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Theorem brecop2 6957
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 4885 . . 3  |-  ( [
<. A ,  B >. ]  .~  R [ <. C ,  D >. ]  .~  ->  ( [ <. A ,  B >. ]  .~  e.  H  /\  [ <. C ,  D >. ]  .~  e.  H ) )
3 brecop2.5 . . . . . . 7  |-  dom  .~  =  ( G  X.  G )
4 ecelqsdm 6933 . . . . . . 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 2496 . . . . 5  |-  ( [
<. A ,  B >. ]  .~  e.  H  ->  <. A ,  B >.  e.  ( G  X.  G
) )
8 opelxp 4867 . . . . 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 6933 . . . . . . 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 2496 . . . . 5  |-  ( [
<. C ,  D >. ]  .~  e.  H  ->  <. C ,  D >.  e.  ( G  X.  G
) )
13 opelxp 4867 . . . . 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 4885 . . . 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 6192 . . . . 5  |-  ( ( A  .+  D )  e.  G  ->  ( A  e.  G  /\  D  e.  G )
)
2219, 20ndmovrcl 6192 . . . . 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 1721   _Vcvv 2916    C_ wss 3280   (/)c0 3588   <.cop 3777   class class class wbr 4172    X. cxp 4835   dom cdm 4837  (class class class)co 6040   [cec 6862   /.cqs 6863
This theorem is referenced by:  ltsrpr  8908
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fv 5421  df-ov 6043  df-ec 6866  df-qs 6870
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