MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  soltmin Structured version   Unicode version

Theorem soltmin 5412
Description: Being less than a minimum, for a general total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
soltmin  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A R if ( B R C ,  B ,  C )  <->  ( A R B  /\  A R C ) ) )

Proof of Theorem soltmin
StepHypRef Expression
1 sopo 4823 . . . . . 6  |-  ( R  Or  X  ->  R  Po  X )
21ad2antrr 725 . . . . 5  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  R  Po  X )
3 simplr1 1038 . . . . . 6  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  A  e.  X )
4 simplr2 1039 . . . . . . 7  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  B  e.  X )
5 simplr3 1040 . . . . . . 7  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  C  e.  X )
6 ifcl 3987 . . . . . . 7  |-  ( ( B  e.  X  /\  C  e.  X )  ->  if ( B R C ,  B ,  C )  e.  X
)
74, 5, 6syl2anc 661 . . . . . 6  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  if ( B R C ,  B ,  C
)  e.  X )
83, 7, 43jca 1176 . . . . 5  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  -> 
( A  e.  X  /\  if ( B R C ,  B ,  C )  e.  X  /\  B  e.  X
) )
9 simpr 461 . . . . 5  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  A R if ( B R C ,  B ,  C ) )
10 simpll 753 . . . . . 6  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  R  Or  X )
11 somin1 5409 . . . . . 6  |-  ( ( R  Or  X  /\  ( B  e.  X  /\  C  e.  X
) )  ->  if ( B R C ,  B ,  C )
( R  u.  _I  ) B )
1210, 4, 5, 11syl12anc 1226 . . . . 5  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  if ( B R C ,  B ,  C
) ( R  u.  _I  ) B )
13 poltletr 5408 . . . . . 6  |-  ( ( R  Po  X  /\  ( A  e.  X  /\  if ( B R C ,  B ,  C )  e.  X  /\  B  e.  X
) )  ->  (
( A R if ( B R C ,  B ,  C
)  /\  if ( B R C ,  B ,  C ) ( R  u.  _I  ) B )  ->  A R B ) )
1413imp 429 . . . . 5  |-  ( ( ( R  Po  X  /\  ( A  e.  X  /\  if ( B R C ,  B ,  C )  e.  X  /\  B  e.  X
) )  /\  ( A R if ( B R C ,  B ,  C )  /\  if ( B R C ,  B ,  C )
( R  u.  _I  ) B ) )  ->  A R B )
152, 8, 9, 12, 14syl22anc 1229 . . . 4  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  A R B )
163, 7, 53jca 1176 . . . . 5  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  -> 
( A  e.  X  /\  if ( B R C ,  B ,  C )  e.  X  /\  C  e.  X
) )
17 somin2 5411 . . . . . 6  |-  ( ( R  Or  X  /\  ( B  e.  X  /\  C  e.  X
) )  ->  if ( B R C ,  B ,  C )
( R  u.  _I  ) C )
1810, 4, 5, 17syl12anc 1226 . . . . 5  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  if ( B R C ,  B ,  C
) ( R  u.  _I  ) C )
19 poltletr 5408 . . . . . 6  |-  ( ( R  Po  X  /\  ( A  e.  X  /\  if ( B R C ,  B ,  C )  e.  X  /\  C  e.  X
) )  ->  (
( A R if ( B R C ,  B ,  C
)  /\  if ( B R C ,  B ,  C ) ( R  u.  _I  ) C )  ->  A R C ) )
2019imp 429 . . . . 5  |-  ( ( ( R  Po  X  /\  ( A  e.  X  /\  if ( B R C ,  B ,  C )  e.  X  /\  C  e.  X
) )  /\  ( A R if ( B R C ,  B ,  C )  /\  if ( B R C ,  B ,  C )
( R  u.  _I  ) C ) )  ->  A R C )
212, 16, 9, 18, 20syl22anc 1229 . . . 4  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  ->  A R C )
2215, 21jca 532 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R if ( B R C ,  B ,  C ) )  -> 
( A R B  /\  A R C ) )
2322ex 434 . 2  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A R if ( B R C ,  B ,  C )  ->  ( A R B  /\  A R C ) ) )
24 breq2 4457 . . 3  |-  ( B  =  if ( B R C ,  B ,  C )  ->  ( A R B  <->  A R if ( B R C ,  B ,  C
) ) )
25 breq2 4457 . . 3  |-  ( C  =  if ( B R C ,  B ,  C )  ->  ( A R C  <->  A R if ( B R C ,  B ,  C
) ) )
2624, 25ifboth 3981 . 2  |-  ( ( A R B  /\  A R C )  ->  A R if ( B R C ,  B ,  C ) )
2723, 26impbid1 203 1  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A R if ( B R C ,  B ,  C )  <->  ( A R B  /\  A R C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1767    u. cun 3479   ifcif 3945   class class class wbr 4453    _I cid 4796    Po wpo 4804    Or wor 4805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012
This theorem is referenced by:  wemaplem2  7984
  Copyright terms: Public domain W3C validator