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Theorem ndmovass 6024
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 6022 . . . . . 6  |-  ( ( A F B )  e.  S  ->  ( A  e.  S  /\  B  e.  S )
)
43anim1i 551 . . . . 5  |-  ( ( ( A F B )  e.  S  /\  C  e.  S )  ->  ( ( A  e.  S  /\  B  e.  S )  /\  C  e.  S ) )
5 df-3an 936 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( ( A  e.  S  /\  B  e.  S
)  /\  C  e.  S ) )
64, 5sylibr 203 . . . 4  |-  ( ( ( A F B )  e.  S  /\  C  e.  S )  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
76con3i 127 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  (
( A F B )  e.  S  /\  C  e.  S )
)
81ndmov 6020 . . 3  |-  ( -.  ( ( A F B )  e.  S  /\  C  e.  S
)  ->  ( ( A F B ) F C )  =  (/) )
97, 8syl 15 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  ( ( A F B ) F C )  =  (/) )
101, 2ndmovrcl 6022 . . . . . 6  |-  ( ( B F C )  e.  S  ->  ( B  e.  S  /\  C  e.  S )
)
1110anim2i 552 . . . . 5  |-  ( ( A  e.  S  /\  ( B F C )  e.  S )  -> 
( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
12 3anass 938 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
1311, 12sylibr 203 . . . 4  |-  ( ( A  e.  S  /\  ( B F C )  e.  S )  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
1413con3i 127 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  ( A  e.  S  /\  ( B F C )  e.  S ) )
151ndmov 6020 . . 3  |-  ( -.  ( A  e.  S  /\  ( B F C )  e.  S )  ->  ( A F ( B F C ) )  =  (/) )
1614, 15syl 15 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  ( A F ( B F C ) )  =  (/) )
179, 16eqtr4d 2331 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   (/)c0 3468    X. cxp 4703   dom cdm 4705  (class class class)co 5874
This theorem is referenced by:  addasspi  8535  mulasspi  8537  addassnq  8598  mulassnq  8599  genpass  8649  addasssr  8726  mulasssr  8728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-dm 4715  df-iota 5235  df-fv 5279  df-ov 5877
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