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Theorem tsrlemax 14378
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 4064 . . 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 4064 . . 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 2316 . . . . . . . . . 10  |-  dom  R  =  dom  R
76istsr 14375 . . . . . . . . 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 14369 . . . . . . . . 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 14377 . . . . . . 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 14369 . . . . . . . . . 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 3629 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 1633    e. wcel 1701    u. cun 3184    C_ wss 3186   ifcif 3599   class class class wbr 4060    X. cxp 4724   `'ccnv 4725   dom cdm 4726   PosetRelcps 14350    TosetRel ctsr 14351
This theorem is referenced by:  ordtbaslem  16974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-res 4738  df-ps 14355  df-tsr 14356
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