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Theorem tsrlemax 15388
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 4294 . . 3  |-  ( C  =  if ( B R C ,  C ,  B )  ->  ( A R C  <->  A R if ( B R C ,  C ,  B
) ) )
21bibi1d 319 . 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 4294 . . 3  |-  ( B  =  if ( B R C ,  C ,  B )  ->  ( A R B  <->  A R if ( B R C ,  C ,  B
) ) )
43bibi1d 319 . 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 384 . . 3  |-  ( A R C  ->  ( A R B  \/  A R C ) )
6 eqid 2441 . . . . . . . . . 10  |-  dom  R  =  dom  R
76istsr 15385 . . . . . . . . 9  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( dom  R  X.  dom  R )  C_  ( R  u.  `' R ) ) )
87simplbi 460 . . . . . . . 8  |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )
9 pstr 15379 . . . . . . . . 9  |-  ( ( R  e.  PosetRel  /\  A R B  /\  B R C )  ->  A R C )
1093expib 1190 . . . . . . . 8  |-  ( R  e.  PosetRel  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
118, 10syl 16 . . . . . . 7  |-  ( R  e.  TosetRel  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
1211adantr 465 . . . . . 6  |-  ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
1312expdimp 437 . . . . 5  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R B )  ->  ( B R C  ->  A R C ) )
1413impancom 440 . . . 4  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  B R C )  ->  ( A R B  ->  A R C ) )
15 idd 24 . . . 4  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  B R C )  ->  ( A R C  ->  A R C ) )
1614, 15jaod 380 . . 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 204 . 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 385 . . 3  |-  ( A R B  ->  ( A R B  \/  A R C ) )
19 idd 24 . . . 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 15387 . . . . . . 7  |-  ( ( R  e.  TosetRel  /\  B  e.  X  /\  C  e.  X )  ->  ( B R C  \/  C R B ) )
22213adant3r1 1196 . . . . . 6  |-  ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B R C  \/  C R B ) )
2322orcanai 904 . . . . 5  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  -.  B R C )  ->  C R B )
24 pstr 15379 . . . . . . . . . 10  |-  ( ( R  e.  PosetRel  /\  A R C  /\  C R B )  ->  A R B )
25243expib 1190 . . . . . . . . 9  |-  ( R  e.  PosetRel  ->  ( ( A R C  /\  C R B )  ->  A R B ) )
268, 25syl 16 . . . . . . . 8  |-  ( R  e.  TosetRel  ->  ( ( A R C  /\  C R B )  ->  A R B ) )
2726adantr 465 . . . . . . 7  |-  ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A R C  /\  C R B )  ->  A R B ) )
2827expdimp 437 . . . . . 6  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R C )  ->  ( C R B  ->  A R B ) )
2928impancom 440 . . . . 5  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  C R B )  ->  ( A R C  ->  A R B ) )
3023, 29syldan 470 . . . 4  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  -.  B R C )  -> 
( A R C  ->  A R B ) )
3119, 30jaod 380 . . 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 204 . 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 3822 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    u. cun 3324    C_ wss 3326   ifcif 3789   class class class wbr 4290    X. cxp 4836   `'ccnv 4837   dom cdm 4838   PosetRelcps 15366    TosetRel ctsr 15367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-res 4850  df-ps 15368  df-tsr 15369
This theorem is referenced by:  ordtbaslem  18790
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