Theorem wwoml3 213

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

Hypotheses

Ref	Expression
wwoml3.1	a ≤ b
wwoml3.2	(b ∩ a⊥ ) = 0

Assertion

Ref	Expression
wwoml3	(a ≡ b) = 1

Proof of Theorem wwoml3

Step	Hyp	Ref	Expression
1		wwoml3.2	. . . . . 6 (b ∩ a⊥ ) = 0
2	1	ax-r1 35	. . . . 5 0 = (b ∩ a⊥ )
3		ancom 74	. . . . 5 (b ∩ a⊥ ) = (a⊥ ∩ b)
4	2, 3	ax-r2 36	. . . 4 0 = (a⊥ ∩ b)
5	4	lor 70	. . 3 (a ∪ 0) = (a ∪ (a⊥ ∩ b))
6	5	rbi 98	. 2 ((a ∪ 0) ≡ b) = ((a ∪ (a⊥ ∩ b)) ≡ b)
7		or0 102	. . 3 (a ∪ 0) = a
8	7	rbi 98	. 2 ((a ∪ 0) ≡ b) = (a ≡ b)
9		wwoml3.1	. . 3 a ≤ b
10	9	wwoml2 212	. 2 ((a ∪ (a⊥ ∩ b)) ≡ b) = 1
11	6, 8, 10	3tr2 64	1 (a ≡ b) = 1

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9

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

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

This theorem is referenced by:  wwfh1  216  wwfh2  217