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Theorem lem4.6.6i0j3 1090
 Description: Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set to 0, and j is set to 3. (Contributed by Roy F. Longton, 1-Jul-2005.)
Assertion
Ref Expression
lem4.6.6i0j3 ((a0 b) ∪ (a3 b)) = (a0 b)

Proof of Theorem lem4.6.6i0j3
StepHypRef Expression
1 leid 148 . . . 4 (ab) ≤ (ab)
2 leao1 162 . . . . . 6 (ab) ≤ (ab)
3 leao1 162 . . . . . 6 (ab ) ≤ (ab)
42, 3lel2or 170 . . . . 5 ((ab) ∪ (ab )) ≤ (ab)
5 lear 161 . . . . 5 (a ∩ (ab)) ≤ (ab)
64, 5lel2or 170 . . . 4 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) ≤ (ab)
71, 6lel2or 170 . . 3 ((ab) ∪ (((ab) ∪ (ab )) ∪ (a ∩ (ab)))) ≤ (ab)
8 leo 158 . . 3 (ab) ≤ ((ab) ∪ (((ab) ∪ (ab )) ∪ (a ∩ (ab))))
97, 8lebi 145 . 2 ((ab) ∪ (((ab) ∪ (ab )) ∪ (a ∩ (ab)))) = (ab)
10 df-i0 43 . . 3 (a0 b) = (ab)
11 df-i3 46 . . 3 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
1210, 112or 72 . 2 ((a0 b) ∪ (a3 b)) = ((ab) ∪ (((ab) ∪ (ab )) ∪ (a ∩ (ab))))
139, 12, 103tr1 63 1 ((a0 b) ∪ (a3 b)) = (a0 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →0 wi0 11   →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-i0 43  df-i3 46  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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