Theorem lem4.6.6i4j2 1099

Description: Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set to 4, and j is set to 2. (Contributed by Roy F. Longton, 3-Jul-05.)

Assertion

Ref	Expression
lem4.6.6i4j2	((a →4 b) ∪ (a →2 b)) = (a →0 b)

Proof of Theorem lem4.6.6i4j2

Step	Hyp	Ref	Expression
1		ax-a3 32	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∪ (b ∪ (a⊥ ∩ b⊥ ))) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (((a⊥ ∪ b) ∩ b⊥ ) ∪ (b ∪ (a⊥ ∩ b⊥ ))))
2		ax-a3 32	. . . . . 6 ((((a⊥ ∪ b) ∩ b⊥ ) ∪ b) ∪ (a⊥ ∩ b⊥ )) = (((a⊥ ∪ b) ∩ b⊥ ) ∪ (b ∪ (a⊥ ∩ b⊥ )))
3	2	ax-r1 35	. . . . 5 (((a⊥ ∪ b) ∩ b⊥ ) ∪ (b ∪ (a⊥ ∩ b⊥ ))) = ((((a⊥ ∪ b) ∩ b⊥ ) ∪ b) ∪ (a⊥ ∩ b⊥ ))
4		ax-a2 31	. . . . . . 7 (((a⊥ ∪ b) ∩ b⊥ ) ∪ b) = (b ∪ ((a⊥ ∪ b) ∩ b⊥ ))
5		ancom 74	. . . . . . . 8 ((a⊥ ∪ b) ∩ b⊥ ) = (b⊥ ∩ (a⊥ ∪ b))
6	5	lor 70	. . . . . . 7 (b ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (b ∪ (b⊥ ∩ (a⊥ ∪ b)))
7		leor 159	. . . . . . . 8 b ≤ (a⊥ ∪ b)
8	7	oml2 451	. . . . . . 7 (b ∪ (b⊥ ∩ (a⊥ ∪ b))) = (a⊥ ∪ b)
9	4, 6, 8	3tr 65	. . . . . 6 (((a⊥ ∪ b) ∩ b⊥ ) ∪ b) = (a⊥ ∪ b)
10	9	ax-r5 38	. . . . 5 ((((a⊥ ∪ b) ∩ b⊥ ) ∪ b) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∪ b) ∪ (a⊥ ∩ b⊥ ))
11	3, 10	ax-r2 36	. . . 4 (((a⊥ ∪ b) ∩ b⊥ ) ∪ (b ∪ (a⊥ ∩ b⊥ ))) = ((a⊥ ∪ b) ∪ (a⊥ ∩ b⊥ ))
12	11	lor 70	. . 3 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (((a⊥ ∪ b) ∩ b⊥ ) ∪ (b ∪ (a⊥ ∩ b⊥ )))) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∪ (a⊥ ∩ b⊥ )))
13		leao4 165	. . . . . 6 (a ∩ b) ≤ (a⊥ ∪ b)
14		leao1 162	. . . . . 6 (a⊥ ∩ b) ≤ (a⊥ ∪ b)
15	13, 14	lel2or 170	. . . . 5 ((a ∩ b) ∪ (a⊥ ∩ b)) ≤ (a⊥ ∪ b)
16		leid 148	. . . . . 6 (a⊥ ∪ b) ≤ (a⊥ ∪ b)
17		leao1 162	. . . . . 6 (a⊥ ∩ b⊥ ) ≤ (a⊥ ∪ b)
18	16, 17	lel2or 170	. . . . 5 ((a⊥ ∪ b) ∪ (a⊥ ∩ b⊥ )) ≤ (a⊥ ∪ b)
19	15, 18	lel2or 170	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∪ (a⊥ ∩ b⊥ ))) ≤ (a⊥ ∪ b)
20		leo 158	. . . . 5 (a⊥ ∪ b) ≤ ((a⊥ ∪ b) ∪ (a⊥ ∩ b⊥ ))
21	20	lerr 150	. . . 4 (a⊥ ∪ b) ≤ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∪ (a⊥ ∩ b⊥ )))
22	19, 21	lebi 145	. . 3 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∪ (a⊥ ∩ b⊥ ))) = (a⊥ ∪ b)
23	1, 12, 22	3tr 65	. 2 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∪ (b ∪ (a⊥ ∩ b⊥ ))) = (a⊥ ∪ b)
24		df-i4 47	. . 3 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
25		df-i2 45	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
26	24, 25	2or 72	. 2 ((a →4 b) ∪ (a →2 b)) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∪ (b ∪ (a⊥ ∩ b⊥ )))
27		df-i0 43	. 2 (a →0 b) = (a⊥ ∪ b)
28	23, 26, 27	3tr1 63	1 ((a →4 b) ∪ (a →2 b)) = (a →0 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₀ wi0 11   →₂ wi2 13   →₄ wi4 15

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  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-i2 45  df-i4 47  df-le1 130  df-le2 131

This theorem is referenced by: (None)