Theorem lem4.6.6i0j2 1089

Description: Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set to 0, and j is set to 2. (Contributed by Roy F. Longton, 1-Jul-2005.)

Assertion

Ref	Expression
lem4.6.6i0j2	((a →0 b) ∪ (a →2 b)) = (a →0 b)

Proof of Theorem lem4.6.6i0j2

Step	Hyp	Ref	Expression
1		leid 148	. . . 4 (a⊥ ∪ b) ≤ (a⊥ ∪ b)
2		leor 159	. . . . 5 b ≤ (a⊥ ∪ b)
3		leao1 162	. . . . 5 (a⊥ ∩ b⊥ ) ≤ (a⊥ ∪ b)
4	2, 3	lel2or 170	. . . 4 (b ∪ (a⊥ ∩ b⊥ )) ≤ (a⊥ ∪ b)
5	1, 4	lel2or 170	. . 3 ((a⊥ ∪ b) ∪ (b ∪ (a⊥ ∩ b⊥ ))) ≤ (a⊥ ∪ b)
6		leo 158	. . 3 (a⊥ ∪ b) ≤ ((a⊥ ∪ b) ∪ (b ∪ (a⊥ ∩ b⊥ )))
7	5, 6	lebi 145	. 2 ((a⊥ ∪ b) ∪ (b ∪ (a⊥ ∩ b⊥ ))) = (a⊥ ∪ b)
8		df-i0 43	. . 3 (a →0 b) = (a⊥ ∪ b)
9		df-i2 45	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
10	8, 9	2or 72	. 2 ((a →0 b) ∪ (a →2 b)) = ((a⊥ ∪ b) ∪ (b ∪ (a⊥ ∩ b⊥ )))
11	7, 10, 8	3tr1 63	1 ((a →0 b) ∪ (a →2 b)) = (a →0 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₀ wi0 11   →₂ wi2 13

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-i0 43  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by: (None)