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Theorem ndmaovass 28066
 Description: Any operation is associative outside its domain. In contrast to ndmovass 6008 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 2347 . . . . . 6 (()) (())
3 opelxp 4719 . . . . . 6 (()) (())
42, 3bitri 240 . . . . 5 (()) (())
5 aovvdm 28045 . . . . . . 7 (())
61eleq2i 2347 . . . . . . . . 9
7 opelxp 4719 . . . . . . . . 9
86, 7bitri 240 . . . . . . . 8
9 df-3an 936 . . . . . . . . 9
109simplbi2 608 . . . . . . . 8
118, 10sylbi 187 . . . . . . 7
125, 11syl 15 . . . . . 6 (())
1312imp 418 . . . . 5 (())
144, 13sylbi 187 . . . 4 (())
1514con3i 127 . . 3 (())
16 ndmaov 28043 . . 3 (()) (( (()) ))
1715, 16syl 15 . 2 (( (()) ))
181eleq2i 2347 . . . . . . 7 (()) (())
19 opelxp 4719 . . . . . . 7 (()) (())
2018, 19bitri 240 . . . . . 6 (()) (())
21 aovvdm 28045 . . . . . . . 8 (())
221eleq2i 2347 . . . . . . . . . 10
23 opelxp 4719 . . . . . . . . . 10
2422, 23bitri 240 . . . . . . . . 9
25 3anass 938 . . . . . . . . . . . 12
2625biimpri 197 . . . . . . . . . . 11
2726a1d 22 . . . . . . . . . 10 (())
2827expcom 424 . . . . . . . . 9 (())
2924, 28sylbi 187 . . . . . . . 8 (())
3021, 29syl 15 . . . . . . 7 (()) (())
3130impcom 419 . . . . . 6 (()) (())
3220, 31sylbi 187 . . . . 5 (()) (())
3332pm2.43i 43 . . . 4 (())
3433con3i 127 . . 3 (())
35 ndmaov 28043 . . 3 (()) (( (()) ))
3634, 35syl 15 . 2 (( (()) ))
3717, 36eqtr4d 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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