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Theorem ndmovdistr 6009
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 6006 . . . . . 6  |-  ( ( B F C )  e.  S  ->  ( B  e.  S  /\  C  e.  S )
)
43anim2i 552 . . . . 5  |-  ( ( A  e.  S  /\  ( B F C )  e.  S )  -> 
( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
5 3anass 938 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
64, 5sylibr 203 . . . 4  |-  ( ( A  e.  S  /\  ( B F 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  e.  S  /\  ( B F C )  e.  S ) )
8 ndmov.6 . . . 4  |-  dom  G  =  ( S  X.  S )
98ndmov 6004 . . 3  |-  ( -.  ( A  e.  S  /\  ( B F C )  e.  S )  ->  ( A G ( B F C ) )  =  (/) )
107, 9syl 15 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  ( A G ( B F C ) )  =  (/) )
118, 2ndmovrcl 6006 . . . . . 6  |-  ( ( A G B )  e.  S  ->  ( A  e.  S  /\  B  e.  S )
)
128, 2ndmovrcl 6006 . . . . . 6  |-  ( ( A G C )  e.  S  ->  ( A  e.  S  /\  C  e.  S )
)
1311, 12anim12i 549 . . . . 5  |-  ( ( ( A G B )  e.  S  /\  ( A G C )  e.  S )  -> 
( ( A  e.  S  /\  B  e.  S )  /\  ( A  e.  S  /\  C  e.  S )
) )
14 anandi 801 . . . . . 6  |-  ( ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) )  <->  ( ( A  e.  S  /\  B  e.  S )  /\  ( A  e.  S  /\  C  e.  S
) ) )
155, 14bitri 240 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( ( A  e.  S  /\  B  e.  S
)  /\  ( A  e.  S  /\  C  e.  S ) ) )
1613, 15sylibr 203 . . . 4  |-  ( ( ( A G B )  e.  S  /\  ( A G C )  e.  S )  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
1716con3i 127 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  (
( A G B )  e.  S  /\  ( A G C )  e.  S ) )
181ndmov 6004 . . 3  |-  ( -.  ( ( A G B )  e.  S  /\  ( A G C )  e.  S )  ->  ( ( A G B ) F ( A G C ) )  =  (/) )
1917, 18syl 15 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  ( ( A G B ) F ( A G C ) )  =  (/) )
2010, 19eqtr4d 2318 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   (/)c0 3455    X. cxp 4687   dom cdm 4689  (class class class)co 5858
This theorem is referenced by:  distrpi  8522  distrnq  8585  distrpr  8652  distrsr  8713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-dm 4699  df-iota 5219  df-fv 5263  df-ov 5861
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