Theorem ska13 233

Description: Soundness theorem for Kalmbach's quantum propositional logic axiom KA13.

Assertion

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

Proof of Theorem ska13

Step	Hyp	Ref	Expression
1		ledio 168	. . . . 5 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ (((a ∩ b) ∪ a⊥ ) ∩ ((a ∩ b) ∪ b⊥ ))
2		lea 152	. . . . 5 (((a ∩ b) ∪ a⊥ ) ∩ ((a ∩ b) ∪ b⊥ )) ≤ ((a ∩ b) ∪ a⊥ )
3	1, 2	letr 129	. . . 4 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ ((a ∩ b) ∪ a⊥ )
4		ancom 68	. . . . . 6 (a ∩ b) = (b ∩ a)
5		lea 152	. . . . . 6 (b ∩ a) ≤ b
6	4, 5	bltr 130	. . . . 5 (a ∩ b) ≤ b
7	6	leror 144	. . . 4 ((a ∩ b) ∪ a⊥ ) ≤ (b ∪ a⊥ )
8	3, 7	letr 129	. . 3 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ (b ∪ a⊥ )
9		dfb 86	. . 3 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
10		ax-a2 30	. . 3 (a⊥ ∪ b) = (b ∪ a⊥ )
11	8, 9, 10	le3tr1 132	. 2 (a ≡ b) ≤ (a⊥ ∪ b)
12	11	sklem 222	1 ((a ≡ b)⊥ ∪ (a⊥ ∪ b)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 9

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

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123

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