Theorem u1lem9a 777

Description: Lemma used in study of orthoarguesian law. Equation 4.11 of [MegPav2000] p. 23. This is the first part of the inequality. (Contributed by NM, 28-Dec-1998.)

Assertion

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

Proof of Theorem u1lem9a

Step	Hyp	Ref	Expression
1		df-i1 44	. . . 4 (a⊥ →1 b) = (a⊥ ⊥ ∪ (a⊥ ∩ b))
2	1	ax-r4 37	. . 3 (a⊥ →1 b)⊥ = (a⊥ ⊥ ∪ (a⊥ ∩ b))⊥
3		anor1 88	. . . 4 (a⊥ ∩ (a⊥ ∩ b)⊥ ) = (a⊥ ⊥ ∪ (a⊥ ∩ b))⊥
4	3	ax-r1 35	. . 3 (a⊥ ⊥ ∪ (a⊥ ∩ b))⊥ = (a⊥ ∩ (a⊥ ∩ b)⊥ )
5	2, 4	ax-r2 36	. 2 (a⊥ →1 b)⊥ = (a⊥ ∩ (a⊥ ∩ b)⊥ )
6		lea 160	. 2 (a⊥ ∩ (a⊥ ∩ b)⊥ ) ≤ a⊥
7	5, 6	bltr 138	1 (a⊥ →1 b)⊥ ≤ a⊥

This proof depends on syntax axioms:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12

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

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

This theorem is used by:  u1lem9ab  779  sadm3  838  oa4uto4g  975  oa4uto4  977  lem4.6.3le1  1084