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Theorem lem4.6.6i2j1 1096
 Description: Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set to 2, and j is set to 1. (Contributed by Roy F. Longton, 1-Jul-2005.)
Assertion
Ref Expression
lem4.6.6i2j1 ((a2 b) ∪ (a1 b)) = (a0 b)

Proof of Theorem lem4.6.6i2j1
StepHypRef Expression
1 leor 159 . . . . 5 b ≤ (ab)
2 leao1 162 . . . . 5 (ab ) ≤ (ab)
31, 2lel2or 170 . . . 4 (b ∪ (ab )) ≤ (ab)
4 lear 161 . . . . 5 (ab) ≤ b
54lelor 166 . . . 4 (a ∪ (ab)) ≤ (ab)
63, 5lel2or 170 . . 3 ((b ∪ (ab )) ∪ (a ∪ (ab))) ≤ (ab)
7 leo 158 . . . . 5 a ≤ (a ∪ (ab))
87lerr 150 . . . 4 a ≤ ((b ∪ (ab )) ∪ (a ∪ (ab)))
9 leo 158 . . . . 5 b ≤ (b ∪ (ab ))
109ler 149 . . . 4 b ≤ ((b ∪ (ab )) ∪ (a ∪ (ab)))
118, 10lel2or 170 . . 3 (ab) ≤ ((b ∪ (ab )) ∪ (a ∪ (ab)))
126, 11lebi 145 . 2 ((b ∪ (ab )) ∪ (a ∪ (ab))) = (ab)
13 df-i2 45 . . 3 (a2 b) = (b ∪ (ab ))
14 df-i1 44 . . 3 (a1 b) = (a ∪ (ab))
1513, 142or 72 . 2 ((a2 b) ∪ (a1 b)) = ((b ∪ (ab )) ∪ (a ∪ (ab)))
16 df-i0 43 . 2 (a0 b) = (ab)
1712, 15, 163tr1 63 1 ((a2 b) ∪ (a1 b)) = (a0 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →0 wi0 11   →1 wi1 12   →2 wi2 13 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-i1 44  df-i2 45  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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