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Theorem soltmin 5276
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 4523 . . . . . 6  |-  ( R  Or  X  ->  R  Po  X )
21ad2antrr 708 . . . . 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 1000 . . . . . 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 1001 . . . . . . 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 1002 . . . . . . 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 3777 . . . . . . 7  |-  ( ( B  e.  X  /\  C  e.  X )  ->  if ( B R C ,  B ,  C )  e.  X
)
74, 5, 6syl2anc 644 . . . . . 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 1135 . . . . 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 449 . . . . 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 732 . . . . . 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 5273 . . . . . 6  |-  ( ( R  Or  X  /\  ( B  e.  X  /\  C  e.  X
) )  ->  if ( B R C ,  B ,  C )
( R  u.  _I  ) B )
1210, 4, 5, 11syl12anc 1183 . . . . 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 5272 . . . . . 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 420 . . . . 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 1186 . . . 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 1135 . . . . 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 5275 . . . . . 6  |-  ( ( R  Or  X  /\  ( B  e.  X  /\  C  e.  X
) )  ->  if ( B R C ,  B ,  C )
( R  u.  _I  ) C )
1810, 4, 5, 17syl12anc 1183 . . . . 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 5272 . . . . . 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 420 . . . . 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 1186 . . . 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 520 . . 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 425 . 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 4219 . . 3  |-  ( B  =  if ( B R C ,  B ,  C )  ->  ( A R B  <->  A R if ( B R C ,  B ,  C
) ) )
25 breq2 4219 . . 3  |-  ( C  =  if ( B R C ,  B ,  C )  ->  ( A R C  <->  A R if ( B R C ,  B ,  C
) ) )
2624, 25ifboth 3772 . 2  |-  ( ( A R B  /\  A R C )  ->  A R if ( B R C ,  B ,  C ) )
2723, 26impbid1 196 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 178    /\ wa 360    /\ w3a 937    e. wcel 1726    u. cun 3320   ifcif 3741   class class class wbr 4215    _I cid 4496    Po wpo 4504    Or wor 4505
This theorem is referenced by:  wemaplem2  7519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888
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