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Theorem ledi 174
Description: Half of distributive law.
Assertion
Ref Expression
ledi ((a ^ b) v (a ^ c)) =< (a ^ (b v c))
Proof of Theorem ledi
StepHypRef Expression
1 anidm 111 . . 3 (((a ^ b) v (a ^ c)) ^ ((a ^ b) v (a ^ c))) = ((a ^ b) v (a ^ c))
21ax-r1 35 . 2 ((a ^ b) v (a ^ c)) = (((a ^ b) v (a ^ c)) ^ ((a ^ b) v (a ^ c)))
3 lea 160 . . . . 5 (a ^ b) =< a
4 lea 160 . . . . 5 (a ^ c) =< a
53, 4le2or 168 . . . 4 ((a ^ b) v (a ^ c)) =< (a v a)
6 oridm 110 . . . 4 (a v a) = a
75, 6lbtr 139 . . 3 ((a ^ b) v (a ^ c)) =< a
8 ancom 74 . . . . 5 (a ^ b) = (b ^ a)
9 lea 160 . . . . 5 (b ^ a) =< b
108, 9bltr 138 . . . 4 (a ^ b) =< b
11 ancom 74 . . . . 5 (a ^ c) = (c ^ a)
12 lea 160 . . . . 5 (c ^ a) =< c
1311, 12bltr 138 . . . 4 (a ^ c) =< c
1410, 13le2or 168 . . 3 ((a ^ b) v (a ^ c)) =< (b v c)
157, 14le2an 169 . 2 (((a ^ b) v (a ^ c)) ^ ((a ^ b) v (a ^ c))) =< (a ^ (b v c))
162, 15bltr 138 1 ((a ^ b) v (a ^ c)) =< (a ^ (b v c))
Colors of variables: term
Syntax hints:   =< wle 2   v 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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