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Theorem ndmaovass 32494
Description: Any operation is associative outside its domain. In contrast to ndmovass 6462 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 2535 . . . . . 6  |-  ( <. (( A F B))  ,  C >.  e.  dom  F  <->  <. (( A F B))  ,  C >.  e.  ( S  X.  S
) )
3 opelxp 5038 . . . . . 6  |-  ( <. (( A F B))  ,  C >.  e.  ( S  X.  S )  <->  ( (( A F B))  e.  S  /\  C  e.  S
) )
42, 3bitri 249 . . . . 5  |-  ( <. (( A F B))  ,  C >.  e.  dom  F  <->  ( (( A F B))  e.  S  /\  C  e.  S
) )
5 aovvdm 32473 . . . . . . 7  |-  ( (( A F B))  e.  S  -> 
<. A ,  B >.  e. 
dom  F )
61eleq2i 2535 . . . . . . . . 9  |-  ( <. A ,  B >.  e. 
dom  F  <->  <. A ,  B >.  e.  ( S  X.  S ) )
7 opelxp 5038 . . . . . . . . 9  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  <->  ( A  e.  S  /\  B  e.  S ) )
86, 7bitri 249 . . . . . . . 8  |-  ( <. A ,  B >.  e. 
dom  F  <->  ( A  e.  S  /\  B  e.  S ) )
9 df-3an 975 . . . . . . . . 9  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( ( A  e.  S  /\  B  e.  S
)  /\  C  e.  S ) )
109simplbi2 625 . . . . . . . 8  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( C  e.  S  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) )
118, 10sylbi 195 . . . . . . 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 429 . . . . 5  |-  ( ( (( A F B))  e.  S  /\  C  e.  S
)  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) )
144, 13sylbi 195 . . . 4  |-  ( <. (( A F B))  ,  C >.  e.  dom  F  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
1514con3i 135 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  <. (( A F B))  ,  C >.  e. 
dom  F )
16 ndmaov 32471 . . 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 2535 . . . . . . 7  |-  ( <. A , (( B F C))  >.  e.  dom  F  <->  <. A , (( B F C))  >.  e.  ( S  X.  S ) )
19 opelxp 5038 . . . . . . 7  |-  ( <. A , (( B F C))  >.  e.  ( S  X.  S )  <->  ( A  e.  S  /\ (( B F C))  e.  S ) )
2018, 19bitri 249 . . . . . 6  |-  ( <. A , (( B F C))  >.  e.  dom  F  <->  ( A  e.  S  /\ (( B F C))  e.  S
) )
21 aovvdm 32473 . . . . . . . 8  |-  ( (( B F C))  e.  S  -> 
<. B ,  C >.  e. 
dom  F )
221eleq2i 2535 . . . . . . . . . 10  |-  ( <. B ,  C >.  e. 
dom  F  <->  <. B ,  C >.  e.  ( S  X.  S ) )
23 opelxp 5038 . . . . . . . . . 10  |-  ( <. B ,  C >.  e.  ( S  X.  S
)  <->  ( B  e.  S  /\  C  e.  S ) )
2422, 23bitri 249 . . . . . . . . 9  |-  ( <. B ,  C >.  e. 
dom  F  <->  ( B  e.  S  /\  C  e.  S ) )
25 3anass 977 . . . . . . . . . . . 12  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
2625biimpri 206 . . . . . . . . . . 11  |-  ( ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) )  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )
)
2726a1d 25 . . . . . . . . . 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 435 . . . . . . . . 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 195 . . . . . . . 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 430 . . . . . 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 195 . . . . 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 47 . . . 4  |-  ( <. A , (( B F C))  >.  e.  dom  F  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
3433con3i 135 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  ->  -.  <. A , (( B F C))  >.  e.  dom  F )
35 ndmaov 32471 . . 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 2501 1  |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  -> (( (( A F B))  F C))  = (( A F (( B F C)) ))  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   _Vcvv 3109   <.cop 4038    X. cxp 5006   dom cdm 5008   ((caov 32403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-opab 4516  df-xp 5014  df-fv 5602  df-dfat 32404  df-afv 32405  df-aov 32406
This theorem is referenced by: (None)
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