Theorem ledio 176

Description: Half of distributive law. (Contributed by NM, 28-Aug-1997.)

Assertion

Ref	Expression
ledio	(a ∪ (b ∩ c)) ≤ ((a ∪ b) ∩ (a ∪ 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 ∪ b)
4		leo 158	. . . . 5 a ≤ (a ∪ c)
5	3, 4	le2an 169	. . . 4 (a ∩ a) ≤ ((a ∪ b) ∩ (a ∪ c))
6	2, 5	bltr 138	. . 3 a ≤ ((a ∪ b) ∩ (a ∪ c))
7		leo 158	. . . . 5 b ≤ (b ∪ a)
8		ax-a2 31	. . . . 5 (b ∪ a) = (a ∪ b)
9	7, 8	lbtr 139	. . . 4 b ≤ (a ∪ b)
10		leo 158	. . . . 5 c ≤ (c ∪ a)
11		ax-a2 31	. . . . 5 (c ∪ a) = (a ∪ c)
12	10, 11	lbtr 139	. . . 4 c ≤ (a ∪ c)
13	9, 12	le2an 169	. . 3 (b ∩ c) ≤ ((a ∪ b) ∩ (a ∪ c))
14	6, 13	le2or 168	. 2 (a ∪ (b ∩ c)) ≤ (((a ∪ b) ∩ (a ∪ c)) ∪ ((a ∪ b) ∩ (a ∪ c)))
15		oridm 110	. 2 (((a ∪ b) ∩ (a ∪ c)) ∪ ((a ∪ b) ∩ (a ∪ c))) = ((a ∪ b) ∩ (a ∪ c))
16	14, 15	lbtr 139	1 (a ∪ (b ∩ c)) ≤ ((a ∪ b) ∩ (a ∪ c))

This proof depends on syntax axioms:   ≤ wle 2   ∪ wo 6   ∩ wa 7

This proof depends on 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 proof depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131

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