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Theorem poinxp 4507
Description: Intersection of partial order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
Assertion
Ref Expression
poinxp  |-  ( R  Po  A  <->  ( R  i^i  ( A  X.  A
) )  Po  A
)

Proof of Theorem poinxp
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 496 . . . . . . . 8  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  x  e.  A )
2 brinxp 4506 . . . . . . . 8  |-  ( ( x  e.  A  /\  x  e.  A )  ->  ( x R x  <-> 
x ( R  i^i  ( A  X.  A
) ) x ) )
31, 1, 2syl2anc 403 . . . . . . 7  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  (
x R x  <->  x ( R  i^i  ( A  X.  A ) ) x ) )
43notbid 627 . . . . . 6  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  ( -.  x R x  <->  -.  x
( R  i^i  ( A  X.  A ) ) x ) )
5 brinxp 4506 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  <-> 
x ( R  i^i  ( A  X.  A
) ) y ) )
65adantr 270 . . . . . . . 8  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  (
x R y  <->  x ( R  i^i  ( A  X.  A ) ) y ) )
7 brinxp 4506 . . . . . . . . 9  |-  ( ( y  e.  A  /\  z  e.  A )  ->  ( y R z  <-> 
y ( R  i^i  ( A  X.  A
) ) z ) )
87adantll 460 . . . . . . . 8  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  (
y R z  <->  y ( R  i^i  ( A  X.  A ) ) z ) )
96, 8anbi12d 457 . . . . . . 7  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  (
( x R y  /\  y R z )  <->  ( x ( R  i^i  ( A  X.  A ) ) y  /\  y ( R  i^i  ( A  X.  A ) ) z ) ) )
10 brinxp 4506 . . . . . . . 8  |-  ( ( x  e.  A  /\  z  e.  A )  ->  ( x R z  <-> 
x ( R  i^i  ( A  X.  A
) ) z ) )
1110adantlr 461 . . . . . . 7  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  (
x R z  <->  x ( R  i^i  ( A  X.  A ) ) z ) )
129, 11imbi12d 232 . . . . . 6  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  (
( ( x R y  /\  y R z )  ->  x R z )  <->  ( (
x ( R  i^i  ( A  X.  A
) ) y  /\  y ( R  i^i  ( A  X.  A
) ) z )  ->  x ( R  i^i  ( A  X.  A ) ) z ) ) )
134, 12anbi12d 457 . . . . 5  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  (
( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) )  <-> 
( -.  x ( R  i^i  ( A  X.  A ) ) x  /\  ( ( x ( R  i^i  ( A  X.  A
) ) y  /\  y ( R  i^i  ( A  X.  A
) ) z )  ->  x ( R  i^i  ( A  X.  A ) ) z ) ) ) )
1413ralbidva 2376 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( A. z  e.  A  ( -.  x R x  /\  (
( x R y  /\  y R z )  ->  x R
z ) )  <->  A. z  e.  A  ( -.  x ( R  i^i  ( A  X.  A
) ) x  /\  ( ( x ( R  i^i  ( A  X.  A ) ) y  /\  y ( R  i^i  ( A  X.  A ) ) z )  ->  x
( R  i^i  ( A  X.  A ) ) z ) ) ) )
1514ralbidva 2376 . . 3  |-  ( x  e.  A  ->  ( A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) )  <->  A. y  e.  A  A. z  e.  A  ( -.  x ( R  i^i  ( A  X.  A ) ) x  /\  ( ( x ( R  i^i  ( A  X.  A ) ) y  /\  y ( R  i^i  ( A  X.  A ) ) z )  ->  x
( R  i^i  ( A  X.  A ) ) z ) ) ) )
1615ralbiia 2392 . 2  |-  ( A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) )  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x ( R  i^i  ( A  X.  A ) ) x  /\  ( ( x ( R  i^i  ( A  X.  A ) ) y  /\  y ( R  i^i  ( A  X.  A ) ) z )  ->  x
( R  i^i  ( A  X.  A ) ) z ) ) )
17 df-po 4123 . 2  |-  ( R  Po  A  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) )
18 df-po 4123 . 2  |-  ( ( 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
) ) x  /\  ( ( x ( R  i^i  ( A  X.  A ) ) y  /\  y ( R  i^i  ( A  X.  A ) ) z )  ->  x
( R  i^i  ( A  X.  A ) ) z ) ) )
1916, 17, 183bitr4i 210 1  |-  ( R  Po  A  <->  ( R  i^i  ( A  X.  A
) )  Po  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1438   A.wral 2359    i^i cin 2998   class class class wbr 3845    Po wpo 4121    X. cxp 4436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-po 4123  df-xp 4444
This theorem is referenced by:  soinxp  4508
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