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Theorem soinxp 4438
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 4437 . . 3  |-  ( R  Po  A  <->  ( R  i^i  ( A  X.  A
) )  Po  A
)
2 brinxp 4436 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  <-> 
x ( R  i^i  ( A  X.  A
) ) y ) )
323adant3 935 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( x R y  <-> 
x ( R  i^i  ( A  X.  A
) ) y ) )
4 brinxp 4436 . . . . . . . . 9  |-  ( ( x  e.  A  /\  z  e.  A )  ->  ( x R z  <-> 
x ( R  i^i  ( A  X.  A
) ) z ) )
543adant2 934 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( x R z  <-> 
x ( R  i^i  ( A  X.  A
) ) z ) )
6 brinxp 4436 . . . . . . . . . 10  |-  ( ( z  e.  A  /\  y  e.  A )  ->  ( z R y  <-> 
z ( R  i^i  ( A  X.  A
) ) y ) )
76ancoms 259 . . . . . . . . 9  |-  ( ( y  e.  A  /\  z  e.  A )  ->  ( z R y  <-> 
z ( R  i^i  ( A  X.  A
) ) y ) )
873adant1 933 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( z R y  <-> 
z ( R  i^i  ( A  X.  A
) ) y ) )
95, 8orbi12d 717 . . . . . . 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 227 . . . . . 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 1116 . . . . 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 2363 . . . 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 2355 . . 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 441 . 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 4062 . 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 4062 . 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 205 1  |-  ( R  Or  A  <->  ( R  i^i  ( A  X.  A
) )  Or  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639    /\ w3a 896    e. wcel 1409   A.wral 2323    i^i cin 2944   class class class wbr 3792    Po wpo 4059    Or wor 4060    X. cxp 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-po 4061  df-iso 4062  df-xp 4379
This theorem is referenced by: (None)
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