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Theorem poinxp 4465
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 4464 . . . . . . . 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 625 . . . . . 6  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  ( -.  x R x  <->  -.  x
( R  i^i  ( A  X.  A ) ) x ) )
5 brinxp 4464 . . . . . . . . 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 4464 . . . . . . . . 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 4464 . . . . . . . 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 2370 . . . 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 2370 . . 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 2386 . 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 4087 . 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 4087 . 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 1434   A.wral 2353    i^i cin 2983   class class class wbr 3811    Po wpo 4085    X. cxp 4399
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-po 4087  df-xp 4407
This theorem is referenced by:  soinxp  4466
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