Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  i3orlem3 GIF version

Theorem i3orlem3 554
 Description: Lemma for Kalmbach implication OR builder.
Assertion
Ref Expression
i3orlem3 c ≤ ((ac) →3 (bc))

Proof of Theorem i3orlem3
StepHypRef Expression
1 ax-a2 31 . . . . . 6 ((ac)c) = (c ∪ (ac) )
21lan 77 . . . . 5 (c ∩ ((ac)c)) = (c ∩ (c ∪ (ac) ))
3 anabs 121 . . . . 5 (c ∩ (c ∪ (ac) )) = c
42, 3ax-r2 36 . . . 4 (c ∩ ((ac)c)) = c
54ax-r1 35 . . 3 c = (c ∩ ((ac)c))
6 leor 159 . . . 4 c ≤ (ac)
7 leor 159 . . . . 5 c ≤ (bc)
87lelor 166 . . . 4 ((ac)c) ≤ ((ac) ∪ (bc))
96, 8le2an 169 . . 3 (c ∩ ((ac)c)) ≤ ((ac) ∩ ((ac) ∪ (bc)))
105, 9bltr 138 . 2 c ≤ ((ac) ∩ ((ac) ∪ (bc)))
11 i3orlem1 552 . 2 ((ac) ∩ ((ac) ∪ (bc))) ≤ ((ac) →3 (bc))
1210, 11letr 137 1 c ≤ ((ac) →3 (bc))
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →3 wi3 14 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 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator