Theorem oml5 431

Description: Orthomodular law.

Assertion

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

Proof of Theorem oml5

Step	Hyp	Ref	Expression
1		oml 427	. . 3 ((a ∩ b) ∪ ((a ∩ b)⊥ ∩ ((a ∩ b) ∪ (b ∪ c)))) = ((a ∩ b) ∪ (b ∪ c))
2		ax-a3 31	. . . . . 6 ((b ∪ (a ∩ b)) ∪ c) = (b ∪ ((a ∩ b) ∪ c))
3		ancom 68	. . . . . . . . 9 (a ∩ b) = (b ∩ a)
4	3	lor 66	. . . . . . . 8 (b ∪ (a ∩ b)) = (b ∪ (b ∩ a))
5		a5b 112	. . . . . . . 8 (b ∪ (b ∩ a)) = b
6	4, 5	ax-r2 35	. . . . . . 7 (b ∪ (a ∩ b)) = b
7	6	ax-r5 37	. . . . . 6 ((b ∪ (a ∩ b)) ∪ c) = (b ∪ c)
8		or12 73	. . . . . 6 (b ∪ ((a ∩ b) ∪ c)) = ((a ∩ b) ∪ (b ∪ c))
9	2, 7, 8	3tr2 61	. . . . 5 (b ∪ c) = ((a ∩ b) ∪ (b ∪ c))
10	9	lan 70	. . . 4 ((a ∩ b)⊥ ∩ (b ∪ c)) = ((a ∩ b)⊥ ∩ ((a ∩ b) ∪ (b ∪ c)))
11	10	lor 66	. . 3 ((a ∩ b) ∪ ((a ∩ b)⊥ ∩ (b ∪ c))) = ((a ∩ b) ∪ ((a ∩ b)⊥ ∩ ((a ∩ b) ∪ (b ∪ c))))
12	2, 8	ax-r2 35	. . 3 ((b ∪ (a ∩ b)) ∪ c) = ((a ∩ b) ∪ (b ∪ c))
13	1, 11, 12	3tr1 60	. 2 ((a ∩ b) ∪ ((a ∩ b)⊥ ∩ (b ∪ c))) = ((b ∪ (a ∩ b)) ∪ c)
14	13, 7	ax-r2 35	1 ((a ∩ b) ∪ ((a ∩ b)⊥ ∩ (b ∪ c))) = (b ∪ c)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7

This theorem is referenced by:  i3th1 525

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41

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