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Theorem otel3xp 4875
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 3968 . . . 4  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
2 3simpa 1027 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( A  e.  X  /\  B  e.  Y
) )
3 opelxp 4869 . . . . . 6  |-  ( <. A ,  B >.  e.  ( X  X.  Y
)  <->  ( A  e.  X  /\  B  e.  Y ) )
42, 3sylibr 217 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
5 simp3 1032 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  C  e.  Z )
6 opelxp 4869 . . . . 5  |-  ( <. <. A ,  B >. ,  C >.  e.  (
( X  X.  Y
)  X.  Z )  <-> 
( <. A ,  B >.  e.  ( X  X.  Y )  /\  C  e.  Z ) )
74, 5, 6sylanbrc 677 . . . 4  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  -> 
<. <. A ,  B >. ,  C >.  e.  ( ( X  X.  Y
)  X.  Z ) )
81, 7syl5eqel 2553 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  -> 
<. A ,  B ,  C >.  e.  ( ( X  X.  Y )  X.  Z ) )
9 eleq1 2537 . . 3  |-  ( T  =  <. A ,  B ,  C >.  ->  ( T  e.  ( ( X  X.  Y )  X.  Z )  <->  <. A ,  B ,  C >.  e.  ( ( X  X.  Y )  X.  Z
) ) )
108, 9syl5ibr 229 . 2  |-  ( T  =  <. A ,  B ,  C >.  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  T  e.  ( ( X  X.  Y )  X.  Z ) ) )
1110imp 436 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   <.cop 3965   <.cotp 3967    X. cxp 4837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-ot 3968  df-opab 4455  df-xp 4845
This theorem is referenced by:  el2wlkonot  25676  el2spthonot  25677  el2spthonot0  25678  el2wlksotot  25689  usg2spot2nb  25872  usgreg2spot  25874
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