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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 ( ), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypothesis
Ref Expression
ndmaov.1
Assertion
Ref Expression
ndmaovass (( (()) )) (( (()) ))

Proof of Theorem ndmaovass
StepHypRef Expression
1 ndmaov.1 . . . . . . 7
21eleq2i 2535 . . . . . 6 (()) (())
3 opelxp 5038 . . . . . 6 (()) (())
42, 3bitri 249 . . . . 5 (()) (())
5 aovvdm 32473 . . . . . . 7 (())
61eleq2i 2535 . . . . . . . . 9
7 opelxp 5038 . . . . . . . . 9
86, 7bitri 249 . . . . . . . 8
9 df-3an 975 . . . . . . . . 9
109simplbi2 625 . . . . . . . 8
118, 10sylbi 195 . . . . . . 7
125, 11syl 16 . . . . . 6 (())
1312imp 429 . . . . 5 (())
144, 13sylbi 195 . . . 4 (())
1514con3i 135 . . 3 (())
16 ndmaov 32471 . . 3 (()) (( (()) ))
1715, 16syl 16 . 2 (( (()) ))
181eleq2i 2535 . . . . . . 7 (()) (())
19 opelxp 5038 . . . . . . 7 (()) (())
2018, 19bitri 249 . . . . . 6 (()) (())
21 aovvdm 32473 . . . . . . . 8 (())
221eleq2i 2535 . . . . . . . . . 10
23 opelxp 5038 . . . . . . . . . 10
2422, 23bitri 249 . . . . . . . . 9
25 3anass 977 . . . . . . . . . . . 12
2625biimpri 206 . . . . . . . . . . 11
2726a1d 25 . . . . . . . . . 10 (())
2827expcom 435 . . . . . . . . 9 (())
2924, 28sylbi 195 . . . . . . . 8 (())
3021, 29syl 16 . . . . . . 7 (()) (())
3130impcom 430 . . . . . 6 (()) (())
3220, 31sylbi 195 . . . . 5 (()) (())
3332pm2.43i 47 . . . 4 (())
3433con3i 135 . . 3 (())
35 ndmaov 32471 . . 3 (()) (( (()) ))
3634, 35syl 16 . 2 (( (()) ))
3717, 36eqtr4d 2501 1 (( (()) )) (( (()) ))
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 369   w3a 973   wceq 1395   wcel 1819  cvv 3109  cop 4038   cxp 5006   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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