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Theorem ledi 174
 Description: Half of distributive law. (Contributed by NM, 28-Aug-1997.)
Assertion
Ref Expression
ledi ((ab) ∪ (ac)) ≤ (a ∩ (bc))

Proof of Theorem ledi
StepHypRef Expression
1 anidm 111 . . 3 (((ab) ∪ (ac)) ∩ ((ab) ∪ (ac))) = ((ab) ∪ (ac))
21ax-r1 35 . 2 ((ab) ∪ (ac)) = (((ab) ∪ (ac)) ∩ ((ab) ∪ (ac)))
3 lea 160 . . . . 5 (ab) ≤ a
4 lea 160 . . . . 5 (ac) ≤ a
53, 4le2or 168 . . . 4 ((ab) ∪ (ac)) ≤ (aa)
6 oridm 110 . . . 4 (aa) = a
75, 6lbtr 139 . . 3 ((ab) ∪ (ac)) ≤ a
8 ancom 74 . . . . 5 (ab) = (ba)
9 lea 160 . . . . 5 (ba) ≤ b
108, 9bltr 138 . . . 4 (ab) ≤ b
11 ancom 74 . . . . 5 (ac) = (ca)
12 lea 160 . . . . 5 (ca) ≤ c
1311, 12bltr 138 . . . 4 (ac) ≤ c
1410, 13le2or 168 . . 3 ((ab) ∪ (ac)) ≤ (bc)
157, 14le2an 169 . 2 (((ab) ∪ (ac)) ∩ ((ab) ∪ (ac))) ≤ (a ∩ (bc))
162, 15bltr 138 1 ((ab) ∪ (ac)) ≤ (a ∩ (bc))
 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:  ledir  175  distlem  188  wwfh1  216  wwfh2  217  ska2  432  fh1  469  fh2  470  i3orlem2  553  distid  887  oadist  1019  oadistb  1020  oadistc  1022  oadistd  1023  4oadist  1044
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