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Theorem otel3xp 5024
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 4025 . . . 4  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
2 3simpa 991 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( A  e.  X  /\  B  e.  Y
) )
3 opelxp 5018 . . . . . 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 996 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  C  e.  Z )
6 opelxp 5018 . . . . 5  |-  ( <. <. A ,  B >. ,  C >.  e.  (
( X  X.  Y
)  X.  Z )  <-> 
( <. A ,  B >.  e.  ( X  X.  Y )  /\  C  e.  Z ) )
74, 5, 6sylanbrc 662 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  -> 
<. <. A ,  B >. ,  C >.  e.  ( ( X  X.  Y
)  X.  Z ) )
81, 7syl5eqel 2546 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  -> 
<. A ,  B ,  C >.  e.  ( ( X  X.  Y )  X.  Z ) )
9 eleq1 2526 . . 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 427 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   <.cop 4022   <.cotp 4024    X. cxp 4986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-ot 4025  df-opab 4498  df-xp 4994
This theorem is referenced by:  el2wlkonot  25071  el2spthonot  25072  el2spthonot0  25073  el2wlksotot  25084  usg2spot2nb  25267  usgreg2spot  25269
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