Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  u5lemana GIF version

Theorem u5lemana 609
 Description: Lemma for relevance implication study.
Assertion
Ref Expression
u5lemana ((a5 b) ∩ a ) = ((ab) ∪ (ab ))

Proof of Theorem u5lemana
StepHypRef Expression
1 df-i5 48 . . 3 (a5 b) = (((ab) ∪ (ab)) ∪ (ab ))
21ran 78 . 2 ((a5 b) ∩ a ) = ((((ab) ∪ (ab)) ∪ (ab )) ∩ a )
3 comanr1 464 . . . . . 6 a C (ab)
43comcom3 454 . . . . 5 a C (ab)
5 comanr1 464 . . . . 5 a C (ab)
64, 5com2or 483 . . . 4 a C ((ab) ∪ (ab))
7 comanr1 464 . . . 4 a C (ab )
86, 7fh1r 473 . . 3 ((((ab) ∪ (ab)) ∪ (ab )) ∩ a ) = ((((ab) ∪ (ab)) ∩ a ) ∪ ((ab ) ∩ a ))
94, 5fh1r 473 . . . . 5 (((ab) ∪ (ab)) ∩ a ) = (((ab) ∩ a ) ∪ ((ab) ∩ a ))
10 ax-a2 31 . . . . . 6 (((ab) ∩ a ) ∪ ((ab) ∩ a )) = (((ab) ∩ a ) ∪ ((ab) ∩ a ))
11 an32 83 . . . . . . . . 9 ((ab) ∩ a ) = ((aa ) ∩ b)
12 anidm 111 . . . . . . . . . 10 (aa ) = a
1312ran 78 . . . . . . . . 9 ((aa ) ∩ b) = (ab)
1411, 13ax-r2 36 . . . . . . . 8 ((ab) ∩ a ) = (ab)
15 an32 83 . . . . . . . . 9 ((ab) ∩ a ) = ((aa ) ∩ b)
16 ancom 74 . . . . . . . . . 10 ((aa ) ∩ b) = (b ∩ (aa ))
17 dff 101 . . . . . . . . . . . . 13 0 = (aa )
1817lan 77 . . . . . . . . . . . 12 (b ∩ 0) = (b ∩ (aa ))
1918ax-r1 35 . . . . . . . . . . 11 (b ∩ (aa )) = (b ∩ 0)
20 an0 108 . . . . . . . . . . 11 (b ∩ 0) = 0
2119, 20ax-r2 36 . . . . . . . . . 10 (b ∩ (aa )) = 0
2216, 21ax-r2 36 . . . . . . . . 9 ((aa ) ∩ b) = 0
2315, 22ax-r2 36 . . . . . . . 8 ((ab) ∩ a ) = 0
2414, 232or 72 . . . . . . 7 (((ab) ∩ a ) ∪ ((ab) ∩ a )) = ((ab) ∪ 0)
25 or0 102 . . . . . . 7 ((ab) ∪ 0) = (ab)
2624, 25ax-r2 36 . . . . . 6 (((ab) ∩ a ) ∪ ((ab) ∩ a )) = (ab)
2710, 26ax-r2 36 . . . . 5 (((ab) ∩ a ) ∪ ((ab) ∩ a )) = (ab)
289, 27ax-r2 36 . . . 4 (((ab) ∪ (ab)) ∩ a ) = (ab)
29 an32 83 . . . . 5 ((ab ) ∩ a ) = ((aa ) ∩ b )
3012ran 78 . . . . 5 ((aa ) ∩ b ) = (ab )
3129, 30ax-r2 36 . . . 4 ((ab ) ∩ a ) = (ab )
3228, 312or 72 . . 3 ((((ab) ∪ (ab)) ∩ a ) ∪ ((ab ) ∩ a )) = ((ab) ∪ (ab ))
338, 32ax-r2 36 . 2 ((((ab) ∪ (ab)) ∪ (ab )) ∩ a ) = ((ab) ∪ (ab ))
342, 33ax-r2 36 1 ((a5 b) ∩ a ) = ((ab) ∪ (ab ))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →5 wi5 16 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-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  u5lemnoa  664
 Copyright terms: Public domain W3C validator