Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ndmaovcom Structured version   Unicode version

Theorem ndmaovcom 28045
 Description: Any operation is commutative outside its domain, analogous to ndmovcom 6234. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypothesis
Ref Expression
ndmaov.1
Assertion
Ref Expression
ndmaovcom (()) (())

Proof of Theorem ndmaovcom
StepHypRef Expression
1 opelxp 4908 . . . 4
2 ndmaov.1 . . . . . 6
32eqcomi 2440 . . . . 5
43eleq2i 2500 . . . 4
51, 4bitr3i 243 . . 3
6 ndmaov 28023 . . 3 (())
75, 6sylnbi 298 . 2 (())
8 ancom 438 . . . 4
9 opelxp 4908 . . . 4
103eleq2i 2500 . . . 4
118, 9, 103bitr2i 265 . . 3
12 ndmaov 28023 . . 3 (())
1311, 12sylnbi 298 . 2 (())
147, 13eqtr4d 2471 1 (()) (())
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   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-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
 Copyright terms: Public domain W3C validator