MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  somincom Structured version   Unicode version

Theorem somincom 5411
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 so2nr 4833 . . . . 5  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  -.  ( A R B  /\  B R A ) )
2 nan 580 . . . . 5  |-  ( ( ( 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 ) )
31, 2mpbi 208 . . . 4  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  -.  B R A )
43iffalsed 3955 . . 3  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  if ( B R A ,  B ,  A )  =  A )
54eqcomd 2465 . 2  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  A R B )  ->  A  =  if ( B R A ,  B ,  A ) )
6 sotric 4835 . . . . 5  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( A R B  <->  -.  ( A  =  B  \/  B R A ) ) )
76con2bid 329 . . . 4  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( A  =  B  \/  B R A )  <->  -.  A R B ) )
8 ifeq2 3949 . . . . . 6  |-  ( A  =  B  ->  if ( B R A ,  B ,  A )  =  if ( B R A ,  B ,  B ) )
9 ifid 3981 . . . . . 6  |-  if ( B R A ,  B ,  B )  =  B
108, 9syl6req 2515 . . . . 5  |-  ( A  =  B  ->  B  =  if ( B R A ,  B ,  A ) )
11 iftrue 3950 . . . . . 6  |-  ( B R A  ->  if ( B R A ,  B ,  A )  =  B )
1211eqcomd 2465 . . . . 5  |-  ( B R A  ->  B  =  if ( B R A ,  B ,  A ) )
1310, 12jaoi 379 . . . 4  |-  ( ( A  =  B  \/  B R A )  ->  B  =  if ( B R A ,  B ,  A ) )
147, 13syl6bir 229 . . 3  |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  ( -.  A R B  ->  B  =  if ( B R A ,  B ,  A ) ) )
1514imp 429 . 2  |-  ( ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X
) )  /\  -.  A R B )  ->  B  =  if ( B R A ,  B ,  A ) )
165, 15ifeqda 3977 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 1395    e. wcel 1819   ifcif 3944   class class class wbr 4456    Or wor 4808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-po 4809  df-so 4810
This theorem is referenced by:  somin2  5412
  Copyright terms: Public domain W3C validator