Theorem nom50 323

Description: Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper.

Assertion

Ref	Expression
nom50	((a v b) ==0 b) = (a ->2 b)

Proof of Theorem nom50

Step	Hyp	Ref	Expression
1		ancom 68	. . . . . . . 8 (b_|_ ^ a_|_) = (a_|_ ^ b_|_)
2		anor3 82	. . . . . . . 8 (a_|_ ^ b_|_) = (a v b)_|_
3	1, 2	ax-r2 35	. . . . . . 7 (b_|_ ^ a_|_) = (a v b)_|_
4	3	lor 66	. . . . . 6 (b_|__|_ v (b_|_ ^ a_|_)) = (b_|__|_ v (a v b)_|_)
5	3	ax-r4 36	. . . . . . 7 (b_|_ ^ a_|_)_|_ = (a v b)_|__|_
6	5	ax-r5 37	. . . . . 6 ((b_|_ ^ a_|_)_|_ v b_|_) = ((a v b)_|__|_ v b_|_)
7	4, 6	2an 72	. . . . 5 ((b_|__|_ v (b_|_ ^ a_|_)) ^ ((b_|_ ^ a_|_)_|_ v b_|_)) = ((b_|__|_ v (a v b)_|_) ^ ((a v b)_|__|_ v b_|_))
8	7	ax-r1 34	. . . 4 ((b_|__|_ v (a v b)_|_) ^ ((a v b)_|__|_ v b_|_)) = ((b_|__|_ v (b_|_ ^ a_|_)) ^ ((b_|_ ^ a_|_)_|_ v b_|_))
9		df-id0 48	. . . 4 (b_|_ ==0 (a v b)_|_) = ((b_|__|_ v (a v b)_|_) ^ ((a v b)_|__|_ v b_|_))
10		df-id0 48	. . . 4 (b_|_ ==0 (b_|_ ^ a_|_)) = ((b_|__|_ v (b_|_ ^ a_|_)) ^ ((b_|_ ^ a_|_)_|_ v b_|_))
11	8, 9, 10	3tr1 60	. . 3 (b_|_ ==0 (a v b)_|_) = (b_|_ ==0 (b_|_ ^ a_|_))
12		nom20 305	. . 3 (b_|_ ==0 (b_|_ ^ a_|_)) = (b_|_ ->1 a_|_)
13	11, 12	ax-r2 35	. 2 (b_|_ ==0 (a v b)_|_) = (b_|_ ->1 a_|_)
14		nomcon0 293	. 2 ((a v b) ==0 b) = (b_|_ ==0 (a v b)_|_)
15		i2i1 259	. 2 (a ->2 b) = (b_|_ ->1 a_|_)
16	13, 14, 15	3tr1 60	1 ((a v b) ==0 b) = (a ->2 b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13   ->2 wi2 14   ==0 wid0 18

This theorem is referenced by:  nom60 329

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-i2 44  df-id0 48  df-le1 122  df-le2 123

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