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Theorem soinxp 4714
Description: Intersection of linear order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
Assertion
Ref Expression
soinxp  |-  ( R  Or  A  <->  ( R  i^i  ( A  X.  A
) )  Or  A
)

Proof of Theorem soinxp
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poinxp 4713 . . 3  |-  ( R  Po  A  <->  ( R  i^i  ( A  X.  A
) )  Po  A
)
2 brinxp 4712 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  <-> 
x ( R  i^i  ( A  X.  A
) ) y ) )
323adant3 1019 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( x R y  <-> 
x ( R  i^i  ( A  X.  A
) ) y ) )
4 brinxp 4712 . . . . . . . . 9  |-  ( ( x  e.  A  /\  z  e.  A )  ->  ( x R z  <-> 
x ( R  i^i  ( A  X.  A
) ) z ) )
543adant2 1018 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( x R z  <-> 
x ( R  i^i  ( A  X.  A
) ) z ) )
6 brinxp 4712 . . . . . . . . . 10  |-  ( ( z  e.  A  /\  y  e.  A )  ->  ( z R y  <-> 
z ( R  i^i  ( A  X.  A
) ) y ) )
76ancoms 268 . . . . . . . . 9  |-  ( ( y  e.  A  /\  z  e.  A )  ->  ( z R y  <-> 
z ( R  i^i  ( A  X.  A
) ) y ) )
873adant1 1017 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( z R y  <-> 
z ( R  i^i  ( A  X.  A
) ) y ) )
95, 8orbi12d 794 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( ( x R z  \/  z R y )  <->  ( x
( R  i^i  ( A  X.  A ) ) z  \/  z ( R  i^i  ( A  X.  A ) ) y ) ) )
103, 9imbi12d 234 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( ( x R y  ->  ( x R z  \/  z R y ) )  <-> 
( x ( R  i^i  ( A  X.  A ) ) y  ->  ( x ( R  i^i  ( A  X.  A ) ) z  \/  z ( R  i^i  ( A  X.  A ) ) y ) ) ) )
11103expb 1206 . . . . 5  |-  ( ( x  e.  A  /\  ( y  e.  A  /\  z  e.  A
) )  ->  (
( x R y  ->  ( x R z  \/  z R y ) )  <->  ( x
( R  i^i  ( A  X.  A ) ) y  ->  ( x
( R  i^i  ( A  X.  A ) ) z  \/  z ( R  i^i  ( A  X.  A ) ) y ) ) ) )
12112ralbidva 2512 . . . 4  |-  ( x  e.  A  ->  ( A. y  e.  A  A. z  e.  A  ( x R y  ->  ( x R z  \/  z R y ) )  <->  A. y  e.  A  A. z  e.  A  ( x
( R  i^i  ( A  X.  A ) ) y  ->  ( x
( R  i^i  ( A  X.  A ) ) z  \/  z ( R  i^i  ( A  X.  A ) ) y ) ) ) )
1312ralbiia 2504 . . 3  |-  ( A. x  e.  A  A. y  e.  A  A. z  e.  A  (
x R y  -> 
( x R z  \/  z R y ) )  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( x
( R  i^i  ( A  X.  A ) ) y  ->  ( x
( R  i^i  ( A  X.  A ) ) z  \/  z ( R  i^i  ( A  X.  A ) ) y ) ) )
141, 13anbi12i 460 . 2  |-  ( ( R  Po  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  (
x R y  -> 
( x R z  \/  z R y ) ) )  <->  ( ( R  i^i  ( A  X.  A ) )  Po  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( x
( R  i^i  ( A  X.  A ) ) y  ->  ( x
( R  i^i  ( A  X.  A ) ) z  \/  z ( R  i^i  ( A  X.  A ) ) y ) ) ) )
15 df-iso 4315 . 2  |-  ( R  Or  A  <->  ( R  Po  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( x R y  ->  (
x R z  \/  z R y ) ) ) )
16 df-iso 4315 . 2  |-  ( ( R  i^i  ( A  X.  A ) )  Or  A  <->  ( ( R  i^i  ( A  X.  A ) )  Po  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( x
( R  i^i  ( A  X.  A ) ) y  ->  ( x
( R  i^i  ( A  X.  A ) ) z  \/  z ( R  i^i  ( A  X.  A ) ) y ) ) ) )
1714, 15, 163bitr4i 212 1  |-  ( R  Or  A  <->  ( R  i^i  ( A  X.  A
) )  Or  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    /\ w3a 980    e. wcel 2160   A.wral 2468    i^i cin 3143   class class class wbr 4018    Po wpo 4312    Or wor 4313    X. cxp 4642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-po 4314  df-iso 4315  df-xp 4650
This theorem is referenced by: (None)
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