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Theorem elicoelioo 26066
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 991 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  A  e.  RR* )
2 simpl2 992 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  B  e.  RR* )
3 simprl 755 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  C  e.  ( A [,) B ) )
4 elico1 11341 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B
) ) )
54biimpa 484 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,) B ) )  ->  ( C  e. 
RR*  /\  A  <_  C  /\  C  <  B
) )
65simp1d 1000 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,) B ) )  ->  C  e.  RR* )
71, 2, 3, 6syl21anc 1217 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  C  e.  RR* )
85simp2d 1001 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,) B ) )  ->  A  <_  C
)
91, 2, 3, 8syl21anc 1217 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  A  <_  C
)
101, 2jca 532 . . . . . . . 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 756 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  -.  C  e.  ( A (,) B ) )
125simp3d 1002 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,) B ) )  ->  C  <  B
)
1310, 3, 12syl2anc 661 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  C  <  B
)
14 elioo1 11338 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <  B ) ) )
1514notbid 294 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  C  e.  ( A (,) B )  <->  -.  ( C  e.  RR*  /\  A  <  C  /\  C  < 
B ) ) )
1615biimpa 484 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A (,) B ) )  ->  -.  ( C  e.  RR*  /\  A  <  C  /\  C  < 
B ) )
17 3anan32 977 . . . . . . . . . . . 12  |-  ( ( C  e.  RR*  /\  A  <  C  /\  C  < 
B )  <->  ( ( C  e.  RR*  /\  C  <  B )  /\  A  <  C ) )
1817notbii 296 . . . . . . . . . . 11  |-  ( -.  ( C  e.  RR*  /\  A  <  C  /\  C  <  B )  <->  -.  (
( C  e.  RR*  /\  C  <  B )  /\  A  <  C
) )
19 imnan 422 . . . . . . . . . . 11  |-  ( ( ( C  e.  RR*  /\  C  <  B )  ->  -.  A  <  C )  <->  -.  ( ( C  e.  RR*  /\  C  <  B )  /\  A  <  C ) )
2018, 19bitr4i 252 . . . . . . . . . 10  |-  ( -.  ( C  e.  RR*  /\  A  <  C  /\  C  <  B )  <->  ( ( C  e.  RR*  /\  C  <  B )  ->  -.  A  <  C ) )
2116, 20sylib 196 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A (,) B ) )  ->  ( ( C  e.  RR*  /\  C  <  B )  ->  -.  A  <  C ) )
2221imp 429 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  C  e.  ( A (,) B
) )  /\  ( C  e.  RR*  /\  C  <  B ) )  ->  -.  A  <  C )
2310, 11, 7, 13, 22syl22anc 1219 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  -.  A  <  C )
24 xeqlelt 26064 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A  =  C  <->  ( A  <_  C  /\  -.  A  <  C ) ) )
2524biimpar 485 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  C  /\  -.  A  <  C
) )  ->  A  =  C )
261, 7, 9, 23, 25syl22anc 1219 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B ) ) )  ->  A  =  C )
2726ex 434 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B
) )  ->  A  =  C ) )
28 eqcom 2443 . . . . 5  |-  ( A  =  C  <->  C  =  A )
2927, 28syl6ib 226 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B
) )  ->  C  =  A ) )
30 pm5.6 903 . . . 4  |-  ( ( ( C  e.  ( A [,) B )  /\  -.  C  e.  ( A (,) B
) )  ->  C  =  A )  <->  ( C  e.  ( A [,) B
)  ->  ( C  e.  ( A (,) B
)  \/  C  =  A ) ) )
3129, 30sylib 196 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( C  e.  ( A [,) B )  ->  ( C  e.  ( A (,) B )  \/  C  =  A ) ) )
32 orcom 387 . . 3  |-  ( ( C  e.  ( A (,) B )  \/  C  =  A )  <-> 
( C  =  A  \/  C  e.  ( A (,) B ) ) )
3331, 32syl6ib 226 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( C  e.  ( A [,) B )  ->  ( C  =  A  \/  C  e.  ( A (,) B ) ) ) )
34 simpr 461 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  C  =  A )
35 simpl1 991 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  A  e.  RR* )
3634, 35eqeltrd 2515 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  C  e.  RR* )
37 xrleid 11125 . . . . . . 7  |-  ( A  e.  RR*  ->  A  <_  A )
3835, 37syl 16 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  A  <_  A )
3938, 34breqtrrd 4316 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  A  <_  C )
40 simpl3 993 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  A  <  B )
4134, 40eqbrtrd 4310 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  C  <  B )
42 simpl2 992 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  B  e.  RR* )
4335, 42, 4syl2anc 661 . . . . 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 1171 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  =  A )  ->  C  e.  ( A [,) B
) )
45 ioossico 26058 . . . . 5  |-  ( A (,) B )  C_  ( A [,) B )
46 simpr 461 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  e.  ( A (,) B
) )  ->  C  e.  ( A (,) B
) )
4745, 46sseldi 3352 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  C  e.  ( A (,) B
) )  ->  C  e.  ( A [,) B
) )
4844, 47jaodan 783 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  /\  ( C  =  A  \/  C  e.  ( A (,) B ) ) )  ->  C  e.  ( A [,) B ) )
4948ex 434 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( C  =  A  \/  C  e.  ( A (,) B ) )  ->  C  e.  ( A [,) B ) ) )
5033, 49impbid 191 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 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4290  (class class class)co 6089   RR*cxr 9415    < clt 9416    <_ cle 9417   (,)cioo 11298   [,)cico 11300
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 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-pre-lttri 9354  ax-pre-lttrn 9355
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-ioo 11302  df-ico 11304
This theorem is referenced by:  xrge0mulc1cn  26369
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