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Theorem infpr 8409
Description: The infimum of a pair. (Contributed by AV, 4-Sep-2020.)
Assertion
Ref Expression
infpr |- ( ( R Or A /\ B e. A /\ C e. A ) -> inf ( { B , C } , A , R ) = if ( B R C , B , C ) )
Proof of Theorem infpr
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 ( B R C , B , C ) e. A )
323adant1 1079 . 2 |- ( ( R Or A /\ B e. A /\ C e. A ) -> if ( B R C , B , C ) e. A )
4 ifpr 4233 . . 3 |- ( ( B e. A /\ C e. A ) -> if ( B R C , B , C ) e. { B , C } )
543adant1 1079 . 2 |- ( ( R Or A /\ B e. A /\ C e. A ) -> if ( B R C , B , C ) e. { B , C } )
6 breq2 4657 . . . . . 6 |- ( B = if ( B R C , B , C ) -> ( B R B <-> B R if ( B R C , B , C ) ) )
76notbid 308 . . . . 5 |- ( B = if ( B R C , B , C ) -> ( -. B R B <-> -. B R if ( B R C , B , C ) ) )
8 breq2 4657 . . . . . 6 |- ( C = if ( B R C , B , C ) -> ( B R C <-> B R if ( B R C , B , C ) ) )
98notbid 308 . . . . 5 |- ( C = if ( B R C , B , C ) -> ( -. B R C <-> -. B R if ( B R C , B , C ) ) )
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 ) /\ B R C ) -> -. B R B )
13 simpr 477 . . . . 5 |- ( ( ( R Or A /\ B e. A /\ C e. A ) /\ -. B R C ) -> -. B R C )
147, 9, 12, 13ifbothda 4123 . . . 4 |- ( ( R Or A /\ B e. A /\ C e. A ) -> -. B R if ( B R C , B , C ) )
15 breq2 4657 . . . . . 6 |- ( B = if ( B R C , B , C ) -> ( C R B <-> C R if ( B R C , B , C ) ) )
1615notbid 308 . . . . 5 |- ( B = if ( B R C , B , C ) -> ( -. C R B <-> -. C R if ( B R C , B , C ) ) )
17 breq2 4657 . . . . . 6 |- ( C = if ( B R C , B , C ) -> ( C R C <-> C R if ( B R C , B , C ) ) )
1817notbid 308 . . . . 5 |- ( C = if ( B R C , B , C ) -> ( -. C R C <-> -. C R if ( B R C , B , C ) ) )
19 so2nr 5059 . . . . . . . 8 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> -. ( B R C /\ C R B ) )
20193impb 1260 . . . . . . 7 |- ( ( R Or A /\ B e. A /\ C e. A ) -> -. ( B R C /\ C R B ) )
21 imnan 438 . . . . . . 7 |- ( ( B R C -> -. C R B ) <-> -. ( B R C /\ C R B ) )
2220, 21sylibr 224 . . . . . 6 |- ( ( R Or A /\ B e. A /\ C e. A ) -> ( B R C -> -. C R B ) )
2322imp 445 . . . . 5 |- ( ( ( R Or A /\ B e. A /\ C e. A ) /\ B R C ) -> -. C R B )
24 sonr 5056 . . . . . . 7 |- ( ( R Or A /\ C e. A ) -> -. C R C )
25243adant2 1080 . . . . . 6 |- ( ( R Or A /\ B e. A /\ C e. A ) -> -. C R C )
2625adantr 481 . . . . 5 |- ( ( ( R Or A /\ B e. A /\ C e. A ) /\ -. B R C ) -> -. C R C )
2716, 18, 23, 26ifbothda 4123 . . . 4 |- ( ( R Or A /\ B e. A /\ C e. A ) -> -. C R if ( B R C , B , C ) )
28 breq1 4656 . . . . . . 7 |- ( y = B -> ( y R if ( B R C , B , C ) <-> B R if ( B R C , B , C ) ) )
2928notbid 308 . . . . . 6 |- ( y = B -> ( -. y R if ( B R C , B , C ) <-> -. B R if ( B R C , B , C ) ) )
30 breq1 4656 . . . . . . 7 |- ( y = C -> ( y R if ( B R C , B , C ) <-> C R if ( B R C , B , C ) ) )
3130notbid 308 . . . . . 6 |- ( y = C -> ( -. y R if ( B R C , B , C ) <-> -. C R if ( B R C , B , C ) ) )
3229, 31ralprg 4234 . . . . 5 |- ( ( B e. A /\ C e. A ) -> ( A. y e. { B , C } -. y R if ( B R C , B , C ) <-> ( -. B R if ( B R C , B , C ) /\ -. C R if ( B R C , B , C ) ) ) )
33323adant1 1079 . . . 4 |- ( ( R Or A /\ B e. A /\ C e. A ) -> ( A. y e. { B , C } -. y R if ( B R C , B , C ) <-> ( -. B R if ( B R C , B , C ) /\ -. C R if ( B R C , B , C ) ) ) )
3414, 27, 33mpbir2and 957 . . 3 |- ( ( R Or A /\ B e. A /\ C e. A ) -> A. y e. { B , C } -. y R if ( B R C , B , C ) )
3534r19.21bi 2932 . 2 |- ( ( ( R Or A /\ B e. A /\ C e. A ) /\ y e. { B , C } ) -> -. y R if ( B R C , B , C ) )
361, 3, 5, 35infmin 8400 1 |- ( ( R Or A /\ B e. A /\ C e. A ) -> inf ( { B , C } , A , R ) = if ( B R C , 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  infcinf 8347
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  ax-sep 4781  ax-nul 4789  ax-pr 4906
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-opab 4713  df-po 5035  df-so 5036  df-cnv 5122  df-iota 5851  df-riota 6611  df-sup 8348  df-inf 8349
This theorem is referenced by:  infsn  8410  liminf10ex  40006
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