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Theorem ndmovass 6236
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 6234 . . . . . 6  |-  ( ( A F B )  e.  S  ->  ( A  e.  S  /\  B  e.  S )
)
43anim1i 553 . . . . 5  |-  ( ( ( A F B )  e.  S  /\  C  e.  S )  ->  ( ( A  e.  S  /\  B  e.  S )  /\  C  e.  S ) )
5 df-3an 939 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( ( A  e.  S  /\  B  e.  S
)  /\  C  e.  S ) )
64, 5sylibr 205 . . . 4  |-  ( ( ( A F B )  e.  S  /\  C  e.  S )  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
76con3i 130 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  (
( A F B )  e.  S  /\  C  e.  S )
)
81ndmov 6232 . . 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 6234 . . . . . 6  |-  ( ( B F C )  e.  S  ->  ( B  e.  S  /\  C  e.  S )
)
1110anim2i 554 . . . . 5  |-  ( ( A  e.  S  /\  ( B F C )  e.  S )  -> 
( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
12 3anass 941 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
1311, 12sylibr 205 . . . 4  |-  ( ( A  e.  S  /\  ( B F C )  e.  S )  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
1413con3i 130 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  ( A  e.  S  /\  ( B F C )  e.  S ) )
151ndmov 6232 . . 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 2472 1  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   (/)c0 3629    X. cxp 4877   dom cdm 4879  (class class class)co 6082
This theorem is referenced by:  addasspi  8773  mulasspi  8775  addassnq  8836  mulassnq  8837  genpass  8887  addasssr  8964  mulasssr  8966
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-xp 4885  df-dm 4889  df-iota 5419  df-fv 5463  df-ov 6085
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