Theorem distlem 180

Description: Distributive law inference (uses OL only).

Hypothesis

Ref	Expression
distlem.1	(a ^ (b v c)) =< b

Assertion

Ref	Expression
distlem	(a ^ (b v c)) = ((a ^ b) v (a ^ c))

Proof of Theorem distlem

Step	Hyp	Ref	Expression
1		lea 152	. . . 4 (a ^ (b v c)) =< a
2		distlem.1	. . . 4 (a ^ (b v c)) =< b
3	1, 2	ler2an 165	. . 3 (a ^ (b v c)) =< (a ^ b)
4		leo 150	. . 3 (a ^ b) =< ((a ^ b) v (a ^ c))
5	3, 4	letr 129	. 2 (a ^ (b v c)) =< ((a ^ b) v (a ^ c))
6		ledi 166	. 2 ((a ^ b) v (a ^ c)) =< (a ^ (b v c))
7	5, 6	lebi 137	1 (a ^ (b v c)) = ((a ^ b) v (a ^ c))

Syntax hints:   = wb 1   =< wle 2   v wo 6   ^ wa 7

This theorem is referenced by:  oadist2a 987

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123

metamath.org