Theorem nom15 304

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

Assertion

Ref	Expression
nom15	(a ->5 (a ^ b)) = (a ->1 b)

Proof of Theorem nom15

Step	Hyp	Ref	Expression
1		anass 69	. . . . . . . 8 ((a ^ a) ^ b) = (a ^ (a ^ b))
2	1	ax-r1 34	. . . . . . 7 (a ^ (a ^ b)) = ((a ^ a) ^ b)
3		anidm 103	. . . . . . . 8 (a ^ a) = a
4	3	ran 71	. . . . . . 7 ((a ^ a) ^ b) = (a ^ b)
5	2, 4	ax-r2 35	. . . . . 6 (a ^ (a ^ b)) = (a ^ b)
6	5	ax-r5 37	. . . . 5 ((a ^ (a ^ b)) v (a_|_ ^ (a ^ b))) = ((a ^ b) v (a_|_ ^ (a ^ b)))
7		ax-a2 30	. . . . 5 ((a ^ b) v (a_|_ ^ (a ^ b))) = ((a_|_ ^ (a ^ b)) v (a ^ b))
8		lear 153	. . . . . 6 (a_|_ ^ (a ^ b)) =< (a ^ b)
9	8	df-le2 123	. . . . 5 ((a_|_ ^ (a ^ b)) v (a ^ b)) = (a ^ b)
10	6, 7, 9	3tr 62	. . . 4 ((a ^ (a ^ b)) v (a_|_ ^ (a ^ b))) = (a ^ b)
11		oran3 85	. . . . . . 7 (a_|_ v b_|_) = (a ^ b)_|_
12	11	lan 70	. . . . . 6 (a_|_ ^ (a_|_ v b_|_)) = (a_|_ ^ (a ^ b)_|_)
13	12	ax-r1 34	. . . . 5 (a_|_ ^ (a ^ b)_|_) = (a_|_ ^ (a_|_ v b_|_))
14		a5c 113	. . . . 5 (a_|_ ^ (a_|_ v b_|_)) = a_|_
15	13, 14	ax-r2 35	. . . 4 (a_|_ ^ (a ^ b)_|_) = a_|_
16	10, 15	2or 67	. . 3 (((a ^ (a ^ b)) v (a_|_ ^ (a ^ b))) v (a_|_ ^ (a ^ b)_|_)) = ((a ^ b) v a_|_)
17		ax-a2 30	. . 3 ((a ^ b) v a_|_) = (a_|_ v (a ^ b))
18	16, 17	ax-r2 35	. 2 (((a ^ (a ^ b)) v (a_|_ ^ (a ^ b))) v (a_|_ ^ (a ^ b)_|_)) = (a_|_ v (a ^ b))
19		df-i5 47	. 2 (a ->5 (a ^ b)) = (((a ^ (a ^ b)) v (a_|_ ^ (a ^ b))) v (a_|_ ^ (a ^ b)_|_))
20		df-i1 43	. 2 (a ->1 b) = (a_|_ v (a ^ b))
21	18, 19, 20	3tr1 60	1 (a ->5 (a ^ b)) = (a ->1 b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13   ->5 wi5 17

This theorem is referenced by:  nom45 322

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-i5 47  df-le1 122  df-le2 123

metamath.org