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Theorem ledio 176
Description: Half of distributive law.
Assertion
Ref Expression
ledio (a v (b ^ c)) =< ((a v b) ^ (a v c))
Proof of Theorem ledio
StepHypRef Expression
1 anidm 111 . . . . 5 (a ^ a) = a
21ax-r1 35 . . . 4 a = (a ^ a)
3 leo 158 . . . . 5 a =< (a v b)
4 leo 158 . . . . 5 a =< (a v c)
53, 4le2an 169 . . . 4 (a ^ a) =< ((a v b) ^ (a v c))
62, 5bltr 138 . . 3 a =< ((a v b) ^ (a v c))
7 leo 158 . . . . 5 b =< (b v a)
8 ax-a2 31 . . . . 5 (b v a) = (a v b)
97, 8lbtr 139 . . . 4 b =< (a v b)
10 leo 158 . . . . 5 c =< (c v a)
11 ax-a2 31 . . . . 5 (c v a) = (a v c)
1210, 11lbtr 139 . . . 4 c =< (a v c)
139, 12le2an 169 . . 3 (b ^ c) =< ((a v b) ^ (a v c))
146, 13le2or 168 . 2 (a v (b ^ c)) =< (((a v b) ^ (a v c)) v ((a v b) ^ (a v c)))
15 oridm 110 . 2 (((a v b) ^ (a v c)) v ((a v b) ^ (a v c))) = ((a v b) ^ (a v c))
1614, 15lbtr 139 1 (a v (b ^ c)) =< ((a v b) ^ (a 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:  ledior  177  ka4lemo  228  ska13  241  wlem1  243
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