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Theorem oadistb 1020
 Description: Distributive law derived from OAL.
Hypotheses
Ref Expression
oadistb.2 d ≤ (a2 b)
oadistb.1 e ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
Assertion
Ref Expression
oadistb (d ∩ (e ∪ ((a2 b) ∩ (a2 c)))) = ((de) ∪ (d ∩ ((a2 b) ∩ (a2 c))))

Proof of Theorem oadistb
StepHypRef Expression
1 oadistb.2 . . . . . . 7 d ≤ (a2 b)
21df2le2 136 . . . . . 6 (d ∩ (a2 b)) = d
32ran 78 . . . . 5 ((d ∩ (a2 b)) ∩ (e ∪ ((a2 b) ∩ (a2 c)))) = (d ∩ (e ∪ ((a2 b) ∩ (a2 c))))
43ax-r1 35 . . . 4 (d ∩ (e ∪ ((a2 b) ∩ (a2 c)))) = ((d ∩ (a2 b)) ∩ (e ∪ ((a2 b) ∩ (a2 c))))
5 anass 76 . . . . 5 ((d ∩ (a2 b)) ∩ (e ∪ ((a2 b) ∩ (a2 c)))) = (d ∩ ((a2 b) ∩ (e ∪ ((a2 b) ∩ (a2 c)))))
6 oadistb.1 . . . . . . 7 e ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
76oagen1 1014 . . . . . 6 ((a2 b) ∩ (e ∪ ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (a2 c))
87lan 77 . . . . 5 (d ∩ ((a2 b) ∩ (e ∪ ((a2 b) ∩ (a2 c))))) = (d ∩ ((a2 b) ∩ (a2 c)))
95, 8ax-r2 36 . . . 4 ((d ∩ (a2 b)) ∩ (e ∪ ((a2 b) ∩ (a2 c)))) = (d ∩ ((a2 b) ∩ (a2 c)))
104, 9ax-r2 36 . . 3 (d ∩ (e ∪ ((a2 b) ∩ (a2 c)))) = (d ∩ ((a2 b) ∩ (a2 c)))
11 leor 159 . . 3 (d ∩ ((a2 b) ∩ (a2 c))) ≤ ((de) ∪ (d ∩ ((a2 b) ∩ (a2 c))))
1210, 11bltr 138 . 2 (d ∩ (e ∪ ((a2 b) ∩ (a2 c)))) ≤ ((de) ∪ (d ∩ ((a2 b) ∩ (a2 c))))
13 ledi 174 . 2 ((de) ∪ (d ∩ ((a2 b) ∩ (a2 c)))) ≤ (d ∩ (e ∪ ((a2 b) ∩ (a2 c))))
1412, 13lebi 145 1 (d ∩ (e ∪ ((a2 b) ∩ (a2 c)))) = ((de) ∪ (d ∩ ((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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