Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  i1abs GIF version

Theorem i1abs 801
 Description: An absorption law for →1 . (Contributed by NM, 21-Feb-2002.)
Assertion
Ref Expression
i1abs ((a1 b) ∪ (ab)) = a

Proof of Theorem i1abs
StepHypRef Expression
1 ud1lem0c 277 . . 3 (a1 b) = (a ∩ (ab ))
21ax-r5 38 . 2 ((a1 b) ∪ (ab)) = ((a ∩ (ab )) ∪ (ab))
3 comanr1 464 . . 3 a C (ab)
4 comorr 184 . . . 4 a C (ab )
54comcom6 459 . . 3 a C (ab )
63, 5fh4r 476 . 2 ((a ∩ (ab )) ∪ (ab)) = ((a ∪ (ab)) ∩ ((ab ) ∪ (ab)))
7 orabs 120 . . . 4 (a ∪ (ab)) = a
8 df-a 40 . . . . . 6 (ab) = (ab )
98lor 70 . . . . 5 ((ab ) ∪ (ab)) = ((ab ) ∪ (ab ) )
10 df-t 41 . . . . . 6 1 = ((ab ) ∪ (ab ) )
1110ax-r1 35 . . . . 5 ((ab ) ∪ (ab ) ) = 1
129, 11ax-r2 36 . . . 4 ((ab ) ∪ (ab)) = 1
137, 122an 79 . . 3 ((a ∪ (ab)) ∩ ((ab ) ∪ (ab))) = (a ∩ 1)
14 an1 106 . . 3 (a ∩ 1) = a
1513, 14ax-r2 36 . 2 ((a ∪ (ab)) ∩ ((ab ) ∪ (ab))) = a
162, 6, 153tr 65 1 ((a1 b) ∪ (ab)) = a
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →1 wi1 12 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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  cancellem  891
 Copyright terms: Public domain W3C validator