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Theorem ndmovass 6380
Description: Any operation is associative outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
Hypotheses
Ref Expression
ndmov.1  |-  dom  F  =  ( S  X.  S )
ndmov.5  |-  -.  (/)  e.  S
Assertion
Ref Expression
ndmovass  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )

Proof of Theorem ndmovass
StepHypRef Expression
1 ndmov.1 . . . . . . 7  |-  dom  F  =  ( S  X.  S )
2 ndmov.5 . . . . . . 7  |-  -.  (/)  e.  S
31, 2ndmovrcl 6378 . . . . . 6  |-  ( ( A F B )  e.  S  ->  ( A  e.  S  /\  B  e.  S )
)
43anim1i 566 . . . . 5  |-  ( ( ( A F B )  e.  S  /\  C  e.  S )  ->  ( ( A  e.  S  /\  B  e.  S )  /\  C  e.  S ) )
5 df-3an 973 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( ( A  e.  S  /\  B  e.  S
)  /\  C  e.  S ) )
64, 5sylibr 212 . . . 4  |-  ( ( ( A F B )  e.  S  /\  C  e.  S )  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
76con3i 135 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  (
( A F B )  e.  S  /\  C  e.  S )
)
81ndmov 6376 . . 3  |-  ( -.  ( ( A F B )  e.  S  /\  C  e.  S
)  ->  ( ( A F B ) F C )  =  (/) )
97, 8syl 16 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  ( ( A F B ) F C )  =  (/) )
101, 2ndmovrcl 6378 . . . . . 6  |-  ( ( B F C )  e.  S  ->  ( B  e.  S  /\  C  e.  S )
)
1110anim2i 567 . . . . 5  |-  ( ( A  e.  S  /\  ( B F C )  e.  S )  -> 
( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
12 3anass 975 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
1311, 12sylibr 212 . . . 4  |-  ( ( A  e.  S  /\  ( B F C )  e.  S )  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
1413con3i 135 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  ( A  e.  S  /\  ( B F C )  e.  S ) )
151ndmov 6376 . . 3  |-  ( -.  ( A  e.  S  /\  ( B F C )  e.  S )  ->  ( A F ( B F C ) )  =  (/) )
1614, 15syl 16 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  ( A F ( B F C ) )  =  (/) )
179, 16eqtr4d 2436 1  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836   (/)c0 3724    X. cxp 4924   dom cdm 4926  (class class class)co 6214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-rab 2751  df-v 3049  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-sn 3958  df-pr 3960  df-op 3964  df-uni 4177  df-br 4381  df-opab 4439  df-xp 4932  df-dm 4936  df-iota 5473  df-fv 5517  df-ov 6217
This theorem is referenced by:  addasspi  9202  mulasspi  9204  addassnq  9265  mulassnq  9266  genpass  9316  addasssr  9394  mulasssr  9396
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