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Theorem somincom 5271
Description: Commutativity of minimum in a total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
somincom  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  if ( A R B ,  A ,  B )  =  if ( B R A ,  B ,  A ) )

Proof of Theorem somincom
StepHypRef Expression
1 iftrue 3745 . . . 4  |-  ( A R B  ->  if ( A R B ,  A ,  B )  =  A )
21adantl 453 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  if ( A R B ,  A ,  B )  =  A )
3 so2nr 4527 . . . . . 6  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  -.  ( A R B  /\  B R A ) )
4 nan 564 . . . . . 6  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  -.  ( A R B  /\  B R A ) )  <-> 
( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X )
)  /\  A R B )  ->  -.  B R A ) )
53, 4mpbi 200 . . . . 5  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  -.  B R A )
6 iffalse 3746 . . . . 5  |-  ( -.  B R A  ->  if ( B R A ,  B ,  A
)  =  A )
75, 6syl 16 . . . 4  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  if ( B R A ,  B ,  A )  =  A )
87eqcomd 2441 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  A  =  if ( B R A ,  B ,  A ) )
92, 8eqtrd 2468 . 2  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  if ( A R B ,  A ,  B )  =  if ( B R A ,  B ,  A ) )
10 iffalse 3746 . . . 4  |-  ( -.  A R B  ->  if ( A R B ,  A ,  B
)  =  B )
1110adantl 453 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  -.  A R B )  ->  if ( A R B ,  A ,  B
)  =  B )
12 sotric 4529 . . . . . 6  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( A R B  <->  -.  ( A  =  B  \/  B R A ) ) )
1312con2bid 320 . . . . 5  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( A  =  B  \/  B R A )  <->  -.  A R B ) )
14 ifeq2 3744 . . . . . . 7  |-  ( A  =  B  ->  if ( B R A ,  B ,  A )  =  if ( B R A ,  B ,  B ) )
15 ifid 3771 . . . . . . 7  |-  if ( B R A ,  B ,  B )  =  B
1614, 15syl6req 2485 . . . . . 6  |-  ( A  =  B  ->  B  =  if ( B R A ,  B ,  A ) )
17 iftrue 3745 . . . . . . 7  |-  ( B R A  ->  if ( B R A ,  B ,  A )  =  B )
1817eqcomd 2441 . . . . . 6  |-  ( B R A  ->  B  =  if ( B R A ,  B ,  A ) )
1916, 18jaoi 369 . . . . 5  |-  ( ( A  =  B  \/  B R A )  ->  B  =  if ( B R A ,  B ,  A ) )
2013, 19syl6bir 221 . . . 4  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( -.  A R B  ->  B  =  if ( B R A ,  B ,  A ) ) )
2120imp 419 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  -.  A R B )  ->  B  =  if ( B R A ,  B ,  A ) )
2211, 21eqtrd 2468 . 2  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  -.  A R B )  ->  if ( A R B ,  A ,  B
)  =  if ( B R A ,  B ,  A )
)
239, 22pm2.61dan 767 1  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  if ( A R B ,  A ,  B )  =  if ( B R A ,  B ,  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   ifcif 3739   class class class wbr 4212    Or wor 4502
This theorem is referenced by:  somin2  5272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-po 4503  df-so 4504
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