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Theorem somincom 5403
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 3945 . . . 4  |-  ( A R B  ->  if ( A R B ,  A ,  B )  =  A )
21adantl 466 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  if ( A R B ,  A ,  B )  =  A )
3 so2nr 4824 . . . . . 6  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  -.  ( A R B  /\  B R A ) )
4 nan 580 . . . . . 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 208 . . . . 5  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  -.  B R A )
6 iffalse 3948 . . . . 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 2475 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  A  =  if ( B R A ,  B ,  A ) )
92, 8eqtrd 2508 . 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 3948 . . . 4  |-  ( -.  A R B  ->  if ( A R B ,  A ,  B
)  =  B )
1110adantl 466 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  -.  A R B )  ->  if ( A R B ,  A ,  B
)  =  B )
12 sotric 4826 . . . . . 6  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( A R B  <->  -.  ( A  =  B  \/  B R A ) ) )
1312con2bid 329 . . . . 5  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( A  =  B  \/  B R A )  <->  -.  A R B ) )
14 ifeq2 3944 . . . . . . 7  |-  ( A  =  B  ->  if ( B R A ,  B ,  A )  =  if ( B R A ,  B ,  B ) )
15 ifid 3976 . . . . . . 7  |-  if ( B R A ,  B ,  B )  =  B
1614, 15syl6req 2525 . . . . . 6  |-  ( A  =  B  ->  B  =  if ( B R A ,  B ,  A ) )
17 iftrue 3945 . . . . . . 7  |-  ( B R A  ->  if ( B R A ,  B ,  A )  =  B )
1817eqcomd 2475 . . . . . 6  |-  ( B R A  ->  B  =  if ( B R A ,  B ,  A ) )
1916, 18jaoi 379 . . . . 5  |-  ( ( A  =  B  \/  B R A )  ->  B  =  if ( B R A ,  B ,  A ) )
2013, 19syl6bir 229 . . . 4  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( -.  A R B  ->  B  =  if ( B R A ,  B ,  A ) ) )
2120imp 429 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  -.  A R B )  ->  B  =  if ( B R A ,  B ,  A ) )
2211, 21eqtrd 2508 . 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 789 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   ifcif 3939   class class class wbr 4447    Or wor 4799
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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
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-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-po 4800  df-so 4801
This theorem is referenced by:  somin2  5404
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