Theorem ledio 168

Description: Half of distributive law.

Assertion

Ref	Expression
ledio	(a ∪ (b ∩ c)) ≤ ((a ∪ b) ∩ (a ∪ c))

Proof of Theorem ledio

Step	Hyp	Ref	Expression
1		anidm 103	. . . . 5 (a ∩ a) = a
2	1	ax-r1 34	. . . 4 a = (a ∩ a)
3		leo 150	. . . . 5 a ≤ (a ∪ b)
4		leo 150	. . . . 5 a ≤ (a ∪ c)
5	3, 4	le2an 161	. . . 4 (a ∩ a) ≤ ((a ∪ b) ∩ (a ∪ c))
6	2, 5	bltr 130	. . 3 a ≤ ((a ∪ b) ∩ (a ∪ c))
7		leo 150	. . . . 5 b ≤ (b ∪ a)
8		ax-a2 30	. . . . 5 (b ∪ a) = (a ∪ b)
9	7, 8	lbtr 131	. . . 4 b ≤ (a ∪ b)
10		leo 150	. . . . 5 c ≤ (c ∪ a)
11		ax-a2 30	. . . . 5 (c ∪ a) = (a ∪ c)
12	10, 11	lbtr 131	. . . 4 c ≤ (a ∪ c)
13	9, 12	le2an 161	. . 3 (b ∩ c) ≤ ((a ∪ b) ∩ (a ∪ c))
14	6, 13	le2or 160	. 2 (a ∪ (b ∩ c)) ≤ (((a ∪ b) ∩ (a ∪ c)) ∪ ((a ∪ b) ∩ (a ∪ c)))
15		oridm 102	. 2 (((a ∪ b) ∩ (a ∪ c)) ∪ ((a ∪ b) ∩ (a ∪ c))) = ((a ∪ b) ∩ (a ∪ c))
16	14, 15	lbtr 131	1 (a ∪ (b ∩ c)) ≤ ((a ∪ b) ∩ (a ∪ c))

Syntax hints:   ≤ wle 2   ∪ wo 6   ∩ wa 7

This theorem is referenced by:  ledior 169  ka4lemo 220  ska13 233  wlem1 235

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

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