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Theorem tsrlemax 14654
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 4218 . . 3  |-  ( C  =  if ( B R C ,  C ,  B )  ->  ( A R C  <->  A R if ( B R C ,  C ,  B
) ) )
21bibi1d 312 . 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 4218 . . 3  |-  ( B  =  if ( B R C ,  C ,  B )  ->  ( A R B  <->  A R if ( B R C ,  C ,  B
) ) )
43bibi1d 312 . 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 375 . . 3  |-  ( A R C  ->  ( A R B  \/  A R C ) )
6 eqid 2438 . . . . . . . . . 10  |-  dom  R  =  dom  R
76istsr 14651 . . . . . . . . 9  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( dom  R  X.  dom  R )  C_  ( R  u.  `' R ) ) )
87simplbi 448 . . . . . . . 8  |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )
9 pstr 14645 . . . . . . . . 9  |-  ( ( R  e.  PosetRel  /\  A R B  /\  B R C )  ->  A R C )
1093expib 1157 . . . . . . . 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 453 . . . . . 6  |-  ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
1312expdimp 428 . . . . 5  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R B )  ->  ( B R C  ->  A R C ) )
1413impancom 429 . . . 4  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  B R C )  ->  ( A R B  ->  A R C ) )
15 idd 23 . . . 4  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  B R C )  ->  ( A R C  ->  A R C ) )
1614, 15jaod 371 . . 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 197 . 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 376 . . 3  |-  ( A R B  ->  ( A R B  \/  A R C ) )
19 idd 23 . . . 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 14653 . . . . . . 7  |-  ( ( R  e.  TosetRel  /\  B  e.  X  /\  C  e.  X )  ->  ( B R C  \/  C R B ) )
22213adant3r1 1163 . . . . . 6  |-  ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B R C  \/  C R B ) )
2322orcanai 881 . . . . 5  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  -.  B R C )  ->  C R B )
24 pstr 14645 . . . . . . . . . 10  |-  ( ( R  e.  PosetRel  /\  A R C  /\  C R B )  ->  A R B )
25243expib 1157 . . . . . . . . 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 453 . . . . . . 7  |-  ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A R C  /\  C R B )  ->  A R B ) )
2827expdimp 428 . . . . . 6  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A R C )  ->  ( C R B  ->  A R B ) )
2928impancom 429 . . . . 5  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  C R B )  ->  ( A R C  ->  A R B ) )
3023, 29syldan 458 . . . 4  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  -.  B R C )  -> 
( A R C  ->  A R B ) )
3119, 30jaod 371 . . 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 197 . 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 3771 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 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    u. cun 3320    C_ wss 3322   ifcif 3741   class class class wbr 4214    X. cxp 4878   `'ccnv 4879   dom cdm 4880   PosetRelcps 14626    TosetRel ctsr 14627
This theorem is referenced by:  ordtbaslem  17254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-res 4892  df-ps 14631  df-tsr 14632
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