Theorem mldual 1124

Description: Dual of modular law. (Contributed by NM, 15-Mar-2010.) (Revised by NM, 31-Mar-2011.)

Assertion

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

Proof of Theorem mldual

Step	Hyp	Ref	Expression
1		anor3 90	. . . . . . 7 (b⊥ ∩ (a ∩ c)⊥ ) = (b ∪ (a ∩ c))⊥
2	1	cm 61	. . . . . 6 (b ∪ (a ∩ c))⊥ = (b⊥ ∩ (a ∩ c)⊥ )
3		oran3 93	. . . . . . . 8 (a⊥ ∪ c⊥ ) = (a ∩ c)⊥
4	3	lan 77	. . . . . . 7 (b⊥ ∩ (a⊥ ∪ c⊥ )) = (b⊥ ∩ (a ∩ c)⊥ )
5	4	ax-r1 35	. . . . . 6 (b⊥ ∩ (a ∩ c)⊥ ) = (b⊥ ∩ (a⊥ ∪ c⊥ ))
6	2, 5	tr 62	. . . . 5 (b ∪ (a ∩ c))⊥ = (b⊥ ∩ (a⊥ ∪ c⊥ ))
7	6	lor 70	. . . 4 (a⊥ ∪ (b ∪ (a ∩ c))⊥ ) = (a⊥ ∪ (b⊥ ∩ (a⊥ ∪ c⊥ )))
8		ml 1123	. . . 4 (a⊥ ∪ (b⊥ ∩ (a⊥ ∪ c⊥ ))) = ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ ))
9		oran3 93	. . . . 5 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
10	9, 3	2an 79	. . . 4 ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ c⊥ )) = ((a ∩ b)⊥ ∩ (a ∩ c)⊥ )
11	7, 8, 10	3tr 65	. . 3 (a⊥ ∪ (b ∪ (a ∩ c))⊥ ) = ((a ∩ b)⊥ ∩ (a ∩ c)⊥ )
12		oran3 93	. . 3 (a⊥ ∪ (b ∪ (a ∩ c))⊥ ) = (a ∩ (b ∪ (a ∩ c)))⊥
13		anor3 90	. . 3 ((a ∩ b)⊥ ∩ (a ∩ c)⊥ ) = ((a ∩ b) ∪ (a ∩ c))⊥
14	11, 12, 13	3tr2 64	. 2 (a ∩ (b ∪ (a ∩ c)))⊥ = ((a ∩ b) ∪ (a ∩ c))⊥
15	14	con1 66	1 (a ∩ (b ∪ (a ∩ c))) = ((a ∩ b) ∪ (a ∩ c))

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ 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  ax-ml 1122

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:  mldual2i  1127  vneulem13  1143  vneulemexp  1148  dp41lemd  1186  dp41leme  1187  dp32  1196  xdp41  1198  xxdp41  1201  xdp45lem  1204  xdp43lem  1205  xdp45  1206  xdp43  1207  3dp43  1208