Theorem wom2 416

Description: Weak orthomodular law for study of weakly orthomodular lattices.

Assertion

Ref	Expression
wom2	a ≤ ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c)))

Proof of Theorem wom2

Step	Hyp	Ref	Expression
1		le1 138	. 2 a ≤ 1
2		conb 114	. . . . . 6 (a ≡ b) = (a⊥ ≡ b⊥ )
3	2	ax-r4 36	. . . . 5 (a ≡ b)⊥ = (a⊥ ≡ b⊥ )⊥
4		oran 79	. . . . . . 7 (a ∪ c) = (a⊥ ∩ c⊥ )⊥
5		oran 79	. . . . . . 7 (b ∪ c) = (b⊥ ∩ c⊥ )⊥
6	4, 5	2bi 91	. . . . . 6 ((a ∪ c) ≡ (b ∪ c)) = ((a⊥ ∩ c⊥ )⊥ ≡ (b⊥ ∩ c⊥ )⊥ )
7		conb 114	. . . . . . 7 ((a⊥ ∩ c⊥ ) ≡ (b⊥ ∩ c⊥ )) = ((a⊥ ∩ c⊥ )⊥ ≡ (b⊥ ∩ c⊥ )⊥ )
8	7	ax-r1 34	. . . . . 6 ((a⊥ ∩ c⊥ )⊥ ≡ (b⊥ ∩ c⊥ )⊥ ) = ((a⊥ ∩ c⊥ ) ≡ (b⊥ ∩ c⊥ ))
9	6, 8	ax-r2 35	. . . . 5 ((a ∪ c) ≡ (b ∪ c)) = ((a⊥ ∩ c⊥ ) ≡ (b⊥ ∩ c⊥ ))
10	3, 9	2or 67	. . . 4 ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c))) = ((a⊥ ≡ b⊥ )⊥ ∪ ((a⊥ ∩ c⊥ ) ≡ (b⊥ ∩ c⊥ )))
11		ska4 415	. . . 4 ((a⊥ ≡ b⊥ )⊥ ∪ ((a⊥ ∩ c⊥ ) ≡ (b⊥ ∩ c⊥ ))) = 1
12	10, 11	ax-r2 35	. . 3 ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c))) = 1
13	12	ax-r1 34	. 2 1 = ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c)))
14	1, 13	lbtr 131	1 a ≤ ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c)))

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

This theorem is referenced by:  ka4ot 417

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-wom 343

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le 121  df-le1 122  df-le2 123  df-cmtr 126

metamath.org