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Theorem iocinif 27464
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 11364 . . . . . . . . 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 1155 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  B )  ->  A  <_  B )
5 iocinioc2 27462 . . . . . . 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 1001 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  ->  B  e.  RR* )
10 simpl1 1000 . . . . . . . 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 9655 . . . . . . . . 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 1228 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  -.  A  <  B )  ->  B  <_  A )
15 3ancoma 981 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  A  e.  RR*  /\  C  e. 
RR* )  <->  ( A  e.  RR*  /\  B  e. 
RR*  /\  C  e.  RR* ) )
16 incom 3676 . . . . . . . . 9  |-  ( ( B (,] C )  i^i  ( A (,] C ) )  =  ( ( A (,] C )  i^i  ( B (,] C ) )
17 iocinioc2 27462 . . . . . . . . 9  |-  ( ( ( B  e.  RR*  /\  A  e.  RR*  /\  C  e.  RR* )  /\  B  <_  A )  ->  (
( B (,] C
)  i^i  ( A (,] C ) )  =  ( A (,] C
) )
1816, 17syl5eqr 2498 . . . . . . . 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 838 . . 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 3964 . 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 974    = wceq 1383    e. wcel 1804    i^i cin 3460   ifcif 3926   class class class wbr 4437  (class class class)co 6281   RR*cxr 9630    < clt 9631    <_ cle 9632   (,]cioc 11539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-pre-lttri 9569  ax-pre-lttrn 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-ioc 11543
This theorem is referenced by:  pnfneige0  27806
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