Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  com3iia GIF version

Theorem com3iia 1100
 Description: The dual of com3ii 457. (Contributed by Roy F. Longton, 3-Jul-05.)
Hypothesis
Ref Expression
com3iia.1 a C b
Assertion
Ref Expression
com3iia (a ∪ (ab)) = (ab)

Proof of Theorem com3iia
StepHypRef Expression
1 comid 187 . . . 4 a C a
21comcom2 183 . . 3 a C a
3 com3iia.1 . . 3 a C b
42, 3fh3 471 . 2 (a ∪ (ab)) = ((aa ) ∩ (ab))
5 lear 161 . . 3 ((aa ) ∩ (ab)) ≤ (ab)
6 ax-a4 33 . . . . 5 ((ab) ∪ (aa )) = (aa )
76df-le1 130 . . . 4 (ab) ≤ (aa )
8 leid 148 . . . 4 (ab) ≤ (ab)
97, 8ler2an 173 . . 3 (ab) ≤ ((aa ) ∩ (ab))
105, 9lebi 145 . 2 ((aa ) ∩ (ab)) = (ab)
114, 10ax-r2 36 1 (a ∪ (ab)) = (ab)
 Colors of variables: term Syntax hints:   = wb 1   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7 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-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator