Theorem ska13 241

Description: Soundness theorem for Kalmbach's quantum propositional logic axiom KA13. (Contributed by NM, 30-Aug-1997.)

Assertion

Ref	Expression
ska13	((a ≡ b)⊥ ∪ (a⊥ ∪ b)) = 1

Proof of Theorem ska13

Step	Hyp	Ref	Expression
1		ledio 176	. . . . 5 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ (((a ∩ b) ∪ a⊥ ) ∩ ((a ∩ b) ∪ b⊥ ))
2		lea 160	. . . . 5 (((a ∩ b) ∪ a⊥ ) ∩ ((a ∩ b) ∪ b⊥ )) ≤ ((a ∩ b) ∪ a⊥ )
3	1, 2	letr 137	. . . 4 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ ((a ∩ b) ∪ a⊥ )
4		ancom 74	. . . . . 6 (a ∩ b) = (b ∩ a)
5		lea 160	. . . . . 6 (b ∩ a) ≤ b
6	4, 5	bltr 138	. . . . 5 (a ∩ b) ≤ b
7	6	leror 152	. . . 4 ((a ∩ b) ∪ a⊥ ) ≤ (b ∪ a⊥ )
8	3, 7	letr 137	. . 3 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ (b ∪ a⊥ )
9		dfb 94	. . 3 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
10		ax-a2 31	. . 3 (a⊥ ∪ b) = (b ∪ a⊥ )
11	8, 9, 10	le3tr1 140	. 2 (a ≡ b) ≤ (a⊥ ∪ b)
12	11	sklem 230	1 ((a ≡ b)⊥ ∪ (a⊥ ∪ b)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8

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-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131

This theorem is referenced by: (None)