Theorem nom24 309

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

Assertion

Ref	Expression
nom24	(a ==4 (a ^ b)) = (a ->1 b)

Proof of Theorem nom24

Step	Hyp	Ref	Expression
1		leo 150	. . . . 5 a_|_ =< (a_|_ v b_|_)
2	1	leror 144	. . . 4 (a_|_ v (a ^ b)) =< ((a_|_ v b_|_) v (a ^ b))
3		oran3 85	. . . . 5 (a_|_ v b_|_) = (a ^ b)_|_
4		anidm 103	. . . . . . . 8 (a ^ a) = a
5	4	ran 71	. . . . . . 7 ((a ^ a) ^ b) = (a ^ b)
6	5	ax-r1 34	. . . . . 6 (a ^ b) = ((a ^ a) ^ b)
7		anass 69	. . . . . 6 ((a ^ a) ^ b) = (a ^ (a ^ b))
8	6, 7	ax-r2 35	. . . . 5 (a ^ b) = (a ^ (a ^ b))
9	3, 8	2or 67	. . . 4 ((a_|_ v b_|_) v (a ^ b)) = ((a ^ b)_|_ v (a ^ (a ^ b)))
10	2, 9	lbtr 131	. . 3 (a_|_ v (a ^ b)) =< ((a ^ b)_|_ v (a ^ (a ^ b)))
11	10	df2le2 128	. 2 ((a_|_ v (a ^ b)) ^ ((a ^ b)_|_ v (a ^ (a ^ b)))) = (a_|_ v (a ^ b))
12		df-id4 52	. 2 (a ==4 (a ^ b)) = ((a_|_ v (a ^ b)) ^ ((a ^ b)_|_ v (a ^ (a ^ b))))
13		df-i1 43	. 2 (a ->1 b) = (a_|_ v (a ^ b))
14	11, 12, 13	3tr1 60	1 (a ==4 (a ^ b)) = (a ->1 b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13   ==4 wid4 22

This theorem is referenced by:  nom31 312  nom53 326

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-id4 52  df-le1 122  df-le2 123

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