Theorem ml 1123

Description: Modular law in equational form. (Contributed by NM, 15-Mar-2010.) (Revised by NM, 31-Mar-2011.)

Assertion

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

Proof of Theorem ml

Step	Hyp	Ref	Expression
1		leo 158	. . . 4 a ≤ (a ∪ b)
2		leo 158	. . . 4 a ≤ (a ∪ c)
3	1, 2	ler2an 173	. . 3 a ≤ ((a ∪ b) ∩ (a ∪ c))
4		leor 159	. . . 4 b ≤ (a ∪ b)
5	4	leran 153	. . 3 (b ∩ (a ∪ c)) ≤ ((a ∪ b) ∩ (a ∪ c))
6	3, 5	lel2or 170	. 2 (a ∪ (b ∩ (a ∪ c))) ≤ ((a ∪ b) ∩ (a ∪ c))
7		ax-ml 1122	. 2 ((a ∪ b) ∩ (a ∪ c)) ≤ (a ∪ (b ∩ (a ∪ c)))
8	6, 7	lebi 145	1 (a ∪ (b ∩ (a ∪ c))) = ((a ∪ b) ∩ (a ∪ c))

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:  mldual  1124  ml2i  1125  vneulem2  1132  vneulem5  1135  vneulem12  1142  vneulemexp  1148