MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  poinxp Unicode version

Theorem poinxp 4753
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 730 . . . . . . . 8  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  x  e.  A )
2 brinxp 4752 . . . . . . . 8  |-  ( ( x  e.  A  /\  x  e.  A )  ->  ( x R x  <-> 
x ( R  i^i  ( A  X.  A
) ) x ) )
31, 1, 2syl2anc 642 . . . . . . 7  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  (
x R x  <->  x ( R  i^i  ( A  X.  A ) ) x ) )
43notbid 285 . . . . . 6  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  ( -.  x R x  <->  -.  x
( R  i^i  ( A  X.  A ) ) x ) )
5 brinxp 4752 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  <-> 
x ( R  i^i  ( A  X.  A
) ) y ) )
65adantr 451 . . . . . . . 8  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  (
x R y  <->  x ( R  i^i  ( A  X.  A ) ) y ) )
7 brinxp 4752 . . . . . . . . 9  |-  ( ( y  e.  A  /\  z  e.  A )  ->  ( y R z  <-> 
y ( R  i^i  ( A  X.  A
) ) z ) )
87adantll 694 . . . . . . . 8  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  (
y R z  <->  y ( R  i^i  ( A  X.  A ) ) z ) )
96, 8anbi12d 691 . . . . . . 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 4752 . . . . . . . 8  |-  ( ( x  e.  A  /\  z  e.  A )  ->  ( x R z  <-> 
x ( R  i^i  ( A  X.  A
) ) z ) )
1110adantlr 695 . . . . . . 7  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  (
x R z  <->  x ( R  i^i  ( A  X.  A ) ) z ) )
129, 11imbi12d 311 . . . . . 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 691 . . . . 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 2559 . . . 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 2559 . . 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 2575 . 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 4314 . 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 4314 . 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 268 1  |-  ( R  Po  A  <->  ( R  i^i  ( A  X.  A
) )  Po  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   A.wral 2543    i^i cin 3151   class class class wbr 4023    Po wpo 4312    X. cxp 4687
This theorem is referenced by:  soinxp  4754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-po 4314  df-xp 4695
  Copyright terms: Public domain W3C validator