Theorem nom22 307

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

Assertion

Ref	Expression
nom22	(a ==2 (a ^ b)) = (a ->1 b)

Proof of Theorem nom22

Step	Hyp	Ref	Expression
1		oran3 85	. . . . . . 7 (a_|_ v b_|_) = (a ^ b)_|_
2	1	lor 66	. . . . . 6 (a v (a_|_ v b_|_)) = (a v (a ^ b)_|_)
3	2	ax-r1 34	. . . . 5 (a v (a ^ b)_|_) = (a v (a_|_ v b_|_))
4		or12 73	. . . . 5 (a v (a_|_ v b_|_)) = (a_|_ v (a v b_|_))
5	3, 4	ax-r2 35	. . . 4 (a v (a ^ b)_|_) = (a_|_ v (a v b_|_))
6		ax-a2 30	. . . . 5 ((a ^ b) v (a_|_ ^ (a ^ b)_|_)) = ((a_|_ ^ (a ^ b)_|_) v (a ^ b))
7	1	lan 70	. . . . . . . 8 (a_|_ ^ (a_|_ v b_|_)) = (a_|_ ^ (a ^ b)_|_)
8	7	ax-r1 34	. . . . . . 7 (a_|_ ^ (a ^ b)_|_) = (a_|_ ^ (a_|_ v b_|_))
9		a5c 113	. . . . . . 7 (a_|_ ^ (a_|_ v b_|_)) = a_|_
10	8, 9	ax-r2 35	. . . . . 6 (a_|_ ^ (a ^ b)_|_) = a_|_
11	10	ax-r5 37	. . . . 5 ((a_|_ ^ (a ^ b)_|_) v (a ^ b)) = (a_|_ v (a ^ b))
12	6, 11	ax-r2 35	. . . 4 ((a ^ b) v (a_|_ ^ (a ^ b)_|_)) = (a_|_ v (a ^ b))
13	5, 12	2an 72	. . 3 ((a v (a ^ b)_|_) ^ ((a ^ b) v (a_|_ ^ (a ^ b)_|_))) = ((a_|_ v (a v b_|_)) ^ (a_|_ v (a ^ b)))
14		ancom 68	. . 3 ((a_|_ v (a v b_|_)) ^ (a_|_ v (a ^ b))) = ((a_|_ v (a ^ b)) ^ (a_|_ v (a v b_|_)))
15		lea 152	. . . . . 6 (a ^ b) =< a
16		leo 150	. . . . . 6 a =< (a v b_|_)
17	15, 16	letr 129	. . . . 5 (a ^ b) =< (a v b_|_)
18	17	lelor 158	. . . 4 (a_|_ v (a ^ b)) =< (a_|_ v (a v b_|_))
19	18	df2le2 128	. . 3 ((a_|_ v (a ^ b)) ^ (a_|_ v (a v b_|_))) = (a_|_ v (a ^ b))
20	13, 14, 19	3tr 62	. 2 ((a v (a ^ b)_|_) ^ ((a ^ b) v (a_|_ ^ (a ^ b)_|_))) = (a_|_ v (a ^ b))
21		df-id2 50	. 2 (a ==2 (a ^ b)) = ((a v (a ^ b)_|_) ^ ((a ^ b) v (a_|_ ^ (a ^ b)_|_)))
22		df-i1 43	. 2 (a ->1 b) = (a_|_ v (a ^ b))
23	20, 21, 22	3tr1 60	1 (a ==2 (a ^ b)) = (a ->1 b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13   ==2 wid2 20

This theorem is referenced by:  nom33 314  nom51 324

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-id2 50  df-le1 122  df-le2 123

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