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 Description: OA distributive law.
Hypotheses
Ref Expression
oadistd.2 e ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
oadistd.3 f ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
oadistd.4 (d ∩ (a2 c)) ≤ f
Assertion
Ref Expression
oadistd (d ∩ (ef)) = ((de) ∪ (df))

StepHypRef Expression
1 oadistd.2 . . . . . . . . . 10 e ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
2 oadistd.3 . . . . . . . . . 10 f ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
31, 2le2or 168 . . . . . . . . 9 (ef) ≤ (((bc) →0 ((a2 b) ∩ (a2 c))) ∪ ((bc) →0 ((a2 b) ∩ (a2 c))))
4 oridm 110 . . . . . . . . 9 (((bc) →0 ((a2 b) ∩ (a2 c))) ∪ ((bc) →0 ((a2 b) ∩ (a2 c)))) = ((bc) →0 ((a2 b) ∩ (a2 c)))
53, 4lbtr 139 . . . . . . . 8 (ef) ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
65lelan 167 . . . . . . 7 (d ∩ (ef)) ≤ (d ∩ ((bc) →0 ((a2 b) ∩ (a2 c))))
76df2le2 136 . . . . . 6 ((d ∩ (ef)) ∩ (d ∩ ((bc) →0 ((a2 b) ∩ (a2 c))))) = (d ∩ (ef))
87ax-r1 35 . . . . 5 (d ∩ (ef)) = ((d ∩ (ef)) ∩ (d ∩ ((bc) →0 ((a2 b) ∩ (a2 c)))))
9 df-i0 43 . . . . . . . 8 ((bc) →0 ((a2 b) ∩ (a2 c))) = ((bc) ∪ ((a2 b) ∩ (a2 c)))
109lan 77 . . . . . . 7 (d ∩ ((bc) →0 ((a2 b) ∩ (a2 c)))) = (d ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))
11 oadistd.1 . . . . . . . 8 d ≤ (a2 b)
12 leo 158 . . . . . . . . 9 (bc) ≤ ((bc) ∪ ((a2 b) ∩ (a2 c)))
139ax-r1 35 . . . . . . . . 9 ((bc) ∪ ((a2 b) ∩ (a2 c))) = ((bc) →0 ((a2 b) ∩ (a2 c)))
1412, 13lbtr 139 . . . . . . . 8 (bc) ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
1511, 14oagen1b 1015 . . . . . . 7 (d ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = (d ∩ (a2 c))
1610, 15ax-r2 36 . . . . . 6 (d ∩ ((bc) →0 ((a2 b) ∩ (a2 c)))) = (d ∩ (a2 c))
1716lan 77 . . . . 5 ((d ∩ (ef)) ∩ (d ∩ ((bc) →0 ((a2 b) ∩ (a2 c))))) = ((d ∩ (ef)) ∩ (d ∩ (a2 c)))
188, 17ax-r2 36 . . . 4 (d ∩ (ef)) = ((d ∩ (ef)) ∩ (d ∩ (a2 c)))
19 lear 161 . . . . 5 ((d ∩ (ef)) ∩ (d ∩ (a2 c))) ≤ (d ∩ (a2 c))
20 oadistd.4 . . . . . . . . 9 (d ∩ (a2 c)) ≤ f
2120df2le2 136 . . . . . . . 8 ((d ∩ (a2 c)) ∩ f) = (d ∩ (a2 c))
2221ax-r1 35 . . . . . . 7 (d ∩ (a2 c)) = ((d ∩ (a2 c)) ∩ f)
23 an32 83 . . . . . . 7 ((d ∩ (a2 c)) ∩ f) = ((df) ∩ (a2 c))
2422, 23ax-r2 36 . . . . . 6 (d ∩ (a2 c)) = ((df) ∩ (a2 c))
25 lea 160 . . . . . 6 ((df) ∩ (a2 c)) ≤ (df)
2624, 25bltr 138 . . . . 5 (d ∩ (a2 c)) ≤ (df)
2719, 26letr 137 . . . 4 ((d ∩ (ef)) ∩ (d ∩ (a2 c))) ≤ (df)
2818, 27bltr 138 . . 3 (d ∩ (ef)) ≤ (df)
29 leor 159 . . 3 (df) ≤ ((de) ∪ (df))
3028, 29letr 137 . 2 (d ∩ (ef)) ≤ ((de) ∪ (df))
31 ledi 174 . 2 ((de) ∪ (df)) ≤ (d ∩ (ef))
3230, 31lebi 145 1 (d ∩ (ef)) = ((de) ∪ (df))
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ 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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