Theorem wwoml2 204

Description: Weak orthomodular law.

Hypothesis

Ref	Expression
wwoml2.1	a ≤ b

Assertion

Ref	Expression
wwoml2	((a ∪ (a⊥ ∩ b)) ≡ b) = 1

Proof of Theorem wwoml2

Step	Hyp	Ref	Expression
1		wwoml2.1	. . . . . . 7 a ≤ b
2	1	df-le2 123	. . . . . 6 (a ∪ b) = b
3	2	ax-r1 34	. . . . 5 b = (a ∪ b)
4	3	lan 70	. . . 4 (a⊥ ∩ b) = (a⊥ ∩ (a ∪ b))
5	4	lor 66	. . 3 (a ∪ (a⊥ ∩ b)) = (a ∪ (a⊥ ∩ (a ∪ b)))
6	5	rbi 90	. 2 ((a ∪ (a⊥ ∩ b)) ≡ (a ∪ b)) = ((a ∪ (a⊥ ∩ (a ∪ b))) ≡ (a ∪ b))
7	2	lbi 89	. 2 ((a ∪ (a⊥ ∩ b)) ≡ (a ∪ b)) = ((a ∪ (a⊥ ∩ b)) ≡ b)
8		woml 203	. 2 ((a ∪ (a⊥ ∩ (a ∪ b))) ≡ (a ∪ b)) = 1
9	6, 7, 8	3tr2 61	1 ((a ∪ (a⊥ ∩ b)) ≡ b) = 1

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 9

This theorem is referenced by:  wwoml3 205

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

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-le2 123

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