Theorem lem4.6.6i2j1 1096

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

Assertion

Ref	Expression
lem4.6.6i2j1	((a →2 b) ∪ (a →1 b)) = (a →0 b)

Proof of Theorem lem4.6.6i2j1

Step	Hyp	Ref	Expression
1		leor 159	. . . . 5 b ≤ (a⊥ ∪ b)
2		leao1 162	. . . . 5 (a⊥ ∩ b⊥ ) ≤ (a⊥ ∪ b)
3	1, 2	lel2or 170	. . . 4 (b ∪ (a⊥ ∩ b⊥ )) ≤ (a⊥ ∪ b)
4		lear 161	. . . . 5 (a ∩ b) ≤ b
5	4	lelor 166	. . . 4 (a⊥ ∪ (a ∩ b)) ≤ (a⊥ ∪ b)
6	3, 5	lel2or 170	. . 3 ((b ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ (a ∩ b))) ≤ (a⊥ ∪ b)
7		leo 158	. . . . 5 a⊥ ≤ (a⊥ ∪ (a ∩ b))
8	7	lerr 150	. . . 4 a⊥ ≤ ((b ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ (a ∩ b)))
9		leo 158	. . . . 5 b ≤ (b ∪ (a⊥ ∩ b⊥ ))
10	9	ler 149	. . . 4 b ≤ ((b ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ (a ∩ b)))
11	8, 10	lel2or 170	. . 3 (a⊥ ∪ b) ≤ ((b ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ (a ∩ b)))
12	6, 11	lebi 145	. 2 ((b ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ (a ∩ b))) = (a⊥ ∪ b)
13		df-i2 45	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
14		df-i1 44	. . 3 (a →1 b) = (a⊥ ∪ (a ∩ b))
15	13, 14	2or 72	. 2 ((a →2 b) ∪ (a →1 b)) = ((b ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ (a ∩ b)))
16		df-i0 43	. 2 (a →0 b) = (a⊥ ∪ b)
17	12, 15, 16	3tr1 63	1 ((a →2 b) ∪ (a →1 b)) = (a →0 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₀ wi0 11   →₁ wi1 12   →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-i1 44  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by: (None)