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Theorem brecop2 6768
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 4753 . . 3  |-  ( [
<. A ,  B >. ]  .~  R [ <. C ,  D >. ]  .~  ->  ( [ <. A ,  B >. ]  .~  e.  H  /\  [ <. C ,  D >. ]  .~  e.  H ) )
3 brecop2.5 . . . . . . 7  |-  dom  .~  =  ( G  X.  G )
4 ecelqsdm 6745 . . . . . . 7  |-  ( ( dom  .~  =  ( G  X.  G )  /\  [ <. A ,  B >. ]  .~  e.  ( ( G  X.  G ) /.  .~  ) )  ->  <. A ,  B >.  e.  ( G  X.  G ) )
53, 4mpan 651 . . . . . 6  |-  ( [
<. A ,  B >. ]  .~  e.  ( ( G  X.  G ) /.  .~  )  ->  <. A ,  B >.  e.  ( G  X.  G
) )
6 brecop2.6 . . . . . 6  |-  H  =  ( ( G  X.  G ) /.  .~  )
75, 6eleq2s 2388 . . . . 5  |-  ( [
<. A ,  B >. ]  .~  e.  H  ->  <. A ,  B >.  e.  ( G  X.  G
) )
8 opelxp 4735 . . . . 5  |-  ( <. A ,  B >.  e.  ( G  X.  G
)  <->  ( A  e.  G  /\  B  e.  G ) )
97, 8sylib 188 . . . 4  |-  ( [
<. A ,  B >. ]  .~  e.  H  -> 
( A  e.  G  /\  B  e.  G
) )
10 ecelqsdm 6745 . . . . . . 7  |-  ( ( dom  .~  =  ( G  X.  G )  /\  [ <. C ,  D >. ]  .~  e.  ( ( G  X.  G ) /.  .~  ) )  ->  <. C ,  D >.  e.  ( G  X.  G ) )
113, 10mpan 651 . . . . . 6  |-  ( [
<. C ,  D >. ]  .~  e.  ( ( G  X.  G ) /.  .~  )  ->  <. C ,  D >.  e.  ( G  X.  G
) )
1211, 6eleq2s 2388 . . . . 5  |-  ( [
<. C ,  D >. ]  .~  e.  H  ->  <. C ,  D >.  e.  ( G  X.  G
) )
13 opelxp 4735 . . . . 5  |-  ( <. C ,  D >.  e.  ( G  X.  G
)  <->  ( C  e.  G  /\  D  e.  G ) )
1412, 13sylib 188 . . . 4  |-  ( [
<. C ,  D >. ]  .~  e.  H  -> 
( C  e.  G  /\  D  e.  G
) )
159, 14anim12i 549 . . 3  |-  ( ( [ <. A ,  B >. ]  .~  e.  H  /\  [ <. C ,  D >. ]  .~  e.  H
)  ->  ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) ) )
162, 15syl 15 . 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 4753 . . . 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 6022 . . . . 5  |-  ( ( A  .+  D )  e.  G  ->  ( A  e.  G  /\  D  e.  G )
)
2219, 20ndmovrcl 6022 . . . . 5  |-  ( ( B  .+  C )  e.  G  ->  ( B  e.  G  /\  C  e.  G )
)
2321, 22anim12i 549 . . . 4  |-  ( ( ( A  .+  D
)  e.  G  /\  ( B  .+  C )  e.  G )  -> 
( ( A  e.  G  /\  D  e.  G )  /\  ( B  e.  G  /\  C  e.  G )
) )
2418, 23syl 15 . . 3  |-  ( ( A  .+  D ) 
.<_  ( B  .+  C
)  ->  ( ( A  e.  G  /\  D  e.  G )  /\  ( B  e.  G  /\  C  e.  G
) ) )
25 an42 798 . . 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 188 . 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 342 1  |-  ( [
<. A ,  B >. ]  .~  R [ <. C ,  D >. ]  .~  <->  ( A  .+  D ) 
.<_  ( B  .+  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   (/)c0 3468   <.cop 3656   class class class wbr 4039    X. cxp 4703   dom cdm 4705  (class class class)co 5874   [cec 6674   /.cqs 6675
This theorem is referenced by:  ltsrpr  8715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-ov 5877  df-ec 6678  df-qs 6682
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