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Theorem ndmovdistr 6236
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 6233 . . . . . 6  |-  ( ( B F C )  e.  S  ->  ( B  e.  S  /\  C  e.  S )
)
43anim2i 553 . . . . 5  |-  ( ( A  e.  S  /\  ( B F C )  e.  S )  -> 
( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
5 3anass 940 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
64, 5sylibr 204 . . . 4  |-  ( ( A  e.  S  /\  ( B F C )  e.  S )  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
76con3i 129 . . 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 6231 . . 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 6233 . . . . . 6  |-  ( ( A G B )  e.  S  ->  ( A  e.  S  /\  B  e.  S )
)
128, 2ndmovrcl 6233 . . . . . 6  |-  ( ( A G C )  e.  S  ->  ( A  e.  S  /\  C  e.  S )
)
1311, 12anim12i 550 . . . . 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 802 . . . . . 6  |-  ( ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) )  <->  ( ( A  e.  S  /\  B  e.  S )  /\  ( A  e.  S  /\  C  e.  S
) ) )
155, 14bitri 241 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( ( A  e.  S  /\  B  e.  S
)  /\  ( A  e.  S  /\  C  e.  S ) ) )
1613, 15sylibr 204 . . . 4  |-  ( ( ( A G B )  e.  S  /\  ( A G C )  e.  S )  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
1716con3i 129 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  (
( A G B )  e.  S  /\  ( A G C )  e.  S ) )
181ndmov 6231 . . 3  |-  ( -.  ( ( A G B )  e.  S  /\  ( A G C )  e.  S )  ->  ( ( A G B ) F ( A G C ) )  =  (/) )
1917, 18syl 16 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  ( ( A G B ) F ( A G C ) )  =  (/) )
2010, 19eqtr4d 2471 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   (/)c0 3628    X. cxp 4876   dom cdm 4878  (class class class)co 6081
This theorem is referenced by:  distrpi  8775  distrnq  8838  distrpr  8905  distrsr  8966
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-dm 4888  df-iota 5418  df-fv 5462  df-ov 6084
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