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Theorem brecop2 6927
Description: Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis. (Contributed by NM, 13-Feb-1996.)
Hypotheses
Ref Expression
brecop2.1  |-  .~  e.  _V
brecop2.5  |-  dom  .~  =  ( G  X.  G )
brecop2.6  |-  H  =  ( ( G  X.  G ) /.  .~  )
brecop2.7  |-  R  C_  ( H  X.  H
)
brecop2.8  |-  .<_  C_  ( G  X.  G )
brecop2.9  |-  -.  (/)  e.  G
brecop2.10  |-  dom  .+  =  ( G  X.  G )
brecop2.11  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( [ <. A ,  B >. ]  .~  R [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  .<_  ( B 
.+  C ) ) )
Assertion
Ref Expression
brecop2  |-  ( [
<. A ,  B >. ]  .~  R [ <. C ,  D >. ]  .~  <->  ( A  .+  D ) 
.<_  ( B  .+  C
) )

Proof of Theorem brecop2
StepHypRef Expression
1 brecop2.7 . . . 4  |-  R  C_  ( H  X.  H
)
21brel 4859 . . 3  |-  ( [
<. A ,  B >. ]  .~  R [ <. C ,  D >. ]  .~  ->  ( [ <. A ,  B >. ]  .~  e.  H  /\  [ <. C ,  D >. ]  .~  e.  H ) )
3 brecop2.5 . . . . . . 7  |-  dom  .~  =  ( G  X.  G )
4 ecelqsdm 6903 . . . . . . 7  |-  ( ( dom  .~  =  ( G  X.  G )  /\  [ <. A ,  B >. ]  .~  e.  ( ( G  X.  G ) /.  .~  ) )  ->  <. A ,  B >.  e.  ( G  X.  G ) )
53, 4mpan 652 . . . . . 6  |-  ( [
<. A ,  B >. ]  .~  e.  ( ( G  X.  G ) /.  .~  )  ->  <. A ,  B >.  e.  ( G  X.  G
) )
6 brecop2.6 . . . . . 6  |-  H  =  ( ( G  X.  G ) /.  .~  )
75, 6eleq2s 2472 . . . . 5  |-  ( [
<. A ,  B >. ]  .~  e.  H  ->  <. A ,  B >.  e.  ( G  X.  G
) )
8 opelxp 4841 . . . . 5  |-  ( <. A ,  B >.  e.  ( G  X.  G
)  <->  ( A  e.  G  /\  B  e.  G ) )
97, 8sylib 189 . . . 4  |-  ( [
<. A ,  B >. ]  .~  e.  H  -> 
( A  e.  G  /\  B  e.  G
) )
10 ecelqsdm 6903 . . . . . . 7  |-  ( ( dom  .~  =  ( G  X.  G )  /\  [ <. C ,  D >. ]  .~  e.  ( ( G  X.  G ) /.  .~  ) )  ->  <. C ,  D >.  e.  ( G  X.  G ) )
113, 10mpan 652 . . . . . 6  |-  ( [
<. C ,  D >. ]  .~  e.  ( ( G  X.  G ) /.  .~  )  ->  <. C ,  D >.  e.  ( G  X.  G
) )
1211, 6eleq2s 2472 . . . . 5  |-  ( [
<. C ,  D >. ]  .~  e.  H  ->  <. C ,  D >.  e.  ( G  X.  G
) )
13 opelxp 4841 . . . . 5  |-  ( <. C ,  D >.  e.  ( G  X.  G
)  <->  ( C  e.  G  /\  D  e.  G ) )
1412, 13sylib 189 . . . 4  |-  ( [
<. C ,  D >. ]  .~  e.  H  -> 
( C  e.  G  /\  D  e.  G
) )
159, 14anim12i 550 . . 3  |-  ( ( [ <. A ,  B >. ]  .~  e.  H  /\  [ <. C ,  D >. ]  .~  e.  H
)  ->  ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) ) )
162, 15syl 16 . 2  |-  ( [
<. A ,  B >. ]  .~  R [ <. C ,  D >. ]  .~  ->  ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
) )
17 brecop2.8 . . . . 5  |-  .<_  C_  ( G  X.  G )
1817brel 4859 . . . 4  |-  ( ( A  .+  D ) 
.<_  ( B  .+  C
)  ->  ( ( A  .+  D )  e.  G  /\  ( B 
.+  C )  e.  G ) )
19 brecop2.10 . . . . . 6  |-  dom  .+  =  ( G  X.  G )
20 brecop2.9 . . . . . 6  |-  -.  (/)  e.  G
2119, 20ndmovrcl 6165 . . . . 5  |-  ( ( A  .+  D )  e.  G  ->  ( A  e.  G  /\  D  e.  G )
)
2219, 20ndmovrcl 6165 . . . . 5  |-  ( ( B  .+  C )  e.  G  ->  ( B  e.  G  /\  C  e.  G )
)
2321, 22anim12i 550 . . . 4  |-  ( ( ( A  .+  D
)  e.  G  /\  ( B  .+  C )  e.  G )  -> 
( ( A  e.  G  /\  D  e.  G )  /\  ( B  e.  G  /\  C  e.  G )
) )
2418, 23syl 16 . . 3  |-  ( ( A  .+  D ) 
.<_  ( B  .+  C
)  ->  ( ( A  e.  G  /\  D  e.  G )  /\  ( B  e.  G  /\  C  e.  G
) ) )
25 an42 799 . . 3  |-  ( ( ( A  e.  G  /\  D  e.  G
)  /\  ( B  e.  G  /\  C  e.  G ) )  <->  ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) ) )
2624, 25sylib 189 . 2  |-  ( ( A  .+  D ) 
.<_  ( B  .+  C
)  ->  ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) ) )
27 brecop2.11 . 2  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( [ <. A ,  B >. ]  .~  R [ <. C ,  D >. ]  .~  <->  ( A  .+  D )  .<_  ( B 
.+  C ) ) )
2816, 26, 27pm5.21nii 343 1  |-  ( [
<. A ,  B >. ]  .~  R [ <. C ,  D >. ]  .~  <->  ( A  .+  D ) 
.<_  ( B  .+  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2892    C_ wss 3256   (/)c0 3564   <.cop 3753   class class class wbr 4146    X. cxp 4809   dom cdm 4811  (class class class)co 6013   [cec 6832   /.cqs 6833
This theorem is referenced by:  ltsrpr  8878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-xp 4817  df-cnv 4819  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fv 5395  df-ov 6016  df-ec 6836  df-qs 6840
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