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Theorem soinxp 4569
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 4568 . . 3  |-  ( R  Po  A  <->  ( R  i^i  ( A  X.  A
) )  Po  A
)
2 brinxp 4567 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  <-> 
x ( R  i^i  ( A  X.  A
) ) y ) )
323adant3 984 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( x R y  <-> 
x ( R  i^i  ( A  X.  A
) ) y ) )
4 brinxp 4567 . . . . . . . . 9  |-  ( ( x  e.  A  /\  z  e.  A )  ->  ( x R z  <-> 
x ( R  i^i  ( A  X.  A
) ) z ) )
543adant2 983 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( x R z  <-> 
x ( R  i^i  ( A  X.  A
) ) z ) )
6 brinxp 4567 . . . . . . . . . 10  |-  ( ( z  e.  A  /\  y  e.  A )  ->  ( z R y  <-> 
z ( R  i^i  ( A  X.  A
) ) y ) )
76ancoms 266 . . . . . . . . 9  |-  ( ( y  e.  A  /\  z  e.  A )  ->  ( z R y  <-> 
z ( R  i^i  ( A  X.  A
) ) y ) )
873adant1 982 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( z R y  <-> 
z ( R  i^i  ( A  X.  A
) ) y ) )
95, 8orbi12d 765 . . . . . . 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 233 . . . . . 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 1165 . . . . 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 2431 . . . 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 2423 . . 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 453 . 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 4179 . 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 4179 . 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 211 1  |-  ( R  Or  A  <->  ( R  i^i  ( A  X.  A
) )  Or  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    /\ w3a 945    e. wcel 1463   A.wral 2390    i^i cin 3036   class class class wbr 3895    Po wpo 4176    Or wor 4177    X. cxp 4497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-po 4178  df-iso 4179  df-xp 4505
This theorem is referenced by: (None)
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