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Theorem ndmovdistr 6363
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 6360 . . . . . 6  |-  ( ( B F C )  e.  S  ->  ( B  e.  S  /\  C  e.  S )
)
43anim2i 569 . . . . 5  |-  ( ( A  e.  S  /\  ( B F C )  e.  S )  -> 
( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
5 3anass 969 . . . . 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 6358 . . 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 6360 . . . . . 6  |-  ( ( A G B )  e.  S  ->  ( A  e.  S  /\  B  e.  S )
)
128, 2ndmovrcl 6360 . . . . . 6  |-  ( ( A G C )  e.  S  ->  ( A  e.  S  /\  C  e.  S )
)
1311, 12anim12i 566 . . . . 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 979 . . . . 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 6358 . . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   (/)c0 3746    X. cxp 4947   dom cdm 4949  (class class class)co 6201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-xp 4955  df-dm 4959  df-iota 5490  df-fv 5535  df-ov 6204
This theorem is referenced by:  distrpi  9179  distrnq  9242  distrpr  9309  distrsr  9370
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