Theorem k1-8b 356

Description: Second part of statement (8) in proof of Theorem 1 of Kalmbach, Orthomodular Lattices, p. 21.

Hypotheses

Ref	Expression
k1-8b.1	y⊥ = ((y⊥ ∩ c) ∪ (y⊥ ∩ c⊥ ))
k1-8b.2	x ≤ c
k1-8b.3	y ≤ c⊥

Assertion

Ref	Expression
k1-8b	y = ((x ∪ y) ∩ c⊥ )

Proof of Theorem k1-8b

Step	Hyp	Ref	Expression
1		k1-8b.1	. . . 4 y⊥ = ((y⊥ ∩ c) ∪ (y⊥ ∩ c⊥ ))
2		ax-a1 30	. . . . . 6 c = c⊥ ⊥
3	2	lan 77	. . . . 5 (y⊥ ∩ c) = (y⊥ ∩ c⊥ ⊥ )
4	3	ror 71	. . . 4 ((y⊥ ∩ c) ∪ (y⊥ ∩ c⊥ )) = ((y⊥ ∩ c⊥ ⊥ ) ∪ (y⊥ ∩ c⊥ ))
5		orcom 73	. . . 4 ((y⊥ ∩ c⊥ ⊥ ) ∪ (y⊥ ∩ c⊥ )) = ((y⊥ ∩ c⊥ ) ∪ (y⊥ ∩ c⊥ ⊥ ))
6	1, 4, 5	3tr 65	. . 3 y⊥ = ((y⊥ ∩ c⊥ ) ∪ (y⊥ ∩ c⊥ ⊥ ))
7		k1-8b.3	. . 3 y ≤ c⊥
8		k1-8b.2	. . . 4 x ≤ c
9	8, 2	lbtr 139	. . 3 x ≤ c⊥ ⊥
10	6, 7, 9	k1-8a 355	. 2 y = ((y ∪ x) ∩ c⊥ )
11		orcom 73	. . 3 (y ∪ x) = (x ∪ y)
12	11	ran 78	. 2 ((y ∪ x) ∩ c⊥ ) = ((x ∪ y) ∩ c⊥ )
13	10, 12	tr 62	1 y = ((x ∪ y) ∩ c⊥ )

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7

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-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131

This theorem is referenced by:  k1-3  358