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Theorem ndmaovdistr 28067
 Description: Any operation is distributive outside its domain. In contrast to ndmovdistr 6009 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 2347 . . . . . 6 (()) (())
3 opelxp 4719 . . . . . 6 (()) (())
42, 3bitri 240 . . . . 5 (()) (())
5 aovvdm 28045 . . . . . . 7 (())
6 ndmaov.1 . . . . . . . . . 10
76eleq2i 2347 . . . . . . . . 9
8 opelxp 4719 . . . . . . . . 9
97, 8bitri 240 . . . . . . . 8
10 3anass 938 . . . . . . . . 9
1110simplbi2com 1364 . . . . . . . 8
129, 11sylbi 187 . . . . . . 7
135, 12syl 15 . . . . . 6 (())
1413impcom 419 . . . . 5 (())
154, 14sylbi 187 . . . 4 (())
1615con3i 127 . . 3 (())
17 ndmaov 28043 . . 3 (()) (( (()) ))
1816, 17syl 15 . 2 (( (()) ))
196eleq2i 2347 . . . . . 6 (()) (()) (()) (())
20 opelxp 4719 . . . . . 6 (()) (()) (()) (())
2119, 20bitri 240 . . . . 5 (()) (()) (()) (())
22 aovvdm 28045 . . . . . . 7 (())
231eleq2i 2347 . . . . . . . . 9
24 opelxp 4719 . . . . . . . . 9
2523, 24bitri 240 . . . . . . . 8
26 aovvdm 28045 . . . . . . . . . 10 (())
271eleq2i 2347 . . . . . . . . . . . 12
28 opelxp 4719 . . . . . . . . . . . 12
2927, 28bitri 240 . . . . . . . . . . 11
30 simpll 730 . . . . . . . . . . . . 13
31 simprr 733 . . . . . . . . . . . . 13
32 simplr 731 . . . . . . . . . . . . 13
3330, 31, 323jca 1132 . . . . . . . . . . . 12
3433ex 423 . . . . . . . . . . 11
3529, 34sylbi 187 . . . . . . . . . 10
3626, 35syl 15 . . . . . . . . 9 (())
3736com12 27 . . . . . . . 8 (())
3825, 37sylbi 187 . . . . . . 7 (())
3922, 38syl 15 . . . . . 6 (()) (())
4039imp 418 . . . . 5 (()) (())
4121, 40sylbi 187 . . . 4 (()) (())
4241con3i 127 . . 3 (()) (())
43 ndmaov 28043 . . 3 (()) (()) (( (()) (()) ))
4442, 43syl 15 . 2 (( (()) (()) ))
4518, 44eqtr4d 2318 1 (( (()) )) (( (()) (()) ))
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 358   w3a 934   wceq 1623   wcel 1684  cvv 2788  cop 3643   cxp 4687   cdm 4689   ((caov 27973 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-dfat 27974  df-afv 27975  df-aov 27976
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