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 Description: Distributive law derived from OAL.
Hypothesis
Ref Expression
oadist.1 d ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
Assertion
Ref Expression
oadist ((a2 b) ∩ (d ∪ ((a2 b) ∩ (a2 c)))) = (((a2 b) ∩ d) ∪ ((a2 b) ∩ ((a2 b) ∩ (a2 c))))

Proof of Theorem oadist
StepHypRef Expression
1 oadist.1 . . . . 5 d ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
21oagen1 1014 . . . 4 ((a2 b) ∩ (d ∪ ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (a2 c))
32bile 142 . . 3 ((a2 b) ∩ (d ∪ ((a2 b) ∩ (a2 c)))) ≤ ((a2 b) ∩ (a2 c))
4 anidm 111 . . . . . . 7 ((a2 b) ∩ (a2 b)) = (a2 b)
54ax-r1 35 . . . . . 6 (a2 b) = ((a2 b) ∩ (a2 b))
65ran 78 . . . . 5 ((a2 b) ∩ (a2 c)) = (((a2 b) ∩ (a2 b)) ∩ (a2 c))
7 anass 76 . . . . 5 (((a2 b) ∩ (a2 b)) ∩ (a2 c)) = ((a2 b) ∩ ((a2 b) ∩ (a2 c)))
86, 7ax-r2 36 . . . 4 ((a2 b) ∩ (a2 c)) = ((a2 b) ∩ ((a2 b) ∩ (a2 c)))
9 leor 159 . . . 4 ((a2 b) ∩ ((a2 b) ∩ (a2 c))) ≤ (((a2 b) ∩ d) ∪ ((a2 b) ∩ ((a2 b) ∩ (a2 c))))
108, 9bltr 138 . . 3 ((a2 b) ∩ (a2 c)) ≤ (((a2 b) ∩ d) ∪ ((a2 b) ∩ ((a2 b) ∩ (a2 c))))
113, 10letr 137 . 2 ((a2 b) ∩ (d ∪ ((a2 b) ∩ (a2 c)))) ≤ (((a2 b) ∩ d) ∪ ((a2 b) ∩ ((a2 b) ∩ (a2 c))))
12 ledi 174 . 2 (((a2 b) ∩ d) ∪ ((a2 b) ∩ ((a2 b) ∩ (a2 c)))) ≤ ((a2 b) ∩ (d ∪ ((a2 b) ∩ (a2 c))))
1311, 12lebi 145 1 ((a2 b) ∩ (d ∪ ((a2 b) ∩ (a2 c)))) = (((a2 b) ∩ d) ∪ ((a2 b) ∩ ((a2 b) ∩ (a2 c))))
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7   →0 wi0 11   →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  ax-3oa 998 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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