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Theorem tsrlemax 15977
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 4460 . . 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 4460 . . 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 2457 . . . . . . . . . 10  |-  dom  R  =  dom  R
76istsr 15974 . . . . . . . . 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 15968 . . . . . . . . 9  |-  ( ( R  e.  PosetRel  /\  A R B  /\  B R C )  ->  A R C )
1093expib 1199 . . . . . . . 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 15976 . . . . . . 7  |-  ( ( R  e.  TosetRel  /\  B  e.  X  /\  C  e.  X )  ->  ( B R C  \/  C R B ) )
22213adant3r1 1205 . . . . . 6  |-  ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B R C  \/  C R B ) )
2322orcanai 913 . . . . 5  |-  ( ( ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  -.  B R C )  ->  C R B )
24 pstr 15968 . . . . . . . . . 10  |-  ( ( R  e.  PosetRel  /\  A R C  /\  C R B )  ->  A R B )
25243expib 1199 . . . . . . . . 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 3979 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 973    = wceq 1395    e. wcel 1819    u. cun 3469    C_ wss 3471   ifcif 3944   class class class wbr 4456    X. cxp 5006   `'ccnv 5007   dom cdm 5008   PosetRelcps 15955    TosetRel ctsr 15956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-res 5020  df-ps 15957  df-tsr 15958
This theorem is referenced by:  ordtbaslem  19816
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