Theorem ska15 244

Description: Soundness theorem for Kalmbach's quantum propositional logic axiom KA15. (Contributed by NM, 2-Nov-1997.)

Assertion

Ref	Expression
ska15	((a →3 b)⊥ ∪ (a⊥ ∪ b)) = 1

Proof of Theorem ska15

Step	Hyp	Ref	Expression
1		df-i3 46	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2		ax-a2 31	. . . . . 6 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b))
3		lea 160	. . . . . . 7 (a⊥ ∩ b⊥ ) ≤ a⊥
4		lear 161	. . . . . . 7 (a⊥ ∩ b) ≤ b
5	3, 4	le2or 168	. . . . . 6 ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b)) ≤ (a⊥ ∪ b)
6	2, 5	bltr 138	. . . . 5 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ (a⊥ ∪ b)
7		lear 161	. . . . 5 (a ∩ (a⊥ ∪ b)) ≤ (a⊥ ∪ b)
8	6, 7	le2or 168	. . . 4 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ≤ ((a⊥ ∪ b) ∪ (a⊥ ∪ b))
9		oridm 110	. . . 4 ((a⊥ ∪ b) ∪ (a⊥ ∪ b)) = (a⊥ ∪ b)
10	8, 9	lbtr 139	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ≤ (a⊥ ∪ b)
11	1, 10	bltr 138	. 2 (a →3 b) ≤ (a⊥ ∪ b)
12	11	sklem 230	1 ((a →3 b)⊥ ∪ (a⊥ ∪ b)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₃ wi3 14

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  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-i3 46  df-le1 130  df-le2 131

This theorem is referenced by:  skmp3  245  mccune3  248