Theorem lem4.6.6i2j4 1097

Description: Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set to 2, and j is set to 4. (Contributed by Roy F. Longton, 1-Jul-2005.)

Assertion

Ref	Expression
lem4.6.6i2j4	((a →2 b) ∪ (a →4 b)) = (a →0 b)

Proof of Theorem lem4.6.6i2j4

Step	Hyp	Ref	Expression
1		ax-a2 31	. . . 4 (b ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ b)
2	1	ax-r5 38	. . 3 ((b ∪ (a⊥ ∩ b⊥ )) ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))) = (((a⊥ ∩ b⊥ ) ∪ b) ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )))
3		ax-a3 32	. . 3 (((a⊥ ∩ b⊥ ) ∪ b) ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))) = ((a⊥ ∩ b⊥ ) ∪ (b ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))))
4		ax-a3 32	. . . . . 6 ((b ∪ ((a ∩ b) ∪ (a⊥ ∩ b))) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (b ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )))
5	4	ax-r1 35	. . . . 5 (b ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))) = ((b ∪ ((a ∩ b) ∪ (a⊥ ∩ b))) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
6	5	lor 70	. . . 4 ((a⊥ ∩ b⊥ ) ∪ (b ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )))) = ((a⊥ ∩ b⊥ ) ∪ ((b ∪ ((a ∩ b) ∪ (a⊥ ∩ b))) ∪ ((a⊥ ∪ b) ∩ b⊥ )))
7		ax-a2 31	. . . . . 6 (b ∪ ((a ∩ b) ∪ (a⊥ ∩ b))) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b)
8	7	ax-r5 38	. . . . 5 ((b ∪ ((a ∩ b) ∪ (a⊥ ∩ b))) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
9	8	lor 70	. . . 4 ((a⊥ ∩ b⊥ ) ∪ ((b ∪ ((a ∩ b) ∪ (a⊥ ∩ b))) ∪ ((a⊥ ∪ b) ∩ b⊥ ))) = ((a⊥ ∩ b⊥ ) ∪ ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b) ∪ ((a⊥ ∪ b) ∩ b⊥ )))
10		ax-a3 32	. . . . . 6 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (b ∪ ((a⊥ ∪ b) ∩ b⊥ )))
11	10	lor 70	. . . . 5 ((a⊥ ∩ b⊥ ) ∪ ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b) ∪ ((a⊥ ∪ b) ∩ b⊥ ))) = ((a⊥ ∩ b⊥ ) ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (b ∪ ((a⊥ ∪ b) ∩ b⊥ ))))
12		ancom 74	. . . . . . . . 9 ((a⊥ ∪ b) ∩ b⊥ ) = (b⊥ ∩ (a⊥ ∪ b))
13	12	lor 70	. . . . . . . 8 (b ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (b ∪ (b⊥ ∩ (a⊥ ∪ b)))
14		ax-a2 31	. . . . . . . . . 10 (a⊥ ∪ b) = (b ∪ a⊥ )
15	14	lan 77	. . . . . . . . 9 (b⊥ ∩ (a⊥ ∪ b)) = (b⊥ ∩ (b ∪ a⊥ ))
16	15	lor 70	. . . . . . . 8 (b ∪ (b⊥ ∩ (a⊥ ∪ b))) = (b ∪ (b⊥ ∩ (b ∪ a⊥ )))
17		oml 445	. . . . . . . . 9 (b ∪ (b⊥ ∩ (b ∪ a⊥ ))) = (b ∪ a⊥ )
18		ax-a2 31	. . . . . . . . 9 (b ∪ a⊥ ) = (a⊥ ∪ b)
19	17, 18	ax-r2 36	. . . . . . . 8 (b ∪ (b⊥ ∩ (b ∪ a⊥ ))) = (a⊥ ∪ b)
20	13, 16, 19	3tr 65	. . . . . . 7 (b ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (a⊥ ∪ b)
21	20	lor 70	. . . . . 6 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (b ∪ ((a⊥ ∪ b) ∩ b⊥ ))) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∪ b))
22	21	lor 70	. . . . 5 ((a⊥ ∩ b⊥ ) ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (b ∪ ((a⊥ ∪ b) ∩ b⊥ )))) = ((a⊥ ∩ b⊥ ) ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∪ b)))
23		leao1 162	. . . . . . 7 (a⊥ ∩ b⊥ ) ≤ (a⊥ ∪ b)
24		leao4 165	. . . . . . . . 9 (a ∩ b) ≤ (a⊥ ∪ b)
25		leao1 162	. . . . . . . . 9 (a⊥ ∩ b) ≤ (a⊥ ∪ b)
26	24, 25	lel2or 170	. . . . . . . 8 ((a ∩ b) ∪ (a⊥ ∩ b)) ≤ (a⊥ ∪ b)
27		leid 148	. . . . . . . 8 (a⊥ ∪ b) ≤ (a⊥ ∪ b)
28	26, 27	lel2or 170	. . . . . . 7 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∪ b)) ≤ (a⊥ ∪ b)
29	23, 28	lel2or 170	. . . . . 6 ((a⊥ ∩ b⊥ ) ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∪ b))) ≤ (a⊥ ∪ b)
30		leor 159	. . . . . . 7 (a⊥ ∪ b) ≤ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∪ b))
31	30	lerr 150	. . . . . 6 (a⊥ ∪ b) ≤ ((a⊥ ∩ b⊥ ) ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∪ b)))
32	29, 31	lebi 145	. . . . 5 ((a⊥ ∩ b⊥ ) ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∪ b))) = (a⊥ ∪ b)
33	11, 22, 32	3tr 65	. . . 4 ((a⊥ ∩ b⊥ ) ∪ ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b) ∪ ((a⊥ ∪ b) ∩ b⊥ ))) = (a⊥ ∪ b)
34	6, 9, 33	3tr 65	. . 3 ((a⊥ ∩ b⊥ ) ∪ (b ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )))) = (a⊥ ∪ b)
35	2, 3, 34	3tr 65	. 2 ((b ∪ (a⊥ ∩ b⊥ )) ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))) = (a⊥ ∪ b)
36		df-i2 45	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
37		df-i4 47	. . 3 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
38	36, 37	2or 72	. 2 ((a →2 b) ∪ (a →4 b)) = ((b ∪ (a⊥ ∩ b⊥ )) ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )))
39		df-i0 43	. 2 (a →0 b) = (a⊥ ∪ b)
40	35, 38, 39	3tr1 63	1 ((a →2 b) ∪ (a →4 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)