Theorem skr0 242

Description: Soundness theorem for Kalmbach's quantum propositional logic axiom KR0. (Contributed by NM, 30-Aug-1997.)

Hypotheses

Ref	Expression
skr0.1	a = 1
skr0.2	(a⊥ ∪ b) = 1

Assertion

Ref	Expression
skr0	b = 1

Proof of Theorem skr0

Step	Hyp	Ref	Expression
1		ax-a2 31	. . 3 (b ∪ 0) = (0 ∪ b)
2		or0 102	. . . 4 (b ∪ 0) = b
3	2	ax-r1 35	. . 3 b = (b ∪ 0)
4		skr0.1	. . . . . 6 a = 1
5	4	ax-r4 37	. . . . 5 a⊥ = 1⊥
6		df-f 42	. . . . . 6 0 = 1⊥
7	6	ax-r1 35	. . . . 5 1⊥ = 0
8	5, 7	ax-r2 36	. . . 4 a⊥ = 0
9	8	ax-r5 38	. . 3 (a⊥ ∪ b) = (0 ∪ b)
10	1, 3, 9	3tr1 63	. 2 b = (a⊥ ∪ b)
11		skr0.2	. 2 (a⊥ ∪ b) = 1
12	10, 11	ax-r2 36	1 b = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6  1wt 8  0wf 9

This theorem was proved from axioms:  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-t 41  df-f 42

This theorem is referenced by:  skmp3  245  wql2lem3  290  wql2lem4  291  id5id0  352  wr5-2v  366  lem3.3.2  1046  wdwom  1106