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Theorem suppr 8377
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 1061 . 2 |- ( ( R Or A /\ B e. A /\ C e. A ) -> R Or A )
2 ifcl 4130 . . 3 |- ( ( B e. A /\ C e. A ) -> if ( C R B , B , C ) e. A )
323adant1 1079 . 2 |- ( ( R Or A /\ B e. A /\ C e. A ) -> if ( C R B , B , C ) e. A )
4 ifpr 4233 . . 3 |- ( ( B e. A /\ C e. A ) -> if ( C R B , B , C ) e. { B , C } )
543adant1 1079 . 2 |- ( ( R Or A /\ B e. A /\ C e. A ) -> if ( C R B , B , C ) e. { B , C } )
6 breq1 4656 . . . . . 6 |- ( B = if ( C R B , B , C ) -> ( B R B <-> if ( C R B , B , C ) R B ) )
76notbid 308 . . . . 5 |- ( B = if ( C R B , B , C ) -> ( -. B R B <-> -. if ( C R B , B , C ) R B ) )
8 breq1 4656 . . . . . 6 |- ( C = if ( C R B , B , C ) -> ( C R B <-> if ( C R B , B , C ) R B ) )
98notbid 308 . . . . 5 |- ( C = if ( C R B , B , C ) -> ( -. C R B <-> -. if ( C R B , B , C ) R B ) )
10 sonr 5056 . . . . . . 7 |- ( ( R Or A /\ B e. A ) -> -. B R B )
11103adant3 1081 . . . . . 6 |- ( ( R Or A /\ B e. A /\ C e. A ) -> -. B R B )
1211adantr 481 . . . . 5 |- ( ( ( R Or A /\ B e. A /\ C e. A ) /\ C R B ) -> -. B R B )
13 simpr 477 . . . . 5 |- ( ( ( R Or A /\ B e. A /\ C e. A ) /\ -. C R B ) -> -. C R B )
147, 9, 12, 13ifbothda 4123 . . . 4 |- ( ( R Or A /\ B e. A /\ C e. A ) -> -. if ( C R B , B , C ) R B )
15 breq1 4656 . . . . . 6 |- ( B = if ( C R B , B , C ) -> ( B R C <-> if ( C R B , B , C ) R C ) )
1615notbid 308 . . . . 5 |- ( B = if ( C R B , B , C ) -> ( -. B R C <-> -. if ( C R B , B , C ) R C ) )
17 breq1 4656 . . . . . 6 |- ( C = if ( C R B , B , C ) -> ( C R C <-> if ( C R B , B , C ) R C ) )
1817notbid 308 . . . . 5 |- ( C = if ( C R B , B , C ) -> ( -. C R C <-> -. if ( C R B , B , C ) R C ) )
19 so2nr 5059 . . . . . . . . 9 |- ( ( R Or A /\ ( C e. A /\ B e. A ) ) -> -. ( C R B /\ B R C ) )
20193impb 1260 . . . . . . . 8 |- ( ( R Or A /\ C e. A /\ B e. A ) -> -. ( C R B /\ B R C ) )
21203com23 1271 . . . . . . 7 |- ( ( R Or A /\ B e. A /\ C e. A ) -> -. ( C R B /\ B R C ) )
22 imnan 438 . . . . . . 7 |- ( ( C R B -> -. B R C ) <-> -. ( C R B /\ B R C ) )
2321, 22sylibr 224 . . . . . 6 |- ( ( R Or A /\ B e. A /\ C e. A ) -> ( C R B -> -. B R C ) )
2423imp 445 . . . . 5 |- ( ( ( R Or A /\ B e. A /\ C e. A ) /\ C R B ) -> -. B R C )
25 sonr 5056 . . . . . . 7 |- ( ( R Or A /\ C e. A ) -> -. C R C )
26253adant2 1080 . . . . . 6 |- ( ( R Or A /\ B e. A /\ C e. A ) -> -. C R C )
2726adantr 481 . . . . 5 |- ( ( ( R Or A /\ B e. A /\ C e. A ) /\ -. C R B ) -> -. C R C )
2816, 18, 24, 27ifbothda 4123 . . . 4 |- ( ( R Or A /\ B e. A /\ C e. A ) -> -. if ( C R B , B , C ) R C )
29 breq2 4657 . . . . . . 7 |- ( y = B -> ( if ( C R B , B , C ) R y <-> if ( C R B , B , C ) R B ) )
3029notbid 308 . . . . . 6 |- ( y = B -> ( -. if ( C R B , B , C ) R y <-> -. if ( C R B , B , C ) R B ) )
31 breq2 4657 . . . . . . 7 |- ( y = C -> ( if ( C R B , B , C ) R y <-> if ( C R B , B , C ) R C ) )
3231notbid 308 . . . . . 6 |- ( y = C -> ( -. if ( C R B , B , C ) R y <-> -. if ( C R B , B , C ) R C ) )
3330, 32ralprg 4234 . . . . 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 1079 . . . 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 957 . . 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 2932 . 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 8373 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 setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ifcif 4086   {cpr 4179   class class class wbr 4653   Or wor 5034   supcsup 8346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-po 5035  df-so 5036  df-iota 5851  df-riota 6611  df-sup 8348
This theorem is referenced by:  supsn  8378  2resupmax  12019  tmsxpsval2  22344  esumsnf  30126  limsup10ex  40005  sge0sn  40596
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