Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  comi12 GIF version

Theorem comi12 707
 Description: Commutation theorem for →1 and →2 .
Assertion
Ref Expression
comi12 (a1 b) C (c2 a)

Proof of Theorem comi12
StepHypRef Expression
1 df-i1 44 . 2 (a1 b) = (a ∪ (ab))
2 lea 160 . . . . . . . 8 (a ∩ (ca ) ) ≤ a
3 leo 158 . . . . . . . 8 a ≤ (a ∪ (ab))
42, 3letr 137 . . . . . . 7 (a ∩ (ca ) ) ≤ (a ∪ (ab))
54lecom 180 . . . . . 6 (a ∩ (ca ) ) C (a ∪ (ab))
65comcom 453 . . . . 5 (a ∪ (ab)) C (a ∩ (ca ) )
7 anor3 90 . . . . 5 (a ∩ (ca ) ) = (a ∪ (ca ))
86, 7cbtr 182 . . . 4 (a ∪ (ab)) C (a ∪ (ca ))
98comcom7 460 . . 3 (a ∪ (ab)) C (a ∪ (ca ))
10 df-i2 45 . . . 4 (c2 a) = (a ∪ (ca ))
1110ax-r1 35 . . 3 (a ∪ (ca )) = (c2 a)
129, 11cbtr 182 . 2 (a ∪ (ab)) C (c2 a)
131, 12bctr 181 1 (a1 b) C (c2 a)
 Colors of variables: term Syntax hints:   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12   →2 wi2 13 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  orbi  842
 Copyright terms: Public domain W3C validator