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Theorem somincom 5082
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 3573 . . . 4  |-  ( A R B  ->  if ( A R B ,  A ,  B )  =  A )
21adantl 452 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  if ( A R B ,  A ,  B )  =  A )
3 so2nr 4340 . . . . . 6  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  -.  ( A R B  /\  B R A ) )
4 nan 563 . . . . . 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 199 . . . . 5  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  -.  B R A )
6 iffalse 3574 . . . . 5  |-  ( -.  B R A  ->  if ( B R A ,  B ,  A
)  =  A )
75, 6syl 15 . . . 4  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  if ( B R A ,  B ,  A )  =  A )
87eqcomd 2290 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  A  =  if ( B R A ,  B ,  A ) )
92, 8eqtrd 2317 . 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 3574 . . . 4  |-  ( -.  A R B  ->  if ( A R B ,  A ,  B
)  =  B )
1110adantl 452 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  -.  A R B )  ->  if ( A R B ,  A ,  B
)  =  B )
12 sotric 4342 . . . . . 6  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( A R B  <->  -.  ( A  =  B  \/  B R A ) ) )
1312con2bid 319 . . . . 5  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( A  =  B  \/  B R A )  <->  -.  A R B ) )
14 ifeq2 3572 . . . . . . 7  |-  ( A  =  B  ->  if ( B R A ,  B ,  A )  =  if ( B R A ,  B ,  B ) )
15 ifid 3599 . . . . . . 7  |-  if ( B R A ,  B ,  B )  =  B
1614, 15syl6req 2334 . . . . . 6  |-  ( A  =  B  ->  B  =  if ( B R A ,  B ,  A ) )
17 iftrue 3573 . . . . . . 7  |-  ( B R A  ->  if ( B R A ,  B ,  A )  =  B )
1817eqcomd 2290 . . . . . 6  |-  ( B R A  ->  B  =  if ( B R A ,  B ,  A ) )
1916, 18jaoi 368 . . . . 5  |-  ( ( A  =  B  \/  B R A )  ->  B  =  if ( B R A ,  B ,  A ) )
2013, 19syl6bir 220 . . . 4  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( -.  A R B  ->  B  =  if ( B R A ,  B ,  A ) ) )
2120imp 418 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  -.  A R B )  ->  B  =  if ( B R A ,  B ,  A ) )
2211, 21eqtrd 2317 . 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 766 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 357    /\ wa 358    = wceq 1625    e. wcel 1686   ifcif 3567   class class class wbr 4025    Or wor 4315
This theorem is referenced by:  somin2  5083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
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 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ral 2550  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-po 4316  df-so 4317
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