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Theorem lem4.6.6i3j0 1096
 Description: Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set to 3, and j is set to 0. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem4.6.6i3j0 ((a3 b) ∪ (a0 b)) = (a0 b)

Proof of Theorem lem4.6.6i3j0
StepHypRef Expression
1 ax-a3 32 . . 3 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∪ (ab)) = (((ab) ∪ (ab )) ∪ ((a ∩ (ab)) ∪ (ab)))
2 ax-a3 32 . . . . 5 (((a ∩ (ab)) ∪ a ) ∪ b) = ((a ∩ (ab)) ∪ (ab))
32ax-r1 35 . . . 4 ((a ∩ (ab)) ∪ (ab)) = (((a ∩ (ab)) ∪ a ) ∪ b)
43lor 70 . . 3 (((ab) ∪ (ab )) ∪ ((a ∩ (ab)) ∪ (ab))) = (((ab) ∪ (ab )) ∪ (((a ∩ (ab)) ∪ a ) ∪ b))
5 ax-a2 31 . . . . . . 7 ((a ∩ (ab)) ∪ a ) = (a ∪ (a ∩ (ab)))
6 omln 446 . . . . . . 7 (a ∪ (a ∩ (ab))) = (ab)
75, 6ax-r2 36 . . . . . 6 ((a ∩ (ab)) ∪ a ) = (ab)
87ax-r5 38 . . . . 5 (((a ∩ (ab)) ∪ a ) ∪ b) = ((ab) ∪ b)
98lor 70 . . . 4 (((ab) ∪ (ab )) ∪ (((a ∩ (ab)) ∪ a ) ∪ b)) = (((ab) ∪ (ab )) ∪ ((ab) ∪ b))
10 leid 148 . . . . . . 7 (ab) ≤ (ab)
11 leor 159 . . . . . . 7 b ≤ (ab)
1210, 11lel2or 170 . . . . . 6 ((ab) ∪ b) ≤ (ab)
13 leo 158 . . . . . 6 (ab) ≤ ((ab) ∪ b)
1412, 13lebi 145 . . . . 5 ((ab) ∪ b) = (ab)
1514lor 70 . . . 4 (((ab) ∪ (ab )) ∪ ((ab) ∪ b)) = (((ab) ∪ (ab )) ∪ (ab))
16 leao1 162 . . . . . 6 (ab) ≤ (ab)
17 leao1 162 . . . . . 6 (ab ) ≤ (ab)
1816, 17lel2or 170 . . . . 5 ((ab) ∪ (ab )) ≤ (ab)
1918df-le2 131 . . . 4 (((ab) ∪ (ab )) ∪ (ab)) = (ab)
209, 15, 193tr 65 . . 3 (((ab) ∪ (ab )) ∪ (((a ∩ (ab)) ∪ a ) ∪ b)) = (ab)
211, 4, 203tr 65 . 2 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∪ (ab)) = (ab)
22 df-i3 46 . . 3 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
23 df-i0 43 . . 3 (a0 b) = (ab)
2422, 232or 72 . 2 ((a3 b) ∪ (a0 b)) = ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∪ (ab))
2521, 24, 233tr1 63 1 ((a3 b) ∪ (a0 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  ax-r3 439 This theorem depends on definitions:  df-b 39  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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