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Theorem otel3xp 30271
Description: An ordered triple is an element of a doubled Cartesian product. (Contributed by Alexander van der Vekens, 26-Feb-2018.)
Assertion
Ref Expression
otel3xp  |-  ( ( T  =  <. A ,  B ,  C >.  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
) )  ->  T  e.  ( ( X  X.  Y )  X.  Z
) )

Proof of Theorem otel3xp
StepHypRef Expression
1 df-ot 3989 . . . 4  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
2 3simpa 985 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( A  e.  X  /\  B  e.  Y
) )
3 opelxp 4972 . . . . . 6  |-  ( <. A ,  B >.  e.  ( X  X.  Y
)  <->  ( A  e.  X  /\  B  e.  Y ) )
42, 3sylibr 212 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
5 simp3 990 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  C  e.  Z )
6 opelxp 4972 . . . . 5  |-  ( <. <. A ,  B >. ,  C >.  e.  (
( X  X.  Y
)  X.  Z )  <-> 
( <. A ,  B >.  e.  ( X  X.  Y )  /\  C  e.  Z ) )
74, 5, 6sylanbrc 664 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  -> 
<. <. A ,  B >. ,  C >.  e.  ( ( X  X.  Y
)  X.  Z ) )
81, 7syl5eqel 2544 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  -> 
<. A ,  B ,  C >.  e.  ( ( X  X.  Y )  X.  Z ) )
9 eleq1 2524 . . 3  |-  ( T  =  <. A ,  B ,  C >.  ->  ( T  e.  ( ( X  X.  Y )  X.  Z )  <->  <. A ,  B ,  C >.  e.  ( ( X  X.  Y )  X.  Z
) ) )
108, 9syl5ibr 221 . 2  |-  ( T  =  <. A ,  B ,  C >.  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  T  e.  ( ( X  X.  Y )  X.  Z ) ) )
1110imp 429 1  |-  ( ( T  =  <. A ,  B ,  C >.  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
) )  ->  T  e.  ( ( X  X.  Y )  X.  Z
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   <.cop 3986   <.cotp 3988    X. cxp 4941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-ot 3989  df-opab 4454  df-xp 4949
This theorem is referenced by:  el2wlkonot  30531  el2spthonot  30532  el2spthonot0  30533  el2wlksotot  30544  usg2spot2nb  30801  usgreg2spot  30803
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