Theorem wwoml2 212

Description: Weak orthomodular law. (Contributed by NM, 2-Sep-1997.)

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 131	. . . . . 6 (a ∪ b) = b
3	2	ax-r1 35	. . . . 5 b = (a ∪ b)
4	3	lan 77	. . . 4 (a⊥ ∩ b) = (a⊥ ∩ (a ∪ b))
5	4	lor 70	. . 3 (a ∪ (a⊥ ∩ b)) = (a ∪ (a⊥ ∩ (a ∪ b)))
6	5	rbi 98	. 2 ((a ∪ (a⊥ ∩ b)) ≡ (a ∪ b)) = ((a ∪ (a⊥ ∩ (a ∪ b))) ≡ (a ∪ b))
7	2	lbi 97	. 2 ((a ∪ (a⊥ ∩ b)) ≡ (a ∪ b)) = ((a ∪ (a⊥ ∩ b)) ≡ b)
8		woml 211	. 2 ((a ∪ (a⊥ ∩ (a ∪ b))) ≡ (a ∪ b)) = 1
9	6, 7, 8	3tr2 64	1 ((a ∪ (a⊥ ∩ b)) ≡ b) = 1

This proof depends on syntax axioms:   = wb 1   ≤ wle 2  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8

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

This proof depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le2 131

This theorem is used by:  wwoml3  213