Theorem go1 335

Description: Lemma for proof of Mayet 8-variable "full" equation from 4-variable Godowski equation.

Assertion

Ref	Expression
go1	((a ^ b) ^ (a ->1 b_|_)) = 0

Proof of Theorem go1

Step	Hyp	Ref	Expression
1		df-i1 43	. . 3 (a ->1 b_|_) = (a_|_ v (a ^ b_|_))
2	1	lan 70	. 2 ((a ^ b) ^ (a ->1 b_|_)) = ((a ^ b) ^ (a_|_ v (a ^ b_|_)))
3		lear 153	. . . . . 6 (a ^ b_|_) =< b_|_
4	3	lelor 158	. . . . 5 (a_|_ v (a ^ b_|_)) =< (a_|_ v b_|_)
5	4	lelan 159	. . . 4 ((a ^ b) ^ (a_|_ v (a ^ b_|_))) =< ((a ^ b) ^ (a_|_ v b_|_))
6		oran3 85	. . . . . 6 (a_|_ v b_|_) = (a ^ b)_|_
7	6	lan 70	. . . . 5 ((a ^ b) ^ (a_|_ v b_|_)) = ((a ^ b) ^ (a ^ b)_|_)
8		dff 93	. . . . . 6 0 = ((a ^ b) ^ (a ^ b)_|_)
9	8	ax-r1 34	. . . . 5 ((a ^ b) ^ (a ^ b)_|_) = 0
10	7, 9	ax-r2 35	. . . 4 ((a ^ b) ^ (a_|_ v b_|_)) = 0
11	5, 10	lbtr 131	. . 3 ((a ^ b) ^ (a_|_ v (a ^ b_|_))) =< 0
12		le0 139	. . 3 0 =< ((a ^ b) ^ (a_|_ v (a ^ b_|_)))
13	11, 12	lebi 137	. 2 ((a ^ b) ^ (a_|_ v (a ^ b_|_))) = 0
14	2, 13	ax-r2 35	1 ((a ^ b) ^ (a ->1 b_|_)) = 0

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  0wf 10   ->1 wi1 13

This theorem is referenced by:  gomaex4 880

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-le1 122  df-le2 123

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