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

Theorem wdcom 1103
 Description: Any two variables (weakly) commute in a WDOL.
Assertion
Ref Expression
wdcom C (a, b) = 1

Proof of Theorem wdcom
StepHypRef Expression
1 df-cmtr 134 . 2 C (a, b) = (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab )))
2 or42 85 . 2 (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab ))) = (((ab) ∪ (ab )) ∪ ((ab ) ∪ (ab)))
3 dfb 94 . . . . 5 (ab) = ((ab) ∪ (ab ))
4 dfb 94 . . . . . 6 (ab ) = ((ab ) ∪ (ab ))
5 ax-a1 30 . . . . . . . . 9 b = b
65lan 77 . . . . . . . 8 (ab) = (ab )
76ax-r1 35 . . . . . . 7 (ab ) = (ab)
87lor 70 . . . . . 6 ((ab ) ∪ (ab )) = ((ab ) ∪ (ab))
94, 8ax-r2 36 . . . . 5 (ab ) = ((ab ) ∪ (ab))
103, 92or 72 . . . 4 ((ab) ∪ (ab )) = (((ab) ∪ (ab )) ∪ ((ab ) ∪ (ab)))
1110ax-r1 35 . . 3 (((ab) ∪ (ab )) ∪ ((ab ) ∪ (ab))) = ((ab) ∪ (ab ))
12 ax-wdol 1102 . . 3 ((ab) ∪ (ab )) = 1
1311, 12ax-r2 36 . 2 (((ab) ∪ (ab )) ∪ ((ab ) ∪ (ab))) = 1
141, 2, 133tr 65 1 C (a, b) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   C wcmtr 29 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-wdol 1102 This theorem depends on definitions:  df-b 39  df-a 40  df-cmtr 134 This theorem is referenced by:  wdwom  1104  wddi1  1105  wddi3  1107
 Copyright terms: Public domain W3C validator