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Theorem lem4.6.6i0j4 1091
Description: Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set to 0, and j is set to 4. (Contributed by Roy F. Longton, 1-Jul-2005.)
Assertion
Ref Expression
lem4.6.6i0j4 ((a0 b) ∪ (a4 b)) = (a0 b)

Proof of Theorem lem4.6.6i0j4
StepHypRef Expression
1 leid 148 . . . 4 (ab) ≤ (ab)
2 leao4 165 . . . . . 6 (ab) ≤ (ab)
3 leao1 162 . . . . . 6 (ab) ≤ (ab)
42, 3lel2or 170 . . . . 5 ((ab) ∪ (ab)) ≤ (ab)
5 lea 160 . . . . 5 ((ab) ∩ b ) ≤ (ab)
64, 5lel2or 170 . . . 4 (((ab) ∪ (ab)) ∪ ((ab) ∩ b )) ≤ (ab)
71, 6lel2or 170 . . 3 ((ab) ∪ (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))) ≤ (ab)
8 leo 158 . . 3 (ab) ≤ ((ab) ∪ (((ab) ∪ (ab)) ∪ ((ab) ∩ b )))
97, 8lebi 145 . 2 ((ab) ∪ (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))) = (ab)
10 df-i0 43 . . 3 (a0 b) = (ab)
11 df-i4 47 . . 3 (a4 b) = (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))
1210, 112or 72 . 2 ((a0 b) ∪ (a4 b)) = ((ab) ∪ (((ab) ∪ (ab)) ∪ ((ab) ∩ b )))
139, 12, 103tr1 63 1 ((a0 b) ∪ (a4 b)) = (a0 b)
Colors of variables: term
Syntax hints:   = wb 1   wn 4  wo 6  wa 7  0 wi0 11  4 wi4 15
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-i4 47  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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