Theorem oml5 449

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

Assertion

Ref	Expression
oml5	((a ∩ b) ∪ ((a ∩ b)⊥ ∩ (b ∪ c))) = (b ∪ c)

Proof of Theorem oml5

Step	Hyp	Ref	Expression
1		oml 445	. . 3 ((a ∩ b) ∪ ((a ∩ b)⊥ ∩ ((a ∩ b) ∪ (b ∪ c)))) = ((a ∩ b) ∪ (b ∪ c))
2		ax-a3 32	. . . . . 6 ((b ∪ (a ∩ b)) ∪ c) = (b ∪ ((a ∩ b) ∪ c))
3		ancom 74	. . . . . . . . 9 (a ∩ b) = (b ∩ a)
4	3	lor 70	. . . . . . . 8 (b ∪ (a ∩ b)) = (b ∪ (b ∩ a))
5		orabs 120	. . . . . . . 8 (b ∪ (b ∩ a)) = b
6	4, 5	ax-r2 36	. . . . . . 7 (b ∪ (a ∩ b)) = b
7	6	ax-r5 38	. . . . . 6 ((b ∪ (a ∩ b)) ∪ c) = (b ∪ c)
8		or12 80	. . . . . 6 (b ∪ ((a ∩ b) ∪ c)) = ((a ∩ b) ∪ (b ∪ c))
9	2, 7, 8	3tr2 64	. . . . 5 (b ∪ c) = ((a ∩ b) ∪ (b ∪ c))
10	9	lan 77	. . . 4 ((a ∩ b)⊥ ∩ (b ∪ c)) = ((a ∩ b)⊥ ∩ ((a ∩ b) ∪ (b ∪ c)))
11	10	lor 70	. . 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)

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-r3 439

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

This theorem is used by:  i3th1  543