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Theorem ndmaovass 28046
 Description: Any operation is associative outside its domain. In contrast to ndmovass 6235 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 2500 . . . . . 6 (()) (())
3 opelxp 4908 . . . . . 6 (()) (())
42, 3bitri 241 . . . . 5 (()) (())
5 aovvdm 28025 . . . . . . 7 (())
61eleq2i 2500 . . . . . . . . 9
7 opelxp 4908 . . . . . . . . 9
86, 7bitri 241 . . . . . . . 8
9 df-3an 938 . . . . . . . . 9
109simplbi2 609 . . . . . . . 8
118, 10sylbi 188 . . . . . . 7
125, 11syl 16 . . . . . 6 (())
1312imp 419 . . . . 5 (())
144, 13sylbi 188 . . . 4 (())
1514con3i 129 . . 3 (())
16 ndmaov 28023 . . 3 (()) (( (()) ))
1715, 16syl 16 . 2 (( (()) ))
181eleq2i 2500 . . . . . . 7 (()) (())
19 opelxp 4908 . . . . . . 7 (()) (())
2018, 19bitri 241 . . . . . 6 (()) (())
21 aovvdm 28025 . . . . . . . 8 (())
221eleq2i 2500 . . . . . . . . . 10
23 opelxp 4908 . . . . . . . . . 10
2422, 23bitri 241 . . . . . . . . 9
25 3anass 940 . . . . . . . . . . . 12
2625biimpri 198 . . . . . . . . . . 11
2726a1d 23 . . . . . . . . . 10 (())
2827expcom 425 . . . . . . . . 9 (())
2924, 28sylbi 188 . . . . . . . 8 (())
3021, 29syl 16 . . . . . . 7 (()) (())
3130impcom 420 . . . . . 6 (()) (())
3220, 31sylbi 188 . . . . 5 (()) (())
3332pm2.43i 45 . . . 4 (())
3433con3i 129 . . 3 (())
35 ndmaov 28023 . . 3 (()) (( (()) ))
3634, 35syl 16 . 2 (( (()) ))
3717, 36eqtr4d 2471 1 (( (()) )) (( (()) ))
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  cvv 2956  cop 3817   cxp 4876   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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