QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  negantlem4 GIF version

Theorem negantlem4 851
Description: Lemma for negated antecedent identity. (Contributed by NM, 6-Aug-2001.)
Hypothesis
Ref Expression
negant.1 (a1 c) = (b1 c)
Assertion
Ref Expression
negantlem4 (a1 c) ≤ (b1 c)

Proof of Theorem negantlem4
StepHypRef Expression
1 df-i1 44 . . 3 (a1 c) = (a ∪ (ac))
2 ax-a1 30 . . . . 5 a = a
32ax-r5 38 . . . 4 (a ∪ (ac)) = (a ∪ (ac))
43ax-r1 35 . . 3 (a ∪ (ac)) = (a ∪ (ac))
51, 4ax-r2 36 . 2 (a1 c) = (a ∪ (ac))
6 negant.1 . . . 4 (a1 c) = (b1 c)
76negantlem2 849 . . 3 a ≤ (b1 c)
86negantlem3 850 . . 3 (ac) ≤ (b1 c)
97, 8lel2or 170 . 2 (a ∪ (ac)) ≤ (b1 c)
105, 9bltr 138 1 (a1 c) ≤ (b1 c)
Colors of variables: term
Syntax hints:   = wb 1  wle 2   wn 4  wo 6  wa 7  1 wi1 12
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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  negant  852
  Copyright terms: Public domain W3C validator