Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  u3lem10 GIF version

Theorem u3lem10 785
 Description: Lemma for unified implication study.
Assertion
Ref Expression
u3lem10 (a3 (a ∩ (ab))) = a

Proof of Theorem u3lem10
StepHypRef Expression
1 df-i3 46 . 2 (a3 (a ∩ (ab))) = (((a ∩ (a ∩ (ab))) ∪ (a ∩ (a ∩ (ab)) )) ∪ (a ∩ (a ∪ (a ∩ (ab)))))
2 anass 76 . . . . . . . 8 ((aa ) ∩ (ab)) = (a ∩ (a ∩ (ab)))
32ax-r1 35 . . . . . . 7 (a ∩ (a ∩ (ab))) = ((aa ) ∩ (ab))
4 anidm 111 . . . . . . . 8 (aa ) = a
54ran 78 . . . . . . 7 ((aa ) ∩ (ab)) = (a ∩ (ab))
63, 5ax-r2 36 . . . . . 6 (a ∩ (a ∩ (ab))) = (a ∩ (ab))
7 anor3 90 . . . . . . . . . . 11 (ab ) = (ab)
87lor 70 . . . . . . . . . 10 (a ∪ (ab )) = (a ∪ (ab) )
9 oran1 91 . . . . . . . . . 10 (a ∪ (ab) ) = (a ∩ (ab))
108, 9ax-r2 36 . . . . . . . . 9 (a ∪ (ab )) = (a ∩ (ab))
1110ax-r1 35 . . . . . . . 8 (a ∩ (ab)) = (a ∪ (ab ))
1211lan 77 . . . . . . 7 (a ∩ (a ∩ (ab)) ) = (a ∩ (a ∪ (ab )))
13 omlan 448 . . . . . . 7 (a ∩ (a ∪ (ab ))) = (ab )
1412, 13ax-r2 36 . . . . . 6 (a ∩ (a ∩ (ab)) ) = (ab )
156, 142or 72 . . . . 5 ((a ∩ (a ∩ (ab))) ∪ (a ∩ (a ∩ (ab)) )) = ((a ∩ (ab)) ∪ (ab ))
16 comanr1 464 . . . . . . 7 a C (ab )
17 comorr 184 . . . . . . . 8 a C (ab)
1817comcom3 454 . . . . . . 7 a C (ab)
1916, 18fh4r 476 . . . . . 6 ((a ∩ (ab)) ∪ (ab )) = ((a ∪ (ab )) ∩ ((ab) ∪ (ab )))
20 orabs 120 . . . . . . . 8 (a ∪ (ab )) = a
217lor 70 . . . . . . . . 9 ((ab) ∪ (ab )) = ((ab) ∪ (ab) )
22 df-t 41 . . . . . . . . . 10 1 = ((ab) ∪ (ab) )
2322ax-r1 35 . . . . . . . . 9 ((ab) ∪ (ab) ) = 1
2421, 23ax-r2 36 . . . . . . . 8 ((ab) ∪ (ab )) = 1
2520, 242an 79 . . . . . . 7 ((a ∪ (ab )) ∩ ((ab) ∪ (ab ))) = (a ∩ 1)
26 an1 106 . . . . . . 7 (a ∩ 1) = a
2725, 26ax-r2 36 . . . . . 6 ((a ∪ (ab )) ∩ ((ab) ∪ (ab ))) = a
2819, 27ax-r2 36 . . . . 5 ((a ∩ (ab)) ∪ (ab )) = a
2915, 28ax-r2 36 . . . 4 ((a ∩ (a ∩ (ab))) ∪ (a ∩ (a ∩ (ab)) )) = a
30 orabs 120 . . . . . 6 (a ∪ (a ∩ (ab))) = a
3130lan 77 . . . . 5 (a ∩ (a ∪ (a ∩ (ab)))) = (aa )
32 ancom 74 . . . . 5 (aa ) = (aa)
3331, 32ax-r2 36 . . . 4 (a ∩ (a ∪ (a ∩ (ab)))) = (aa)
3429, 332or 72 . . 3 (((a ∩ (a ∩ (ab))) ∪ (a ∩ (a ∩ (ab)) )) ∪ (a ∩ (a ∪ (a ∩ (ab))))) = (a ∪ (aa))
35 orabs 120 . . 3 (a ∪ (aa)) = a
3634, 35ax-r2 36 . 2 (((a ∩ (a ∩ (ab))) ∪ (a ∩ (a ∩ (ab)) )) ∪ (a ∩ (a ∪ (a ∩ (ab))))) = a
371, 36ax-r2 36 1 (a3 (a ∩ (ab))) = a
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →3 wi3 14 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator