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Theorem iocinif 28313
Description: Relate intersection of two open-below, closed-above intervals with the same upper bound with a conditional construct. (Contributed by Thierry Arnoux, 7-Aug-2017.)
Assertion
Ref Expression
iocinif  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  if ( A  < 
B ,  ( B (,] C ) ,  ( A (,] C
) ) )

Proof of Theorem iocinif
StepHypRef Expression
1 exmid 416 . . 3  |-  ( A  <  B  \/  -.  A  <  B )
2 xrltle 11399 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  A  <_  B ) )
32imp 430 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  <_  B
)
433adantl3 1163 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  B )  ->  A  <_  B )
5 iocinioc2 28311 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <_  B )  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( B (,] C
) )
64, 5syldan 472 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  B )  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( B (,] C
) )
76ex 435 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <  B  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( B (,] C
) ) )
87ancld 555 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <  B  ->  ( A  <  B  /\  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( B (,] C
) ) ) )
9 simpl2 1009 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  ->  B  e.  RR* )
10 simpl1 1008 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  ->  A  e.  RR* )
11 simpr 462 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  ->  -.  A  <  B )
12 xrlenlt 9650 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
1312biimpar 487 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  A  e.  RR* )  /\  -.  A  <  B
)  ->  B  <_  A )
149, 10, 11, 13syl21anc 1263 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  ->  B  <_  A )
15 3ancoma 989 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  A  e.  RR*  /\  C  e. 
RR* )  <->  ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* ) )
16 incom 3598 . . . . . . . . 9  |-  ( ( B (,] C )  i^i  ( A (,] C ) )  =  ( ( A (,] C )  i^i  ( B (,] C ) )
17 iocinioc2 28311 . . . . . . . . 9  |-  ( ( ( B  e.  RR*  /\  A  e.  RR*  /\  C  e.  RR* )  /\  B  <_  A )  ->  (
( B (,] C
)  i^i  ( A (,] C ) )  =  ( A (,] C
) )
1816, 17syl5eqr 2476 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  A  e.  RR*  /\  C  e.  RR* )  /\  B  <_  A )  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( A (,] C
) )
1915, 18sylanbr 475 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  B  <_  A )  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( A (,] C
) )
2014, 19syldan 472 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  -> 
( ( A (,] C )  i^i  ( B (,] C ) )  =  ( A (,] C ) )
2120ex 435 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( -.  A  <  B  -> 
( ( A (,] C )  i^i  ( B (,] C ) )  =  ( A (,] C ) ) )
2221ancld 555 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( -.  A  <  B  -> 
( -.  A  < 
B  /\  ( ( A (,] C )  i^i  ( B (,] C
) )  =  ( A (,] C ) ) ) )
238, 22orim12d 846 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <  B  \/  -.  A  <  B
)  ->  ( ( A  <  B  /\  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( B (,] C
) )  \/  ( -.  A  <  B  /\  ( ( A (,] C )  i^i  ( B (,] C ) )  =  ( A (,] C ) ) ) ) )
241, 23mpi 20 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <  B  /\  ( ( A (,] C )  i^i  ( B (,] C ) )  =  ( B (,] C ) )  \/  ( -.  A  < 
B  /\  ( ( A (,] C )  i^i  ( B (,] C
) )  =  ( A (,] C ) ) ) )
25 eqif 3892 . 2  |-  ( ( ( A (,] C
)  i^i  ( B (,] C ) )  =  if ( A  < 
B ,  ( B (,] C ) ,  ( A (,] C
) )  <->  ( ( A  <  B  /\  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( B (,] C
) )  \/  ( -.  A  <  B  /\  ( ( A (,] C )  i^i  ( B (,] C ) )  =  ( A (,] C ) ) ) )
2624, 25sylibr 215 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  if ( A  < 
B ,  ( B (,] C ) ,  ( A (,] C
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    i^i cin 3378   ifcif 3854   class class class wbr 4366  (class class class)co 6249   RR*cxr 9625    < clt 9626    <_ cle 9627   (,]cioc 11587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-pre-lttri 9564  ax-pre-lttrn 9565
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-po 4717  df-so 4718  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-ioc 11591
This theorem is referenced by:  pnfneige0  28709
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