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Theorem soltmin 5082
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 4331 . . . . . 6  |-  ( R  Or  X  ->  R  Po  X )
21ad2antrr 706 . . . . 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 997 . . . . . 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 998 . . . . . . 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 999 . . . . . . 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 3601 . . . . . . 7  |-  ( ( B  e.  X  /\  C  e.  X )  ->  if ( B R C ,  B ,  C )  e.  X
)
74, 5, 6syl2anc 642 . . . . . 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 1132 . . . . 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 447 . . . . 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 730 . . . . . 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 5079 . . . . . 6  |-  ( ( R  Or  X  /\  ( B  e.  X  /\  C  e.  X
) )  ->  if ( B R C ,  B ,  C )
( R  u.  _I  ) B )
1210, 4, 5, 11syl12anc 1180 . . . . 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 5078 . . . . . 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 418 . . . . 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 1183 . . . 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 1132 . . . . 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 5081 . . . . . 6  |-  ( ( R  Or  X  /\  ( B  e.  X  /\  C  e.  X
) )  ->  if ( B R C ,  B ,  C )
( R  u.  _I  ) C )
1810, 4, 5, 17syl12anc 1180 . . . . 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 5078 . . . . . 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 418 . . . . 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 1183 . . . 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 518 . . 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 423 . 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 4027 . . 3  |-  ( B  =  if ( B R C ,  B ,  C )  ->  ( A R B  <->  A R if ( B R C ,  B ,  C
) ) )
25 breq2 4027 . . 3  |-  ( C  =  if ( B R C ,  B ,  C )  ->  ( A R C  <->  A R if ( B R C ,  B ,  C
) ) )
2624, 25ifboth 3596 . 2  |-  ( ( A R B  /\  A R C )  ->  A R if ( B R C ,  B ,  C ) )
2723, 26impbid1 194 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684    u. cun 3150   ifcif 3565   class class class wbr 4023    _I cid 4304    Po wpo 4312    Or wor 4313
This theorem is referenced by:  wemaplem2  7262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696
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