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Theorem ledio 176
 Description: Half of distributive law.
Assertion
Ref Expression
ledio (a ∪ (bc)) ≤ ((ab) ∩ (ac))

Proof of Theorem ledio
StepHypRef Expression
1 anidm 111 . . . . 5 (aa) = a
21ax-r1 35 . . . 4 a = (aa)
3 leo 158 . . . . 5 a ≤ (ab)
4 leo 158 . . . . 5 a ≤ (ac)
53, 4le2an 169 . . . 4 (aa) ≤ ((ab) ∩ (ac))
62, 5bltr 138 . . 3 a ≤ ((ab) ∩ (ac))
7 leo 158 . . . . 5 b ≤ (ba)
8 ax-a2 31 . . . . 5 (ba) = (ab)
97, 8lbtr 139 . . . 4 b ≤ (ab)
10 leo 158 . . . . 5 c ≤ (ca)
11 ax-a2 31 . . . . 5 (ca) = (ac)
1210, 11lbtr 139 . . . 4 c ≤ (ac)
139, 12le2an 169 . . 3 (bc) ≤ ((ab) ∩ (ac))
146, 13le2or 168 . 2 (a ∪ (bc)) ≤ (((ab) ∩ (ac)) ∪ ((ab) ∩ (ac)))
15 oridm 110 . 2 (((ab) ∩ (ac)) ∪ ((ab) ∩ (ac))) = ((ab) ∩ (ac))
1614, 15lbtr 139 1 (a ∪ (bc)) ≤ ((ab) ∩ (ac))
 Colors of variables: term Syntax hints:   ≤ wle 2   ∪ wo 6   ∩ wa 7 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:  ledior  177  ka4lemo  228  ska13  241  wlem1  243
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