Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  oacom2 GIF version

Theorem oacom2 1012
 Description: Commutation law requiring OA.
Hypothesis
Ref Expression
oacom2.1 d ≤ ((a2 b) ∩ ((bc) →0 ((a2 b) ∩ (a2 c))))
Assertion
Ref Expression
oacom2 d C ((a2 b) ∩ (a2 c))

Proof of Theorem oacom2
StepHypRef Expression
1 oacom2.1 . . . 4 d ≤ ((a2 b) ∩ ((bc) →0 ((a2 b) ∩ (a2 c))))
2 lear 161 . . . 4 ((a2 b) ∩ ((bc) →0 ((a2 b) ∩ (a2 c)))) ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
31, 2letr 137 . . 3 d ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
43lecom 180 . 2 d C ((bc) →0 ((a2 b) ∩ (a2 c)))
5 lea 160 . . . 4 (d ∩ ((bc) →0 ((a2 b) ∩ (a2 c)))) ≤ d
6 lea 160 . . . . 5 ((a2 b) ∩ ((bc) →0 ((a2 b) ∩ (a2 c)))) ≤ (a2 b)
71, 6letr 137 . . . 4 d ≤ (a2 b)
85, 7letr 137 . . 3 (d ∩ ((bc) →0 ((a2 b) ∩ (a2 c)))) ≤ (a2 b)
98lecom 180 . 2 (d ∩ ((bc) →0 ((a2 b) ∩ (a2 c)))) C (a2 b)
104, 9oacom 1011 1 d C ((a2 b) ∩ (a2 c))
 Colors of variables: term Syntax hints:   ≤ wle 2   C wc 3   ∪ wo 6   ∩ wa 7   →0 wi0 11   →2 wi2 13 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  ax-3oa 998 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator