Theorem lem3.3.3lem2 1050

Description: Lemma for lem3.3.3 1052. (Contributed by Roy F. Longton, 27-Jun-2005.) (Revised by Roy F. Longton, 3-Jul-2005.)

Assertion

Ref	Expression
lem3.3.3lem2	(a ≡5 b) ≤ (b →1 a)

Proof of Theorem lem3.3.3lem2

Step	Hyp	Ref	Expression
1		lear 161	. . . 4 (a⊥ ∩ b⊥ ) ≤ b⊥
2	1	leror 152	. . 3 ((a⊥ ∩ b⊥ ) ∪ (a ∩ b)) ≤ (b⊥ ∪ (a ∩ b))
3		ax-a2 31	. . 3 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (a ∩ b))
4		ancom 74	. . . 4 (b ∩ a) = (a ∩ b)
5	4	lor 70	. . 3 (b⊥ ∪ (b ∩ a)) = (b⊥ ∪ (a ∩ b))
6	2, 3, 5	le3tr1 140	. 2 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ (b⊥ ∪ (b ∩ a))
7		df-id5 1047	. 2 (a ≡5 b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
8		df-i1 44	. 2 (b →1 a) = (b⊥ ∪ (b ∩ a))
9	6, 7, 8	le3tr1 140	1 (a ≡5 b) ≤ (b →1 a)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   ≡₅ wid5 22

This theorem was proved from 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 theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-id5 1047

This theorem is referenced by:  lem3.3.3lem3  1051