Theorem nom21 306

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

Assertion

Ref	Expression
nom21	(a ==1 (a ^ b)) = (a ->1 b)

Proof of Theorem nom21

Step	Hyp	Ref	Expression
1		ancom 68	. . 3 ((a_|_ v (a v b_|_)) ^ (a_|_ v (a ^ b))) = ((a_|_ v (a ^ b)) ^ (a_|_ v (a v b_|_)))
2		or12 73	. . . . 5 (a_|_ v (a v b_|_)) = (a v (a_|_ v b_|_))
3		oran3 85	. . . . . 6 (a_|_ v b_|_) = (a ^ b)_|_
4	3	lor 66	. . . . 5 (a v (a_|_ v b_|_)) = (a v (a ^ b)_|_)
5	2, 4	ax-r2 35	. . . 4 (a_|_ v (a v b_|_)) = (a v (a ^ b)_|_)
6		anidm 103	. . . . . . . 8 (a ^ a) = a
7	6	ran 71	. . . . . . 7 ((a ^ a) ^ b) = (a ^ b)
8	7	ax-r1 34	. . . . . 6 (a ^ b) = ((a ^ a) ^ b)
9		anass 69	. . . . . 6 ((a ^ a) ^ b) = (a ^ (a ^ b))
10	8, 9	ax-r2 35	. . . . 5 (a ^ b) = (a ^ (a ^ b))
11	10	lor 66	. . . 4 (a_|_ v (a ^ b)) = (a_|_ v (a ^ (a ^ b)))
12	5, 11	2an 72	. . 3 ((a_|_ v (a v b_|_)) ^ (a_|_ v (a ^ b))) = ((a v (a ^ b)_|_) ^ (a_|_ v (a ^ (a ^ b))))
13		lea 152	. . . . . 6 (a ^ b) =< a
14		leo 150	. . . . . 6 a =< (a v b_|_)
15	13, 14	letr 129	. . . . 5 (a ^ b) =< (a v b_|_)
16	15	lelor 158	. . . 4 (a_|_ v (a ^ b)) =< (a_|_ v (a v b_|_))
17	16	df2le2 128	. . 3 ((a_|_ v (a ^ b)) ^ (a_|_ v (a v b_|_))) = (a_|_ v (a ^ b))
18	1, 12, 17	3tr2 61	. 2 ((a v (a ^ b)_|_) ^ (a_|_ v (a ^ (a ^ b)))) = (a_|_ v (a ^ b))
19		df-id1 49	. 2 (a ==1 (a ^ b)) = ((a v (a ^ b)_|_) ^ (a_|_ v (a ^ (a ^ b))))
20		df-i1 43	. 2 (a ->1 b) = (a_|_ v (a ^ b))
21	18, 19, 20	3tr1 60	1 (a ==1 (a ^ b)) = (a ->1 b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13   ==1 wid1 19

This theorem is referenced by:  nom34 315  nom52 325

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-id1 49  df-le1 122  df-le2 123

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