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Theorem iocinif 26070
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 415 . . 3  |-  ( A  <  B  \/  -.  A  <  B )
2 xrltle 11125 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  A  <_  B ) )
32imp 429 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  <_  B
)
433adantl3 1146 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  B )  ->  A  <_  B )
5 iocinioc2 26068 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <_  B )  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( B (,] C
) )
64, 5syldan 470 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  B )  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( B (,] C
) )
76ex 434 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <  B  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( B (,] C
) ) )
87ancld 553 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <  B  ->  ( A  <  B  /\  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( B (,] C
) ) ) )
9 simpl2 992 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  ->  B  e.  RR* )
10 simpl1 991 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  ->  A  e.  RR* )
11 simpr 461 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  ->  -.  A  <  B )
12 xrlenlt 9441 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
1312biimpar 485 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  A  e.  RR* )  /\  -.  A  <  B
)  ->  B  <_  A )
149, 10, 11, 13syl21anc 1217 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  ->  B  <_  A )
15 3ancoma 972 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  A  e.  RR*  /\  C  e. 
RR* )  <->  ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* ) )
16 incom 3542 . . . . . . . . 9  |-  ( ( B (,] C )  i^i  ( A (,] C ) )  =  ( ( A (,] C )  i^i  ( B (,] C ) )
17 iocinioc2 26068 . . . . . . . . 9  |-  ( ( ( B  e.  RR*  /\  A  e.  RR*  /\  C  e.  RR* )  /\  B  <_  A )  ->  (
( B (,] C
)  i^i  ( A (,] C ) )  =  ( A (,] C
) )
1816, 17syl5eqr 2488 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  A  e.  RR*  /\  C  e.  RR* )  /\  B  <_  A )  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( A (,] C
) )
1915, 18sylanbr 473 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  B  <_  A )  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( A (,] C
) )
2014, 19syldan 470 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  -> 
( ( A (,] C )  i^i  ( B (,] C ) )  =  ( A (,] C ) )
2120ex 434 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( -.  A  <  B  -> 
( ( A (,] C )  i^i  ( B (,] C ) )  =  ( A (,] C ) ) )
2221ancld 553 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( -.  A  <  B  -> 
( -.  A  < 
B  /\  ( ( A (,] C )  i^i  ( B (,] C
) )  =  ( A (,] C ) ) ) )
238, 22orim12d 834 . . 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 17 . 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 3826 . 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 212 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 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    i^i cin 3326   ifcif 3790   class class class wbr 4291  (class class class)co 6090   RR*cxr 9416    < clt 9417    <_ cle 9418   (,]cioc 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 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-pre-lttri 9355  ax-pre-lttrn 9356
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-po 4640  df-so 4641  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-ioc 11304
This theorem is referenced by:  pnfneige0  26380
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