Theorem nom20 305

Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.

Assertion

Ref	Expression
nom20	(a ==0 (a ^ b)) = (a ->1 b)

Proof of Theorem nom20

Step	Hyp	Ref	Expression
1		lea 152	. . . . . 6 (a ^ b) =< a
2		leor 151	. . . . . 6 a =< (b_|_ v a)
3	1, 2	letr 129	. . . . 5 (a ^ b) =< (b_|_ v a)
4	3	lelor 158	. . . 4 (a_|_ v (a ^ b)) =< (a_|_ v (b_|_ v a))
5		ax-a3 31	. . . . . 6 ((a_|_ v b_|_) v a) = (a_|_ v (b_|_ v a))
6	5	ax-r1 34	. . . . 5 (a_|_ v (b_|_ v a)) = ((a_|_ v b_|_) v a)
7		oran3 85	. . . . . 6 (a_|_ v b_|_) = (a ^ b)_|_
8	7	ax-r5 37	. . . . 5 ((a_|_ v b_|_) v a) = ((a ^ b)_|_ v a)
9	6, 8	ax-r2 35	. . . 4 (a_|_ v (b_|_ v a)) = ((a ^ b)_|_ v a)
10	4, 9	lbtr 131	. . 3 (a_|_ v (a ^ b)) =< ((a ^ b)_|_ v a)
11	10	df2le2 128	. 2 ((a_|_ v (a ^ b)) ^ ((a ^ b)_|_ v a)) = (a_|_ v (a ^ b))
12		df-id0 48	. 2 (a ==0 (a ^ b)) = ((a_|_ v (a ^ b)) ^ ((a ^ b)_|_ v a))
13		df-i1 43	. 2 (a ->1 b) = (a_|_ v (a ^ b))
14	11, 12, 13	3tr1 60	1 (a ==0 (a ^ b)) = (a ->1 b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13   ==0 wid0 18

This theorem is referenced by:  nom30 311  nom50 323

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

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i1 43  df-id0 48  df-le1 122  df-le2 123

metamath.org