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Theorem ndmaovdistr 28047
Description: Any operation is distributive outside its domain. In contrast to ndmovdistr 6236 where it is required that the operation's domain doesn't contain the empty set (
-.  (/)  e.  S), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypotheses
Ref Expression
ndmaov.1  |-  dom  F  =  ( S  X.  S )
ndmaov.6  |-  dom  G  =  ( S  X.  S )
Assertion
Ref Expression
ndmaovdistr  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  -> (( A G (( B F C)) ))  = (( (( A G B))  F (( A G C)) ))  )

Proof of Theorem ndmaovdistr
StepHypRef Expression
1 ndmaov.6 . . . . . . 7  |-  dom  G  =  ( S  X.  S )
21eleq2i 2500 . . . . . 6  |-  ( <. A , (( B F C))  >.  e.  dom  G  <->  <. A , (( B F C))  >.  e.  ( S  X.  S ) )
3 opelxp 4908 . . . . . 6  |-  ( <. A , (( B F C))  >.  e.  ( S  X.  S )  <->  ( A  e.  S  /\ (( B F C))  e.  S ) )
42, 3bitri 241 . . . . 5  |-  ( <. A , (( B F C))  >.  e.  dom  G  <->  ( A  e.  S  /\ (( B F C))  e.  S
) )
5 aovvdm 28025 . . . . . . 7  |-  ( (( B F C))  e.  S  -> 
<. B ,  C >.  e. 
dom  F )
6 ndmaov.1 . . . . . . . . . 10  |-  dom  F  =  ( S  X.  S )
76eleq2i 2500 . . . . . . . . 9  |-  ( <. B ,  C >.  e. 
dom  F  <->  <. B ,  C >.  e.  ( S  X.  S ) )
8 opelxp 4908 . . . . . . . . 9  |-  ( <. B ,  C >.  e.  ( S  X.  S
)  <->  ( B  e.  S  /\  C  e.  S ) )
97, 8bitri 241 . . . . . . . 8  |-  ( <. B ,  C >.  e. 
dom  F  <->  ( B  e.  S  /\  C  e.  S ) )
10 3anass 940 . . . . . . . . 9  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
1110simplbi2com 1383 . . . . . . . 8  |-  ( ( B  e.  S  /\  C  e.  S )  ->  ( A  e.  S  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) )
129, 11sylbi 188 . . . . . . 7  |-  ( <. B ,  C >.  e. 
dom  F  ->  ( A  e.  S  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )
) )
135, 12syl 16 . . . . . 6  |-  ( (( B F C))  e.  S  ->  ( A  e.  S  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) )
1413impcom 420 . . . . 5  |-  ( ( A  e.  S  /\ (( B F C))  e.  S
)  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) )
154, 14sylbi 188 . . . 4  |-  ( <. A , (( B F C))  >.  e.  dom  G  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
1615con3i 129 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  <. A , (( B F C))  >.  e.  dom  G )
17 ndmaov 28023 . . 3  |-  ( -. 
<. A , (( B F C))  >.  e.  dom  G  -> (( A G (( B F C)) ))  =  _V )
1816, 17syl 16 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  -> (( A G (( B F C)) ))  =  _V )
196eleq2i 2500 . . . . . 6  |-  ( <. (( A G B))  , (( A G C))  >.  e.  dom  F  <->  <. (( A G B))  , (( A G C))  >.  e.  ( S  X.  S ) )
20 opelxp 4908 . . . . . 6  |-  ( <. (( A G B))  , (( A G C))  >.  e.  ( S  X.  S )  <-> 
( (( A G B))  e.  S  /\ (( A G C))  e.  S
) )
2119, 20bitri 241 . . . . 5  |-  ( <. (( A G B))  , (( A G C))  >.  e.  dom  F  <-> 
( (( A G B))  e.  S  /\ (( A G C))  e.  S
) )
22 aovvdm 28025 . . . . . . 7  |-  ( (( A G B))  e.  S  -> 
<. A ,  B >.  e. 
dom  G )
231eleq2i 2500 . . . . . . . . 9  |-  ( <. A ,  B >.  e. 
dom  G  <->  <. A ,  B >.  e.  ( S  X.  S ) )
24 opelxp 4908 . . . . . . . . 9  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  <->  ( A  e.  S  /\  B  e.  S ) )
2523, 24bitri 241 . . . . . . . 8  |-  ( <. A ,  B >.  e. 
dom  G  <->  ( A  e.  S  /\  B  e.  S ) )
26 aovvdm 28025 . . . . . . . . . 10  |-  ( (( A G C))  e.  S  -> 
<. A ,  C >.  e. 
dom  G )
271eleq2i 2500 . . . . . . . . . . . 12  |-  ( <. A ,  C >.  e. 
dom  G  <->  <. A ,  C >.  e.  ( S  X.  S ) )
28 opelxp 4908 . . . . . . . . . . . 12  |-  ( <. A ,  C >.  e.  ( S  X.  S
)  <->  ( A  e.  S  /\  C  e.  S ) )
2927, 28bitri 241 . . . . . . . . . . 11  |-  ( <. A ,  C >.  e. 
dom  G  <->  ( A  e.  S  /\  C  e.  S ) )
30 simpll 731 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  ( A  e.  S  /\  B  e.  S ) )  ->  A  e.  S )
31 simprr 734 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  ( A  e.  S  /\  B  e.  S ) )  ->  B  e.  S )
32 simplr 732 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  ( A  e.  S  /\  B  e.  S ) )  ->  C  e.  S )
3330, 31, 323jca 1134 . . . . . . . . . . . 12  |-  ( ( ( A  e.  S  /\  C  e.  S
)  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
3433ex 424 . . . . . . . . . . 11  |-  ( ( A  e.  S  /\  C  e.  S )  ->  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )
) )
3529, 34sylbi 188 . . . . . . . . . 10  |-  ( <. A ,  C >.  e. 
dom  G  ->  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) )
3626, 35syl 16 . . . . . . . . 9  |-  ( (( A G C))  e.  S  ->  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )
) )
3736com12 29 . . . . . . . 8  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( (( A G C))  e.  S  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )
) )
3825, 37sylbi 188 . . . . . . 7  |-  ( <. A ,  B >.  e. 
dom  G  ->  ( (( A G C))  e.  S  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) )
3922, 38syl 16 . . . . . 6  |-  ( (( A G B))  e.  S  ->  ( (( A G C))  e.  S  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )
) )
4039imp 419 . . . . 5  |-  ( ( (( A G B))  e.  S  /\ (( A G C))  e.  S )  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
4121, 40sylbi 188 . . . 4  |-  ( <. (( A G B))  , (( A G C))  >.  e.  dom  F  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) )
4241con3i 129 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  <. (( A G B))  , (( A G C))  >.  e.  dom  F
)
43 ndmaov 28023 . . 3  |-  ( -. 
<. (( A G B))  , (( A G C))  >.  e.  dom  F  -> (( (( A G B))  F (( A G C)) ))  =  _V )
4442, 43syl 16 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  -> (( (( A G B))  F (( A G C)) )) 
=  _V )
4518, 44eqtr4d 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   _Vcvv 2956   <.cop 3817    X. cxp 4876   dom cdm 4878   ((caov 27949
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-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-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-opab 4267  df-xp 4884  df-fv 5462  df-dfat 27950  df-afv 27951  df-aov 27952
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