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Theorem poinxp 4941
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 731 . . . . . . . 8  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  x  e.  A )
2 brinxp 4940 . . . . . . . 8  |-  ( ( x  e.  A  /\  x  e.  A )  ->  ( x R x  <-> 
x ( R  i^i  ( A  X.  A
) ) x ) )
31, 1, 2syl2anc 643 . . . . . . 7  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  (
x R x  <->  x ( R  i^i  ( A  X.  A ) ) x ) )
43notbid 286 . . . . . 6  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  ( -.  x R x  <->  -.  x
( R  i^i  ( A  X.  A ) ) x ) )
5 brinxp 4940 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  <-> 
x ( R  i^i  ( A  X.  A
) ) y ) )
65adantr 452 . . . . . . . 8  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  (
x R y  <->  x ( R  i^i  ( A  X.  A ) ) y ) )
7 brinxp 4940 . . . . . . . . 9  |-  ( ( y  e.  A  /\  z  e.  A )  ->  ( y R z  <-> 
y ( R  i^i  ( A  X.  A
) ) z ) )
87adantll 695 . . . . . . . 8  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  (
y R z  <->  y ( R  i^i  ( A  X.  A ) ) z ) )
96, 8anbi12d 692 . . . . . . 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 4940 . . . . . . . 8  |-  ( ( x  e.  A  /\  z  e.  A )  ->  ( x R z  <-> 
x ( R  i^i  ( A  X.  A
) ) z ) )
1110adantlr 696 . . . . . . 7  |-  ( ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A )  ->  (
x R z  <->  x ( R  i^i  ( A  X.  A ) ) z ) )
129, 11imbi12d 312 . . . . . 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 692 . . . . 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 2721 . . . 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 2721 . . 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 2737 . 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 4503 . 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 4503 . 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 269 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 177    /\ wa 359    e. wcel 1725   A.wral 2705    i^i cin 3319   class class class wbr 4212    Po wpo 4501    X. cxp 4876
This theorem is referenced by:  soinxp  4942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-po 4503  df-xp 4884
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