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Theorem ndmaovass 27394
Description: Any operation is associative outside its domain. In contrast to ndmovass 6095 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 2422 . . . . . 6  |-  ( <. (( A F B))  ,  C >.  e.  dom  F  <->  <. (( A F B))  ,  C >.  e.  ( S  X.  S
) )
3 opelxp 4801 . . . . . 6  |-  ( <. (( A F B))  ,  C >.  e.  ( S  X.  S )  <->  ( (( A F B))  e.  S  /\  C  e.  S
) )
42, 3bitri 240 . . . . 5  |-  ( <. (( A F B))  ,  C >.  e.  dom  F  <->  ( (( A F B))  e.  S  /\  C  e.  S
) )
5 aovvdm 27373 . . . . . . 7  |-  ( (( A F B))  e.  S  -> 
<. A ,  B >.  e. 
dom  F )
61eleq2i 2422 . . . . . . . . 9  |-  ( <. A ,  B >.  e. 
dom  F  <->  <. A ,  B >.  e.  ( S  X.  S ) )
7 opelxp 4801 . . . . . . . . 9  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  <->  ( A  e.  S  /\  B  e.  S ) )
86, 7bitri 240 . . . . . . . 8  |-  ( <. A ,  B >.  e. 
dom  F  <->  ( A  e.  S  /\  B  e.  S ) )
9 df-3an 936 . . . . . . . . 9  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( ( A  e.  S  /\  B  e.  S
)  /\  C  e.  S ) )
109simplbi2 608 . . . . . . . 8  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( C  e.  S  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) )
118, 10sylbi 187 . . . . . . 7  |-  ( <. A ,  B >.  e. 
dom  F  ->  ( C  e.  S  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )
) )
125, 11syl 15 . . . . . 6  |-  ( (( A F B))  e.  S  ->  ( C  e.  S  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) )
1312imp 418 . . . . 5  |-  ( ( (( A F B))  e.  S  /\  C  e.  S
)  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) )
144, 13sylbi 187 . . . 4  |-  ( <. (( A F B))  ,  C >.  e.  dom  F  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
1514con3i 127 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  <. (( A F B))  ,  C >.  e. 
dom  F )
16 ndmaov 27371 . . 3  |-  ( -. 
<. (( A F B))  ,  C >.  e.  dom  F  -> (( (( A F B))  F C))  =  _V )
1715, 16syl 15 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  -> (( (( A F B))  F C))  =  _V )
181eleq2i 2422 . . . . . . 7  |-  ( <. A , (( B F C))  >.  e.  dom  F  <->  <. A , (( B F C))  >.  e.  ( S  X.  S ) )
19 opelxp 4801 . . . . . . 7  |-  ( <. A , (( B F C))  >.  e.  ( S  X.  S )  <->  ( A  e.  S  /\ (( B F C))  e.  S ) )
2018, 19bitri 240 . . . . . 6  |-  ( <. A , (( B F C))  >.  e.  dom  F  <->  ( A  e.  S  /\ (( B F C))  e.  S
) )
21 aovvdm 27373 . . . . . . . 8  |-  ( (( B F C))  e.  S  -> 
<. B ,  C >.  e. 
dom  F )
221eleq2i 2422 . . . . . . . . . 10  |-  ( <. B ,  C >.  e. 
dom  F  <->  <. B ,  C >.  e.  ( S  X.  S ) )
23 opelxp 4801 . . . . . . . . . 10  |-  ( <. B ,  C >.  e.  ( S  X.  S
)  <->  ( B  e.  S  /\  C  e.  S ) )
2422, 23bitri 240 . . . . . . . . 9  |-  ( <. B ,  C >.  e. 
dom  F  <->  ( B  e.  S  /\  C  e.  S ) )
25 3anass 938 . . . . . . . . . . . 12  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
2625biimpri 197 . . . . . . . . . . 11  |-  ( ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) )  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )
)
2726a1d 22 . . . . . . . . . 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 424 . . . . . . . . 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 187 . . . . . . . 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 15 . . . . . . 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 419 . . . . . 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 187 . . . . 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 43 . . . 4  |-  ( <. A , (( B F C))  >.  e.  dom  F  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
3433con3i 127 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  <. A , (( B F C))  >.  e.  dom  F )
35 ndmaov 27371 . . 3  |-  ( -. 
<. A , (( B F C))  >.  e.  dom  F  -> (( A F (( B F C)) ))  =  _V )
3634, 35syl 15 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  -> (( A F (( B F C)) ))  =  _V )
3717, 36eqtr4d 2393 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 358    /\ w3a 934    = wceq 1642    e. wcel 1710   _Vcvv 2864   <.cop 3719    X. cxp 4769   dom cdm 4771   ((caov 27296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-opab 4159  df-xp 4777  df-fv 5345  df-dfat 27297  df-afv 27298  df-aov 27299
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