Theorem modexp 1152

Description: Expansion by modular law. (Contributed by NM, 10-Apr-2012.)

Assertion

Ref	Expression
modexp	(a ∩ (b ∪ c)) = (a ∩ (b ∪ (c ∩ (a ∪ b))))

Proof of Theorem modexp

Step	Hyp	Ref	Expression
1		anass 76	. 2 ((a ∩ (a ∪ b)) ∩ (b ∪ c)) = (a ∩ ((a ∪ b) ∩ (b ∪ c)))
2		anabs 121	. . 3 (a ∩ (a ∪ b)) = a
3	2	ran 78	. 2 ((a ∩ (a ∪ b)) ∩ (b ∪ c)) = (a ∩ (b ∪ c))
4		ancom 74	. . . 4 ((a ∪ b) ∩ (b ∪ c)) = ((b ∪ c) ∩ (a ∪ b))
5		leor 159	. . . . 5 b ≤ (a ∪ b)
6	5	mlduali 1128	. . . 4 ((b ∪ c) ∩ (a ∪ b)) = (b ∪ (c ∩ (a ∪ b)))
7	4, 6	tr 62	. . 3 ((a ∪ b) ∩ (b ∪ c)) = (b ∪ (c ∩ (a ∪ b)))
8	7	lan 77	. 2 (a ∩ ((a ∪ b) ∩ (b ∪ c))) = (a ∩ (b ∪ (c ∩ (a ∪ b))))
9	1, 3, 8	3tr2 64	1 (a ∩ (b ∪ c)) = (a ∩ (b ∪ (c ∩ (a ∪ b))))

Syntax hints:   = wb 1   ∪ 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  ax-ml 1122

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: (None)