A  <  C )
2310, 11, 7, 13, 22syl22anc 1270 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  -.  A  <  C )
24 xeqlelt 28370 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A  =  C  <->  ( A  <_  C  /\  -.  A  <  C ) ) )
2524biimpar 488 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  C  /\  -.  A  <  C
) )  ->  A  =  C )
261, 7, 9, 23, 25syl22anc 1270 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  A  =  C )
2726ex 436 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B
) )  ->  A  =  C ) )
28 eqcom 2460 . . . . 5  |-  ( A  =  C  <->  C  =  A )
2927, 28syl6ib 230 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B
) )  ->  C  =  A ) )
30 pm5.6 924 . . . 4  |-  ( ( ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B
) )  ->  C  =  A )  <->  ( C  e.  ( A [,) B
)  ->  ( C  e.  ( A (,) B
)  \/  C  =  A ) ) )
3129, 30sylib 200 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( C  e.  ( A [,) B )  ->  ( C  e.  ( A (,) B )  \/  C  =  A ) ) )
32 orcom 389 . . 3  |-  ( ( C  e.  ( A (,) B )  \/  C  =  A )  <-> 
( C  =  A  \/  C  e.  ( A (,) B ) ) )
3331, 32syl6ib 230 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( C  e.  ( A [,) B )  ->  ( C  =  A  \/  C  e.  ( A (,) B ) ) ) )
34 simpr 463 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  C  =  A )
35 simpl1 1012 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  A  e.  RR* )
3634, 35eqeltrd 2531 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  C  e.  RR* )
37 xrleid 11456 . . . . . . 7  |-  ( A  e.  RR*  ->  A  <_  A )
3835, 37syl 17 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  A  <_  A )
3938, 34breqtrrd 4432 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  A  <_  C )
40 simpl3 1014 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  A  <  B )
4134, 40eqbrtrd 4426 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  C  <  B )
42 simpl2 1013 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  B  e.  RR* )
4335, 42, 4syl2anc 667 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B
) ) )
4436, 39, 41, 43mpbir3and 1192 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  C  e.  ( A [,) B
) )
45 ioossico 11730 . . . . 5  |-  ( A (,) B )  C_  ( A [,) B )
46 simpr 463 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  e.  ( A (,) B
) )  ->  C  e.  ( A (,) B
) )
4745, 46sseldi 3432 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  e.  ( A (,) B
) )  ->  C  e.  ( A [,) B
) )
4844, 47jaodan 795 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  =  A  \/  C  e.  ( A (,) B ) ) )  ->  C  e.  ( A [,) B ) )
4948ex 436 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( C  =  A  \/  C  e.  ( A (,) B ) )  ->  C  e.  ( A [,) B ) ) )
5033, 49impbid 194 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( C  e.  ( A [,) B )  <->  ( C  =  A  \/  C  e.  ( A (,) B
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   class class class wbr 4405  (class class class)co 6295   RR*cxr 9679    < clt 9680    <_ cle 9681   (,)cioo 11642   [,)cico 11644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-pre-lttri 9618  ax-pre-lttrn 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-po 4758  df-so 4759  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-ioo 11646  df-ico 11648
This theorem is referenced by:  xrge0mulc1cn  28759
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