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⊥ ∪ (a ∩ b⊥ ))
2	1	lan 70	. 2 ((a ∩ b) ∩ (a →1 b⊥ )) = ((a ∩ b) ∩ (a⊥ ∪ (a ∩ b⊥ )))
3		lear 153	. . . . . 6 (a ∩ b⊥ ) ≤ b⊥
4	3	lelor 158	. . . . 5 (a⊥ ∪ (a ∩ b⊥ )) ≤ (a⊥ ∪ b⊥ )
5	4	lelan 159	. . . 4 ((a ∩ b) ∩ (a⊥ ∪ (a ∩ b⊥ ))) ≤ ((a ∩ b) ∩ (a⊥ ∪ b⊥ ))
6		oran3 85	. . . . . 6 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
7	6	lan 70	. . . . 5 ((a ∩ b) ∩ (a⊥ ∪ 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⊥ ∪ b⊥ )) = 0
11	5, 10	lbtr 131	. . 3 ((a ∩ b) ∩ (a⊥ ∪ (a ∩ b⊥ ))) ≤ 0
12		le0 139	. . 3 0 ≤ ((a ∩ b) ∩ (a⊥ ∪ (a ∩ b⊥ )))
13	11, 12	lebi 137	. 2 ((a ∩ b) ∩ (a⊥ ∪ (a ∩ b⊥ ))) = 0
14	2, 13	ax-r2 35	1 ((a ∩ b) ∩ (a →1 b⊥ )) = 0

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ 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

metamath.org