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Theorem ndmaovdistr 28047
 Description: Any operation is distributive outside its domain. In contrast to ndmovdistr 6236 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.)
Hypotheses
Ref Expression
ndmaov.1
ndmaov.6
Assertion
Ref Expression
ndmaovdistr (( (()) )) (( (()) (()) ))

Proof of Theorem ndmaovdistr
StepHypRef Expression
1 ndmaov.6 . . . . . . 7
21eleq2i 2500 . . . . . 6 (()) (())
3 opelxp 4908 . . . . . 6 (()) (())
42, 3bitri 241 . . . . 5 (()) (())
5 aovvdm 28025 . . . . . . 7 (())
6 ndmaov.1 . . . . . . . . . 10
76eleq2i 2500 . . . . . . . . 9
8 opelxp 4908 . . . . . . . . 9
97, 8bitri 241 . . . . . . . 8
10 3anass 940 . . . . . . . . 9
1110simplbi2com 1383 . . . . . . . 8
129, 11sylbi 188 . . . . . . 7
135, 12syl 16 . . . . . 6 (())
1413impcom 420 . . . . 5 (())
154, 14sylbi 188 . . . 4 (())
1615con3i 129 . . 3 (())
17 ndmaov 28023 . . 3 (()) (( (()) ))
1816, 17syl 16 . 2 (( (()) ))
196eleq2i 2500 . . . . . 6 (()) (()) (()) (())
20 opelxp 4908 . . . . . 6 (()) (()) (()) (())
2119, 20bitri 241 . . . . 5 (()) (()) (()) (())
22 aovvdm 28025 . . . . . . 7 (())
231eleq2i 2500 . . . . . . . . 9
24 opelxp 4908 . . . . . . . . 9
2523, 24bitri 241 . . . . . . . 8
26 aovvdm 28025 . . . . . . . . . 10 (())
271eleq2i 2500 . . . . . . . . . . . 12
28 opelxp 4908 . . . . . . . . . . . 12
2927, 28bitri 241 . . . . . . . . . . 11
30 simpll 731 . . . . . . . . . . . . 13
31 simprr 734 . . . . . . . . . . . . 13
32 simplr 732 . . . . . . . . . . . . 13
3330, 31, 323jca 1134 . . . . . . . . . . . 12
3433ex 424 . . . . . . . . . . 11
3529, 34sylbi 188 . . . . . . . . . 10
3626, 35syl 16 . . . . . . . . 9 (())
3736com12 29 . . . . . . . 8 (())
3825, 37sylbi 188 . . . . . . 7 (())
3922, 38syl 16 . . . . . 6 (()) (())
4039imp 419 . . . . 5 (()) (())
4121, 40sylbi 188 . . . 4 (()) (())
4241con3i 129 . . 3 (()) (())
43 ndmaov 28023 . . 3 (()) (()) (( (()) (()) ))
4442, 43syl 16 . 2 (( (()) (()) ))
4518, 44eqtr4d 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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