Theorem u1lem9a 759

Description: Lemma used in study of orthoarguesian law.

Assertion

Ref	Expression
u1lem9a	(a⊥ →1 b)⊥ ≤ a⊥

Proof of Theorem u1lem9a

Step	Hyp	Ref	Expression
1		df-i1 43	. . . 4 (a⊥ →1 b) = (a⊥ ⊥ ∪ (a⊥ ∩ b))
2	1	ax-r4 36	. . 3 (a⊥ →1 b)⊥ = (a⊥ ⊥ ∪ (a⊥ ∩ b))⊥
3		anor1 80	. . . 4 (a⊥ ∩ (a⊥ ∩ b)⊥ ) = (a⊥ ⊥ ∪ (a⊥ ∩ b))⊥
4	3	ax-r1 34	. . 3 (a⊥ ⊥ ∪ (a⊥ ∩ b))⊥ = (a⊥ ∩ (a⊥ ∩ b)⊥ )
5	2, 4	ax-r2 35	. 2 (a⊥ →1 b)⊥ = (a⊥ ∩ (a⊥ ∩ b)⊥ )
6		lea 152	. 2 (a⊥ ∩ (a⊥ ∩ b)⊥ ) ≤ a⊥
7	5, 6	bltr 130	1 (a⊥ →1 b)⊥ ≤ a⊥

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13

This theorem is referenced by:  u1lem9ab 761  sadm3 820  oa4uto4g 955  oa4uto4 957

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-i1 43  df-le1 122  df-le2 123

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