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Theorem lem4.6.6i4j2 1101
 Description: Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set to 4, and j is set to 2. (Contributed by Roy F. Longton, 2-Jul-2005.)
Assertion
Ref Expression
lem4.6.6i4j2 ((a4 b) ∪ (a2 b)) = (a0 b)

Proof of Theorem lem4.6.6i4j2
StepHypRef Expression
1 ax-a3 32 . . 3 ((((ab) ∪ (ab)) ∪ ((ab) ∩ b )) ∪ (b ∪ (ab ))) = (((ab) ∪ (ab)) ∪ (((ab) ∩ b ) ∪ (b ∪ (ab ))))
2 ax-a3 32 . . . . . 6 ((((ab) ∩ b ) ∪ b) ∪ (ab )) = (((ab) ∩ b ) ∪ (b ∪ (ab )))
32ax-r1 35 . . . . 5 (((ab) ∩ b ) ∪ (b ∪ (ab ))) = ((((ab) ∩ b ) ∪ b) ∪ (ab ))
4 ax-a2 31 . . . . . . 7 (((ab) ∩ b ) ∪ b) = (b ∪ ((ab) ∩ b ))
5 ancom 74 . . . . . . . 8 ((ab) ∩ b ) = (b ∩ (ab))
65lor 70 . . . . . . 7 (b ∪ ((ab) ∩ b )) = (b ∪ (b ∩ (ab)))
7 leor 159 . . . . . . . 8 b ≤ (ab)
87oml2 451 . . . . . . 7 (b ∪ (b ∩ (ab))) = (ab)
94, 6, 83tr 65 . . . . . 6 (((ab) ∩ b ) ∪ b) = (ab)
109ax-r5 38 . . . . 5 ((((ab) ∩ b ) ∪ b) ∪ (ab )) = ((ab) ∪ (ab ))
113, 10ax-r2 36 . . . 4 (((ab) ∩ b ) ∪ (b ∪ (ab ))) = ((ab) ∪ (ab ))
1211lor 70 . . 3 (((ab) ∪ (ab)) ∪ (((ab) ∩ b ) ∪ (b ∪ (ab )))) = (((ab) ∪ (ab)) ∪ ((ab) ∪ (ab )))
13 leao4 165 . . . . . 6 (ab) ≤ (ab)
14 leao1 162 . . . . . 6 (ab) ≤ (ab)
1513, 14lel2or 170 . . . . 5 ((ab) ∪ (ab)) ≤ (ab)
16 leid 148 . . . . . 6 (ab) ≤ (ab)
17 leao1 162 . . . . . 6 (ab ) ≤ (ab)
1816, 17lel2or 170 . . . . 5 ((ab) ∪ (ab )) ≤ (ab)
1915, 18lel2or 170 . . . 4 (((ab) ∪ (ab)) ∪ ((ab) ∪ (ab ))) ≤ (ab)
20 leo 158 . . . . 5 (ab) ≤ ((ab) ∪ (ab ))
2120lerr 150 . . . 4 (ab) ≤ (((ab) ∪ (ab)) ∪ ((ab) ∪ (ab )))
2219, 21lebi 145 . . 3 (((ab) ∪ (ab)) ∪ ((ab) ∪ (ab ))) = (ab)
231, 12, 223tr 65 . 2 ((((ab) ∪ (ab)) ∪ ((ab) ∩ b )) ∪ (b ∪ (ab ))) = (ab)
24 df-i4 47 . . 3 (a4 b) = (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))
25 df-i2 45 . . 3 (a2 b) = (b ∪ (ab ))
2624, 252or 72 . 2 ((a4 b) ∪ (a2 b)) = ((((ab) ∪ (ab)) ∪ ((ab) ∩ b )) ∪ (b ∪ (ab )))
27 df-i0 43 . 2 (a0 b) = (ab)
2823, 26, 273tr1 63 1 ((a4 b) ∪ (a2 b)) = (a0 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →0 wi0 11   →2 wi2 13   →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  ax-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-i2 45  df-i4 47  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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