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 Description: Distributive law.
Hypotheses
Ref Expression
oadistc.1 d ≤ ((a2 b) ∩ (a2 c))
oadistc.2 ((a2 b) ∩ ((bc)d)) ≤ (((a2 b) ∩ (bc) ) ∪ d)
Assertion
Ref Expression
oadistc ((a2 b) ∩ ((bc)d)) = (((a2 b) ∩ (bc) ) ∪ ((a2 b) ∩ d))

StepHypRef Expression
1 oadistc.2 . . 3 ((a2 b) ∩ ((bc)d)) ≤ (((a2 b) ∩ (bc) ) ∪ d)
2 oadistc.1 . . . . . . . 8 d ≤ ((a2 b) ∩ (a2 c))
3 lea 160 . . . . . . . 8 ((a2 b) ∩ (a2 c)) ≤ (a2 b)
42, 3letr 137 . . . . . . 7 d ≤ (a2 b)
54df2le2 136 . . . . . 6 (d ∩ (a2 b)) = d
65ax-r1 35 . . . . 5 d = (d ∩ (a2 b))
7 ancom 74 . . . . 5 (d ∩ (a2 b)) = ((a2 b) ∩ d)
86, 7ax-r2 36 . . . 4 d = ((a2 b) ∩ d)
98lor 70 . . 3 (((a2 b) ∩ (bc) ) ∪ d) = (((a2 b) ∩ (bc) ) ∪ ((a2 b) ∩ d))
101, 9lbtr 139 . 2 ((a2 b) ∩ ((bc)d)) ≤ (((a2 b) ∩ (bc) ) ∪ ((a2 b) ∩ d))
11 ledi 174 . 2 (((a2 b) ∩ (bc) ) ∪ ((a2 b) ∩ d)) ≤ ((a2 b) ∩ ((bc)d))
1210, 11lebi 145 1 ((a2 b) ∩ ((bc)d)) = (((a2 b) ∩ (bc) ) ∪ ((a2 b) ∩ d))
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →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-le1 130  df-le2 131 This theorem is referenced by: (None)
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