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Theorem ndmaovass 27937
Description: Any operation is associative outside its domain. In contrast to ndmovass 6194 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.)
Hypothesis
Ref Expression
ndmaov.1  |-  dom  F  =  ( S  X.  S )
Assertion
Ref Expression
ndmaovass  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  -> (( (( A F B))  F C))  = (( A F (( B F C)) ))  )

Proof of Theorem ndmaovass
StepHypRef Expression
1 ndmaov.1 . . . . . . 7  |-  dom  F  =  ( S  X.  S )
21eleq2i 2468 . . . . . 6  |-  ( <. (( A F B))  ,  C >.  e.  dom  F  <->  <. (( A F B))  ,  C >.  e.  ( S  X.  S
) )
3 opelxp 4867 . . . . . 6  |-  ( <. (( A F B))  ,  C >.  e.  ( S  X.  S )  <->  ( (( A F B))  e.  S  /\  C  e.  S
) )
42, 3bitri 241 . . . . 5  |-  ( <. (( A F B))  ,  C >.  e.  dom  F  <->  ( (( A F B))  e.  S  /\  C  e.  S
) )
5 aovvdm 27916 . . . . . . 7  |-  ( (( A F B))  e.  S  -> 
<. A ,  B >.  e. 
dom  F )
61eleq2i 2468 . . . . . . . . 9  |-  ( <. A ,  B >.  e. 
dom  F  <->  <. A ,  B >.  e.  ( S  X.  S ) )
7 opelxp 4867 . . . . . . . . 9  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  <->  ( A  e.  S  /\  B  e.  S ) )
86, 7bitri 241 . . . . . . . 8  |-  ( <. A ,  B >.  e. 
dom  F  <->  ( A  e.  S  /\  B  e.  S ) )
9 df-3an 938 . . . . . . . . 9  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( ( A  e.  S  /\  B  e.  S
)  /\  C  e.  S ) )
109simplbi2 609 . . . . . . . 8  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( C  e.  S  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) )
118, 10sylbi 188 . . . . . . 7  |-  ( <. A ,  B >.  e. 
dom  F  ->  ( C  e.  S  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )
) )
125, 11syl 16 . . . . . 6  |-  ( (( A F B))  e.  S  ->  ( C  e.  S  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) )
1312imp 419 . . . . 5  |-  ( ( (( A F B))  e.  S  /\  C  e.  S
)  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) )
144, 13sylbi 188 . . . 4  |-  ( <. (( A F B))  ,  C >.  e.  dom  F  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
1514con3i 129 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  <. (( A F B))  ,  C >.  e. 
dom  F )
16 ndmaov 27914 . . 3  |-  ( -. 
<. (( A F B))  ,  C >.  e.  dom  F  -> (( (( A F B))  F C))  =  _V )
1715, 16syl 16 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  -> (( (( A F B))  F C))  =  _V )
181eleq2i 2468 . . . . . . 7  |-  ( <. A , (( B F C))  >.  e.  dom  F  <->  <. A , (( B F C))  >.  e.  ( S  X.  S ) )
19 opelxp 4867 . . . . . . 7  |-  ( <. A , (( B F C))  >.  e.  ( S  X.  S )  <->  ( A  e.  S  /\ (( B F C))  e.  S ) )
2018, 19bitri 241 . . . . . 6  |-  ( <. A , (( B F C))  >.  e.  dom  F  <->  ( A  e.  S  /\ (( B F C))  e.  S
) )
21 aovvdm 27916 . . . . . . . 8  |-  ( (( B F C))  e.  S  -> 
<. B ,  C >.  e. 
dom  F )
221eleq2i 2468 . . . . . . . . . 10  |-  ( <. B ,  C >.  e. 
dom  F  <->  <. B ,  C >.  e.  ( S  X.  S ) )
23 opelxp 4867 . . . . . . . . . 10  |-  ( <. B ,  C >.  e.  ( S  X.  S
)  <->  ( B  e.  S  /\  C  e.  S ) )
2422, 23bitri 241 . . . . . . . . 9  |-  ( <. B ,  C >.  e. 
dom  F  <->  ( B  e.  S  /\  C  e.  S ) )
25 3anass 940 . . . . . . . . . . . 12  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
2625biimpri 198 . . . . . . . . . . 11  |-  ( ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) )  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )
)
2726a1d 23 . . . . . . . . . 10  |-  ( ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) )  ->  ( <. A , (( B F C))  >.  e.  dom  F  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) )
2827expcom 425 . . . . . . . . 9  |-  ( ( B  e.  S  /\  C  e.  S )  ->  ( A  e.  S  ->  ( <. A , (( B F C))  >.  e.  dom  F  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) ) ) )
2924, 28sylbi 188 . . . . . . . 8  |-  ( <. B ,  C >.  e. 
dom  F  ->  ( A  e.  S  ->  ( <. A , (( B F C))  >.  e.  dom  F  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) ) )
3021, 29syl 16 . . . . . . 7  |-  ( (( B F C))  e.  S  ->  ( A  e.  S  ->  ( <. A , (( B F C))  >.  e.  dom  F  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) ) ) )
3130impcom 420 . . . . . 6  |-  ( ( A  e.  S  /\ (( B F C))  e.  S
)  ->  ( <. A , (( B F C)) 
>.  e.  dom  F  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) )
3220, 31sylbi 188 . . . . 5  |-  ( <. A , (( B F C))  >.  e.  dom  F  ->  ( <. A , (( B F C))  >.  e.  dom  F  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) ) )
3332pm2.43i 45 . . . 4  |-  ( <. A , (( B F C))  >.  e.  dom  F  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
3433con3i 129 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  <. A , (( B F C))  >.  e.  dom  F )
35 ndmaov 27914 . . 3  |-  ( -. 
<. A , (( B F C))  >.  e.  dom  F  -> (( A F (( B F C)) ))  =  _V )
3634, 35syl 16 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  -> (( A F (( B F C)) ))  =  _V )
3717, 36eqtr4d 2439 1  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  -> (( (( A F B))  F C))  = (( A F (( B F C)) ))  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916   <.cop 3777    X. cxp 4835   dom cdm 4837   ((caov 27840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-opab 4227  df-xp 4843  df-fv 5421  df-dfat 27841  df-afv 27842  df-aov 27843
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