Theorem oml5a 450

Description: Orthomodular law. (Contributed by NM, 16-Nov-1997.)

Assertion

Ref	Expression
oml5a	((a ∪ b) ∩ ((a ∪ b)⊥ ∪ (b ∩ c))) = (b ∩ c)

Proof of Theorem oml5a

Step	Hyp	Ref	Expression
1		omla 447	. . 3 ((a ∪ b) ∩ ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ (b ∩ c)))) = ((a ∪ b) ∩ (b ∩ c))
2		anass 76	. . . . . 6 ((b ∩ (a ∪ b)) ∩ c) = (b ∩ ((a ∪ b) ∩ c))
3		ax-a2 31	. . . . . . . . 9 (a ∪ b) = (b ∪ a)
4	3	lan 77	. . . . . . . 8 (b ∩ (a ∪ b)) = (b ∩ (b ∪ a))
5		anabs 121	. . . . . . . 8 (b ∩ (b ∪ a)) = b
6	4, 5	ax-r2 36	. . . . . . 7 (b ∩ (a ∪ b)) = b
7	6	ran 78	. . . . . 6 ((b ∩ (a ∪ b)) ∩ c) = (b ∩ c)
8		an12 81	. . . . . 6 (b ∩ ((a ∪ b) ∩ c)) = ((a ∪ b) ∩ (b ∩ c))
9	2, 7, 8	3tr2 64	. . . . 5 (b ∩ c) = ((a ∪ b) ∩ (b ∩ c))
10	9	lor 70	. . . 4 ((a ∪ b)⊥ ∪ (b ∩ c)) = ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ (b ∩ c)))
11	10	lan 77	. . 3 ((a ∪ b) ∩ ((a ∪ b)⊥ ∪ (b ∩ c))) = ((a ∪ b) ∩ ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ (b ∩ c))))
12	2, 8	ax-r2 36	. . 3 ((b ∩ (a ∪ b)) ∩ c) = ((a ∪ b) ∩ (b ∩ c))
13	1, 11, 12	3tr1 63	. 2 ((a ∪ b) ∩ ((a ∪ b)⊥ ∪ (b ∩ c))) = ((b ∩ (a ∪ b)) ∩ c)
14	13, 7	ax-r2 36	1 ((a ∪ b) ∩ ((a ∪ b)⊥ ∪ (b ∩ c))) = (b ∩ c)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ 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-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42

This theorem is referenced by: (None)