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Theorem ndmovdistr 6437
Description: Any operation is distributive 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
ndmov.6  |-  dom  G  =  ( S  X.  S )
Assertion
Ref Expression
ndmovdistr  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  ( A G ( B F C ) )  =  ( ( A G B ) F ( A G C ) ) )

Proof of Theorem ndmovdistr
StepHypRef Expression
1 ndmov.1 . . . . . . 7  |-  dom  F  =  ( S  X.  S )
2 ndmov.5 . . . . . . 7  |-  -.  (/)  e.  S
31, 2ndmovrcl 6434 . . . . . 6  |-  ( ( B F C )  e.  S  ->  ( B  e.  S  /\  C  e.  S )
)
43anim2i 567 . . . . 5  |-  ( ( A  e.  S  /\  ( B F C )  e.  S )  -> 
( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
5 3anass 975 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
64, 5sylibr 212 . . . 4  |-  ( ( A  e.  S  /\  ( B F 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  e.  S  /\  ( B F C )  e.  S ) )
8 ndmov.6 . . . 4  |-  dom  G  =  ( S  X.  S )
98ndmov 6432 . . 3  |-  ( -.  ( A  e.  S  /\  ( B F C )  e.  S )  ->  ( A G ( B F C ) )  =  (/) )
107, 9syl 16 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  ( A G ( B F C ) )  =  (/) )
118, 2ndmovrcl 6434 . . . . . 6  |-  ( ( A G B )  e.  S  ->  ( A  e.  S  /\  B  e.  S )
)
128, 2ndmovrcl 6434 . . . . . 6  |-  ( ( A G C )  e.  S  ->  ( A  e.  S  /\  C  e.  S )
)
1311, 12anim12i 564 . . . . 5  |-  ( ( ( A G B )  e.  S  /\  ( A G C )  e.  S )  -> 
( ( A  e.  S  /\  B  e.  S )  /\  ( A  e.  S  /\  C  e.  S )
) )
14 anandi3 985 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( ( A  e.  S  /\  B  e.  S
)  /\  ( A  e.  S  /\  C  e.  S ) ) )
1513, 14sylibr 212 . . . 4  |-  ( ( ( A G B )  e.  S  /\  ( A G C )  e.  S )  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
1615con3i 135 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  (
( A G B )  e.  S  /\  ( A G C )  e.  S ) )
171ndmov 6432 . . 3  |-  ( -.  ( ( A G B )  e.  S  /\  ( A G C )  e.  S )  ->  ( ( A G B ) F ( A G C ) )  =  (/) )
1816, 17syl 16 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  ( ( A G B ) F ( A G C ) )  =  (/) )
1910, 18eqtr4d 2498 1  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  ( A G ( B F C ) )  =  ( ( A G B ) F ( A G C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   (/)c0 3783    X. cxp 4986   dom cdm 4988  (class class class)co 6270
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-8 1825  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-pow 4615  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-eu 2288  df-mo 2289  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-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-dm 4998  df-iota 5534  df-fv 5578  df-ov 6273
This theorem is referenced by:  distrpi  9265  distrnq  9328  distrpr  9395  distrsr  9457
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