Theorem go1 343

Description: Lemma for proof of Mayet 8-variable "full" equation from 4-variable Godowski equation. (Contributed by NM, 19-Nov-1999.)

Assertion

Ref	Expression
go1	((a ∩ b) ∩ (a →1 b⊥ )) = 0

Proof of Theorem go1

Step	Hyp	Ref	Expression
1		df-i1 44	. . 3 (a →1 b⊥ ) = (a⊥ ∪ (a ∩ b⊥ ))
2	1	lan 77	. 2 ((a ∩ b) ∩ (a →1 b⊥ )) = ((a ∩ b) ∩ (a⊥ ∪ (a ∩ b⊥ )))
3		lear 161	. . . . . 6 (a ∩ b⊥ ) ≤ b⊥
4	3	lelor 166	. . . . 5 (a⊥ ∪ (a ∩ b⊥ )) ≤ (a⊥ ∪ b⊥ )
5	4	lelan 167	. . . 4 ((a ∩ b) ∩ (a⊥ ∪ (a ∩ b⊥ ))) ≤ ((a ∩ b) ∩ (a⊥ ∪ b⊥ ))
6		oran3 93	. . . . . 6 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
7	6	lan 77	. . . . 5 ((a ∩ b) ∩ (a⊥ ∪ b⊥ )) = ((a ∩ b) ∩ (a ∩ b)⊥ )
8		dff 101	. . . . . 6 0 = ((a ∩ b) ∩ (a ∩ b)⊥ )
9	8	ax-r1 35	. . . . 5 ((a ∩ b) ∩ (a ∩ b)⊥ ) = 0
10	7, 9	ax-r2 36	. . . 4 ((a ∩ b) ∩ (a⊥ ∪ b⊥ )) = 0
11	5, 10	lbtr 139	. . 3 ((a ∩ b) ∩ (a⊥ ∪ (a ∩ b⊥ ))) ≤ 0
12		le0 147	. . 3 0 ≤ ((a ∩ b) ∩ (a⊥ ∪ (a ∩ b⊥ )))
13	11, 12	lebi 145	. 2 ((a ∩ b) ∩ (a⊥ ∪ (a ∩ b⊥ ))) = 0
14	2, 13	ax-r2 36	1 ((a ∩ b) ∩ (a →1 b⊥ )) = 0

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →₁ wi1 12

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This proof depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131

This theorem is used by:  gomaex4  900