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Theorem soltmin 5532
Description: Being less than a minimum, for a general total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
soltmin |- ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A R if ( B R C , B , C ) <-> ( A R B /\ A R C ) ) )
Proof of Theorem soltmin
StepHypRef Expression
1 sopo 5052 . . . . . 6 |- ( R Or X -> R Po X )
21ad2antrr 762 . . . . 5 |- ( ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A R if ( B R C , B , C ) ) -> R Po X )
3 simplr1 1103 . . . . . 6 |- ( ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A R if ( B R C , B , C ) ) -> A e. X )
4 simplr2 1104 . . . . . . 7 |- ( ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A R if ( B R C , B , C ) ) -> B e. X )
5 simplr3 1105 . . . . . . 7 |- ( ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A R if ( B R C , B , C ) ) -> C e. X )
64, 5ifcld 4131 . . . . . 6 |- ( ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A R if ( B R C , B , C ) ) -> if ( B R C , B , C ) e. X )
73, 6, 43jca 1242 . . . . 5 |- ( ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A R if ( B R C , B , C ) ) -> ( A e. X /\ if ( B R C , B , C ) e. X /\ B e. X ) )
8 simpr 477 . . . . 5 |- ( ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A R if ( B R C , B , C ) ) -> A R if ( B R C , B , C ) )
9 simpll 790 . . . . . 6 |- ( ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A R if ( B R C , B , C ) ) -> R Or X )
10 somin1 5529 . . . . . 6 |- ( ( R Or X /\ ( B e. X /\ C e. X ) ) -> if ( B R C , B , C ) ( R u. _I ) B )
119, 4, 5, 10syl12anc 1324 . . . . 5 |- ( ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A R if ( B R C , B , C ) ) -> if ( B R C , B , C ) ( R u. _I ) B )
12 poltletr 5528 . . . . . 6 |- ( ( R Po X /\ ( A e. X /\ if ( B R C , B , C ) e. X /\ B e. X ) ) -> ( ( A R if ( B R C , B , C ) /\ if ( B R C , B , C ) ( R u. _I ) B ) -> A R B ) )
1312imp 445 . . . . 5 |- ( ( ( R Po X /\ ( A e. X /\ if ( B R C , B , C ) e. X /\ B e. X ) ) /\ ( A R if ( B R C , B , C ) /\ if ( B R C , B , C ) ( R u. _I ) B ) ) -> A R B )
142, 7, 8, 11, 13syl22anc 1327 . . . 4 |- ( ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A R if ( B R C , B , C ) ) -> A R B )
153, 6, 53jca 1242 . . . . 5 |- ( ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A R if ( B R C , B , C ) ) -> ( A e. X /\ if ( B R C , B , C ) e. X /\ C e. X ) )
16 somin2 5531 . . . . . 6 |- ( ( R Or X /\ ( B e. X /\ C e. X ) ) -> if ( B R C , B , C ) ( R u. _I ) C )
179, 4, 5, 16syl12anc 1324 . . . . 5 |- ( ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A R if ( B R C , B , C ) ) -> if ( B R C , B , C ) ( R u. _I ) C )
18 poltletr 5528 . . . . . 6 |- ( ( R Po X /\ ( A e. X /\ if ( B R C , B , C ) e. X /\ C e. X ) ) -> ( ( A R if ( B R C , B , C ) /\ if ( B R C , B , C ) ( R u. _I ) C ) -> A R C ) )
1918imp 445 . . . . 5 |- ( ( ( R Po X /\ ( A e. X /\ if ( B R C , B , C ) e. X /\ C e. X ) ) /\ ( A R if ( B R C , B , C ) /\ if ( B R C , B , C ) ( R u. _I ) C ) ) -> A R C )
202, 15, 8, 17, 19syl22anc 1327 . . . 4 |- ( ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A R if ( B R C , B , C ) ) -> A R C )
2114, 20jca 554 . . 3 |- ( ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A R if ( B R C , B , C ) ) -> ( A R B /\ A R C ) )
2221ex 450 . 2 |- ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A R if ( B R C , B , C ) -> ( A R B /\ A R C ) ) )
23 breq2 4657 . . 3 |- ( B = if ( B R C , B , C ) -> ( A R B <-> A R if ( B R C , B , C ) ) )
24 breq2 4657 . . 3 |- ( C = if ( B R C , B , C ) -> ( A R C <-> A R if ( B R C , B , C ) ) )
2523, 24ifboth 4124 . 2 |- ( ( A R B /\ A R C ) -> A R if ( B R C , B , C ) )
2622, 25impbid1 215 1 |- ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A R if ( B R C , B , C ) <-> ( A R B /\ A R C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   u. cun 3572   ifcif 4086   class class class wbr 4653   _I cid 5023   Po wpo 5033   Or wor 5034
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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121
This theorem is referenced by:  wemaplem2  8452
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