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Theorem somincom 5204
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 3681 . . . 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 4461 . . . . . 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 3682 . . . . 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 2385 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  A  =  if ( B R A ,  B ,  A ) )
92, 8eqtrd 2412 . 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 3682 . . . 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 4463 . . . . . 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 3680 . . . . . . 7  |-  ( A  =  B  ->  if ( B R A ,  B ,  A )  =  if ( B R A ,  B ,  B ) )
15 ifid 3707 . . . . . . 7  |-  if ( B R A ,  B ,  B )  =  B
1614, 15syl6req 2429 . . . . . 6  |-  ( A  =  B  ->  B  =  if ( B R A ,  B ,  A ) )
17 iftrue 3681 . . . . . . 7  |-  ( B R A  ->  if ( B R A ,  B ,  A )  =  B )
1817eqcomd 2385 . . . . . 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 2412 . 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 1649    e. wcel 1717   ifcif 3675   class class class wbr 4146    Or wor 4436
This theorem is referenced by:  somin2  5205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-po 4437  df-so 4438
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