Theorem nom14 303

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

Assertion

Ref	Expression
nom14	(a ->4 (a ^ b)) = (a ->1 b)

Proof of Theorem nom14

Step	Hyp	Ref	Expression
1		ax-a2 30	. . . . 5 ((a ^ (a ^ b)) v (a_|_ ^ (a ^ b))) = ((a_|_ ^ (a ^ b)) v (a ^ (a ^ b)))
2		anass 69	. . . . . . . 8 ((a ^ a) ^ b) = (a ^ (a ^ b))
3	2	ax-r1 34	. . . . . . 7 (a ^ (a ^ b)) = ((a ^ a) ^ b)
4		anidm 103	. . . . . . . 8 (a ^ a) = a
5	4	ran 71	. . . . . . 7 ((a ^ a) ^ b) = (a ^ b)
6	3, 5	ax-r2 35	. . . . . 6 (a ^ (a ^ b)) = (a ^ b)
7	6	lor 66	. . . . 5 ((a_|_ ^ (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	1, 7, 9	3tr 62	. . . 4 ((a ^ (a ^ b)) v (a_|_ ^ (a ^ b))) = (a ^ b)
11	10	ax-r5 37	. . 3 (((a ^ (a ^ b)) v (a_|_ ^ (a ^ b))) v ((a_|_ v (a ^ b)) ^ (a ^ b)_|_)) = ((a ^ b) v ((a_|_ v (a ^ b)) ^ (a ^ b)_|_))
12		leo 150	. . . . 5 (a ^ b) =< ((a ^ b) v a_|_)
13		lea 152	. . . . . 6 ((a_|_ v (a ^ b)) ^ (a ^ b)_|_) =< (a_|_ v (a ^ b))
14		ax-a2 30	. . . . . 6 (a_|_ v (a ^ b)) = ((a ^ b) v a_|_)
15	13, 14	lbtr 131	. . . . 5 ((a_|_ v (a ^ b)) ^ (a ^ b)_|_) =< ((a ^ b) v a_|_)
16	12, 15	lel2or 162	. . . 4 ((a ^ b) v ((a_|_ v (a ^ b)) ^ (a ^ b)_|_)) =< ((a ^ b) v a_|_)
17		leo 150	. . . . . 6 a_|_ =< (a_|_ v (a ^ b))
18		lea 152	. . . . . . 7 (a ^ b) =< a
19	18	lecon 146	. . . . . 6 a_|_ =< (a ^ b)_|_
20	17, 19	ler2an 165	. . . . 5 a_|_ =< ((a_|_ v (a ^ b)) ^ (a ^ b)_|_)
21	20	lelor 158	. . . 4 ((a ^ b) v a_|_) =< ((a ^ b) v ((a_|_ v (a ^ b)) ^ (a ^ b)_|_))
22	16, 21	lebi 137	. . 3 ((a ^ b) v ((a_|_ v (a ^ b)) ^ (a ^ b)_|_)) = ((a ^ b) v a_|_)
23		ax-a2 30	. . 3 ((a ^ b) v a_|_) = (a_|_ v (a ^ b))
24	11, 22, 23	3tr 62	. 2 (((a ^ (a ^ b)) v (a_|_ ^ (a ^ b))) v ((a_|_ v (a ^ b)) ^ (a ^ b)_|_)) = (a_|_ v (a ^ b))
25		df-i4 46	. 2 (a ->4 (a ^ b)) = (((a ^ (a ^ b)) v (a_|_ ^ (a ^ b))) v ((a_|_ v (a ^ b)) ^ (a ^ b)_|_))
26		df-i1 43	. 2 (a ->1 b) = (a_|_ v (a ^ b))
27	24, 25, 26	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 wi4 16

This theorem is referenced by:  nom43 320

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-i4 46  df-le1 122  df-le2 123

metamath.org