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 Description: Pre-distributive law. Note that the inference of the second hypothesis from the first may be an OM theorem. (Contributed by NM, 30-Nov-1998.)
Hypotheses
Ref Expression
oadistc0.1 d ≤ ((a2 b) ∩ (a2 c))
oadistc0.2 ((a2 c) ∩ ((a2 b) ∩ ((bc)d))) ≤ (((a2 b) ∩ (bc) ) ∪ d)
Assertion
Ref Expression
oadistc0 ((a2 b) ∩ ((bc)d)) = (((a2 b) ∩ (bc) ) ∪ d)

Proof of Theorem oadistc0
StepHypRef Expression
1 ancom 74 . . . . 5 ((a2 c) ∩ ((a2 b) ∩ ((bc)d))) = (((a2 b) ∩ ((bc)d)) ∩ (a2 c))
2 oadistc0.1 . . . . . . . . 9 d ≤ ((a2 b) ∩ (a2 c))
32lelor 166 . . . . . . . 8 ((bc)d) ≤ ((bc) ∪ ((a2 b) ∩ (a2 c)))
43lelan 167 . . . . . . 7 ((a2 b) ∩ ((bc)d)) ≤ ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))
5 oal2 999 . . . . . . 7 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ≤ (a2 c)
64, 5letr 137 . . . . . 6 ((a2 b) ∩ ((bc)d)) ≤ (a2 c)
76df2le2 136 . . . . 5 (((a2 b) ∩ ((bc)d)) ∩ (a2 c)) = ((a2 b) ∩ ((bc)d))
81, 7ax-r2 36 . . . 4 ((a2 c) ∩ ((a2 b) ∩ ((bc)d))) = ((a2 b) ∩ ((bc)d))
98ax-r1 35 . . 3 ((a2 b) ∩ ((bc)d)) = ((a2 c) ∩ ((a2 b) ∩ ((bc)d)))
10 oadistc0.2 . . 3 ((a2 c) ∩ ((a2 b) ∩ ((bc)d))) ≤ (((a2 b) ∩ (bc) ) ∪ d)
119, 10bltr 138 . 2 ((a2 b) ∩ ((bc)d)) ≤ (((a2 b) ∩ (bc) ) ∪ d)
12 ledior 177 . . 3 (((a2 b) ∩ (bc) ) ∪ d) ≤ (((a2 b) ∪ d) ∩ ((bc)d))
13 ax-a2 31 . . . . 5 ((a2 b) ∪ d) = (d ∪ (a2 b))
14 lea 160 . . . . . . 7 ((a2 b) ∩ (a2 c)) ≤ (a2 b)
152, 14letr 137 . . . . . 6 d ≤ (a2 b)
1615df-le2 131 . . . . 5 (d ∪ (a2 b)) = (a2 b)
1713, 16ax-r2 36 . . . 4 ((a2 b) ∪ d) = (a2 b)
1817ran 78 . . 3 (((a2 b) ∪ d) ∩ ((bc)d)) = ((a2 b) ∩ ((bc)d))
1912, 18lbtr 139 . 2 (((a2 b) ∩ (bc) ) ∪ d) ≤ ((a2 b) ∩ ((bc)d))
2011, 19lebi 145 1 ((a2 b) ∩ ((bc)d)) = (((a2 b) ∩ (bc) ) ∪ 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  ax-3oa 998 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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