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Theorem suppr 7399
Description: The supremum of a pair. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
suppr  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  sup ( { B ,  C } ,  A ,  R )  =  if ( C R B ,  B ,  C
) )

Proof of Theorem suppr
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simp1 957 . 2  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  R  Or  A )
2 ifcl 3711 . . 3  |-  ( ( B  e.  A  /\  C  e.  A )  ->  if ( C R B ,  B ,  C )  e.  A
)
323adant1 975 . 2  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  if ( C R B ,  B ,  C )  e.  A
)
4 ifpr 3792 . . 3  |-  ( ( B  e.  A  /\  C  e.  A )  ->  if ( C R B ,  B ,  C )  e.  { B ,  C }
)
543adant1 975 . 2  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  if ( C R B ,  B ,  C )  e.  { B ,  C }
)
6 breq1 4149 . . . . . 6  |-  ( B  =  if ( C R B ,  B ,  C )  ->  ( B R B  <->  if ( C R B ,  B ,  C ) R B ) )
76notbid 286 . . . . 5  |-  ( B  =  if ( C R B ,  B ,  C )  ->  ( -.  B R B  <->  -.  if ( C R B ,  B ,  C ) R B ) )
8 breq1 4149 . . . . . 6  |-  ( C  =  if ( C R B ,  B ,  C )  ->  ( C R B  <->  if ( C R B ,  B ,  C ) R B ) )
98notbid 286 . . . . 5  |-  ( C  =  if ( C R B ,  B ,  C )  ->  ( -.  C R B  <->  -.  if ( C R B ,  B ,  C ) R B ) )
10 sonr 4458 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
11103adant3 977 . . . . . 6  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  B R B )
1211adantr 452 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  C R B )  ->  -.  B R B )
13 simpr 448 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  -.  C R B )  ->  -.  C R B )
147, 9, 12, 13ifbothda 3705 . . . 4  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  if ( C R B ,  B ,  C ) R B )
15 breq1 4149 . . . . . 6  |-  ( B  =  if ( C R B ,  B ,  C )  ->  ( B R C  <->  if ( C R B ,  B ,  C ) R C ) )
1615notbid 286 . . . . 5  |-  ( B  =  if ( C R B ,  B ,  C )  ->  ( -.  B R C  <->  -.  if ( C R B ,  B ,  C ) R C ) )
17 breq1 4149 . . . . . 6  |-  ( C  =  if ( C R B ,  B ,  C )  ->  ( C R C  <->  if ( C R B ,  B ,  C ) R C ) )
1817notbid 286 . . . . 5  |-  ( C  =  if ( C R B ,  B ,  C )  ->  ( -.  C R C  <->  -.  if ( C R B ,  B ,  C ) R C ) )
19 so2nr 4461 . . . . . . . . 9  |-  ( ( R  Or  A  /\  ( C  e.  A  /\  B  e.  A
) )  ->  -.  ( C R B  /\  B R C ) )
20193impb 1149 . . . . . . . 8  |-  ( ( R  Or  A  /\  C  e.  A  /\  B  e.  A )  ->  -.  ( C R B  /\  B R C ) )
21203com23 1159 . . . . . . 7  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  ( C R B  /\  B R C ) )
22 imnan 412 . . . . . . 7  |-  ( ( C R B  ->  -.  B R C )  <->  -.  ( C R B  /\  B R C ) )
2321, 22sylibr 204 . . . . . 6  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  ( C R B  ->  -.  B R C ) )
2423imp 419 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  C R B )  ->  -.  B R C )
25 sonr 4458 . . . . . . 7  |-  ( ( R  Or  A  /\  C  e.  A )  ->  -.  C R C )
26253adant2 976 . . . . . 6  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  C R C )
2726adantr 452 . . . . 5  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  -.  C R B )  ->  -.  C R C )
2816, 18, 24, 27ifbothda 3705 . . . 4  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  -.  if ( C R B ,  B ,  C ) R C )
29 breq2 4150 . . . . . . 7  |-  ( y  =  B  ->  ( if ( C R B ,  B ,  C
) R y  <->  if ( C R B ,  B ,  C ) R B ) )
3029notbid 286 . . . . . 6  |-  ( y  =  B  ->  ( -.  if ( C R B ,  B ,  C ) R y  <->  -.  if ( C R B ,  B ,  C ) R B ) )
31 breq2 4150 . . . . . . 7  |-  ( y  =  C  ->  ( if ( C R B ,  B ,  C
) R y  <->  if ( C R B ,  B ,  C ) R C ) )
3231notbid 286 . . . . . 6  |-  ( y  =  C  ->  ( -.  if ( C R B ,  B ,  C ) R y  <->  -.  if ( C R B ,  B ,  C ) R C ) )
3330, 32ralprg 3793 . . . . 5  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( A. y  e. 
{ B ,  C }  -.  if ( C R B ,  B ,  C ) R y  <-> 
( -.  if ( C R B ,  B ,  C ) R B  /\  -.  if ( C R B ,  B ,  C ) R C ) ) )
34333adant1 975 . . . 4  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  ( A. y  e. 
{ B ,  C }  -.  if ( C R B ,  B ,  C ) R y  <-> 
( -.  if ( C R B ,  B ,  C ) R B  /\  -.  if ( C R B ,  B ,  C ) R C ) ) )
3514, 28, 34mpbir2and 889 . . 3  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  A. y  e.  { B ,  C }  -.  if ( C R B ,  B ,  C ) R y )
3635r19.21bi 2740 . 2  |-  ( ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A
)  /\  y  e.  { B ,  C }
)  ->  -.  if ( C R B ,  B ,  C ) R y )
371, 3, 5, 36supmax 7396 1  |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A )  ->  sup ( { B ,  C } ,  A ,  R )  =  if ( C R B ,  B ,  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2642   ifcif 3675   {cpr 3751   class class class wbr 4146    Or wor 4436   supcsup 7373
This theorem is referenced by:  supsn  7400  tmsxpsval2  18452  esumsn  24245
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-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  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-uni 3951  df-br 4147  df-po 4437  df-so 4438  df-iota 5351  df-riota 6478  df-sup 7374
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