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Theorem tsrlemax 14329
Description: Two ways of saying a number is less than or equal to the maximum of two others. (Contributed by Mario Carneiro, 9-Sep-2015.)
Hypothesis
Ref Expression
istsr.1  |-  X  =  dom  R
Assertion
Ref Expression
tsrlemax  |-  ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A R if ( B R C ,  C ,  B )  <->  ( A R B  \/  A R C ) ) )

Proof of Theorem tsrlemax
StepHypRef Expression
1 breq2 4027 . . 3  |-  ( C  =  if ( B R C ,  C ,  B )  ->  ( A R C  <->  A R if ( B R C ,  C ,  B
) ) )
21bibi1d 310 . 2  |-  ( C  =  if ( B R C ,  C ,  B )  ->  (
( A R C  <-> 
( A R B  \/  A R C ) )  <->  ( A R if ( B R C ,  C ,  B )  <->  ( A R B  \/  A R C ) ) ) )
3 breq2 4027 . . 3  |-  ( B  =  if ( B R C ,  C ,  B )  ->  ( A R B  <->  A R if ( B R C ,  C ,  B
) ) )
43bibi1d 310 . 2  |-  ( B  =  if ( B R C ,  C ,  B )  ->  (
( A R B  <-> 
( A R B  \/  A R C ) )  <->  ( A R if ( B R C ,  C ,  B )  <->  ( A R B  \/  A R C ) ) ) )
5 olc 373 . . 3  |-  ( A R C  ->  ( A R B  \/  A R C ) )
6 eqid 2283 . . . . . . . . . 10  |-  dom  R  =  dom  R
76istsr 14326 . . . . . . . . 9  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( dom  R  X.  dom  R )  C_  ( R  u.  `' R ) ) )
87simplbi 446 . . . . . . . 8  |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )
9 pstr 14320 . . . . . . . . 9  |-  ( ( R  e.  PosetRel  /\  A R B  /\  B R C )  ->  A R C )
1093expib 1154 . . . . . . . 8  |-  ( R  e.  PosetRel  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
118, 10syl 15 . . . . . . 7  |-  ( R  e.  TosetRel  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
1211adantr 451 . . . . . 6  |-  ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
1312expdimp 426 . . . . 5  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R B )  ->  ( B R C  ->  A R C ) )
1413impancom 427 . . . 4  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  B R C )  ->  ( A R B  ->  A R C ) )
15 idd 21 . . . 4  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  B R C )  ->  ( A R C  ->  A R C ) )
1614, 15jaod 369 . . 3  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  B R C )  ->  (
( A R B  \/  A R C )  ->  A R C ) )
175, 16impbid2 195 . 2  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  B R C )  ->  ( A R C  <->  ( A R B  \/  A R C ) ) )
18 orc 374 . . 3  |-  ( A R B  ->  ( A R B  \/  A R C ) )
19 idd 21 . . . 4  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  -.  B R C )  -> 
( A R B  ->  A R B ) )
20 istsr.1 . . . . . . . 8  |-  X  =  dom  R
2120tsrlin 14328 . . . . . . 7  |-  ( ( R  e.  TosetRel  /\  B  e.  X  /\  C  e.  X )  ->  ( B R C  \/  C R B ) )
22213adant3r1 1160 . . . . . 6  |-  ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B R C  \/  C R B ) )
2322orcanai 879 . . . . 5  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  -.  B R C )  ->  C R B )
24 pstr 14320 . . . . . . . . . 10  |-  ( ( R  e.  PosetRel  /\  A R C  /\  C R B )  ->  A R B )
25243expib 1154 . . . . . . . . 9  |-  ( R  e.  PosetRel  ->  ( ( A R C  /\  C R B )  ->  A R B ) )
268, 25syl 15 . . . . . . . 8  |-  ( R  e.  TosetRel  ->  ( ( A R C  /\  C R B )  ->  A R B ) )
2726adantr 451 . . . . . . 7  |-  ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A R C  /\  C R B )  ->  A R B ) )
2827expdimp 426 . . . . . 6  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R C )  ->  ( C R B  ->  A R B ) )
2928impancom 427 . . . . 5  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  C R B )  ->  ( A R C  ->  A R B ) )
3023, 29syldan 456 . . . 4  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  -.  B R C )  -> 
( A R C  ->  A R B ) )
3119, 30jaod 369 . . 3  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  -.  B R C )  -> 
( ( A R B  \/  A R C )  ->  A R B ) )
3218, 31impbid2 195 . 2  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  -.  B R C )  -> 
( A R B  <-> 
( A R B  \/  A R C ) ) )
332, 4, 17, 32ifbothda 3595 1  |-  ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A R if ( B R C ,  C ,  B )  <->  ( A R B  \/  A R C ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    u. cun 3150    C_ wss 3152   ifcif 3565   class class class wbr 4023    X. cxp 4687   `'ccnv 4688   dom cdm 4689   PosetRelcps 14301    TosetRel ctsr 14302
This theorem is referenced by:  ordtbaslem  16918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-ps 14306  df-tsr 14307
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