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Theorem iocinif 27826
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 413 . . 3  |-  ( A  <  B  \/  -.  A  <  B )
2 xrltle 11358 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  A  <_  B ) )
32imp 427 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <  B )  ->  A  <_  B
)
433adantl3 1152 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  B )  ->  A  <_  B )
5 iocinioc2 27824 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <_  B )  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( B (,] C
) )
64, 5syldan 468 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  B )  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( B (,] C
) )
76ex 432 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <  B  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( B (,] C
) ) )
87ancld 551 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <  B  ->  ( A  <  B  /\  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( B (,] C
) ) ) )
9 simpl2 998 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  ->  B  e.  RR* )
10 simpl1 997 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  ->  A  e.  RR* )
11 simpr 459 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  ->  -.  A  <  B )
12 xrlenlt 9641 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
1312biimpar 483 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  A  e.  RR* )  /\  -.  A  <  B
)  ->  B  <_  A )
149, 10, 11, 13syl21anc 1225 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  ->  B  <_  A )
15 3ancoma 978 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  A  e.  RR*  /\  C  e. 
RR* )  <->  ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* ) )
16 incom 3677 . . . . . . . . 9  |-  ( ( B (,] C )  i^i  ( A (,] C ) )  =  ( ( A (,] C )  i^i  ( B (,] C ) )
17 iocinioc2 27824 . . . . . . . . 9  |-  ( ( ( B  e.  RR*  /\  A  e.  RR*  /\  C  e.  RR* )  /\  B  <_  A )  ->  (
( B (,] C
)  i^i  ( A (,] C ) )  =  ( A (,] C
) )
1816, 17syl5eqr 2509 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  A  e.  RR*  /\  C  e.  RR* )  /\  B  <_  A )  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( A (,] C
) )
1915, 18sylanbr 471 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  B  <_  A )  ->  (
( A (,] C
)  i^i  ( B (,] C ) )  =  ( A (,] C
) )
2014, 19syldan 468 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  -> 
( ( A (,] C )  i^i  ( B (,] C ) )  =  ( A (,] C ) )
2120ex 432 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( -.  A  <  B  -> 
( ( A (,] C )  i^i  ( B (,] C ) )  =  ( A (,] C ) ) )
2221ancld 551 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( -.  A  <  B  -> 
( -.  A  < 
B  /\  ( ( A (,] C )  i^i  ( B (,] C
) )  =  ( A (,] C ) ) ) )
238, 22orim12d 836 . . 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 3967 . 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 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    i^i cin 3460   ifcif 3929   class class class wbr 4439  (class class class)co 6270   RR*cxr 9616    < clt 9617    <_ cle 9618   (,]cioc 11533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-ioc 11537
This theorem is referenced by:  pnfneige0  28168
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