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

Step	Hyp	Ref	Expression
1		anidm 111	. . . . 5 (a ^ a) = a
2	1	ax-r1 35	. . . 4 a = (a ^ a)
3		leo 158	. . . . 5 a =< (a v b)
4		leo 158	. . . . 5 a =< (a v c)
5	3, 4	le2an 169	. . . 4 (a ^ a) =< ((a v b) ^ (a v c))
6	2, 5	bltr 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)
9	7, 8	lbtr 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)
12	10, 11	lbtr 139	. . . 4 c =< (a v c)
13	9, 12	le2an 169	. . 3 (b ^ c) =< ((a v b) ^ (a v c))
14	6, 13	le2or 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))
16	14, 15	lbtr 139	1 (a v (b ^ c)) =< ((a v b) ^ (a v c))

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

This theorem is referenced by:  ledior 177  ka4lemo 228  ska13 241  wlem1 243

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

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