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Theorem iocinif 27248
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 11346 . . . . . . . . 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 1149 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  B )  ->  A  <_  B )
5 iocinioc2 27246 . . . . . . 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 995 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  ->  B  e.  RR* )
10 simpl1 994 . . . . . . . 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 9643 . . . . . . . . 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 1222 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  ->  B  <_  A )
15 3ancoma 975 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  A  e.  RR*  /\  C  e. 
RR* )  <->  ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* ) )
16 incom 3686 . . . . . . . . 9  |-  ( ( B (,] C )  i^i  ( A (,] C ) )  =  ( ( A (,] C )  i^i  ( B (,] C ) )
17 iocinioc2 27246 . . . . . . . . 9  |-  ( ( ( B  e.  RR*  /\  A  e.  RR*  /\  C  e.  RR* )  /\  B  <_  A )  ->  (
( B (,] C
)  i^i  ( A (,] C ) )  =  ( A (,] C
) )
1816, 17syl5eqr 2517 . . . . . . . 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 835 . . 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 3972 . 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 968    = wceq 1374    e. wcel 1762    i^i cin 3470   ifcif 3934   class class class wbr 4442  (class class class)co 6277   RR*cxr 9618    < clt 9619    <_ cle 9620   (,]cioc 11521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-pre-lttri 9557  ax-pre-lttrn 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-ioc 11525
This theorem is referenced by:  pnfneige0  27557
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