Theorem lem4.6.6i1j3 1094

Description: Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set to 1, and j is set to 3. (Contributed by Roy F. Longton, 1-Jul-2005.)

Assertion

Ref	Expression
lem4.6.6i1j3	((a →1 b) ∪ (a →3 b)) = (a →0 b)

Proof of Theorem lem4.6.6i1j3

Step	Hyp	Ref	Expression
1		leo 158	. . . . . . . 8 a⊥ ≤ (a⊥ ∪ (a ∩ b))
2	1	ler 149	. . . . . . 7 a⊥ ≤ ((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
3	2	lecom 180	. . . . . 6 a⊥ C ((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
4	3	comcom6 459	. . . . 5 a C ((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
5	4	comcom 453	. . . 4 ((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) C a
6		lear 161	. . . . . . 7 (a ∩ b) ≤ b
7	6	lelor 166	. . . . . 6 (a⊥ ∪ (a ∩ b)) ≤ (a⊥ ∪ b)
8		lea 160	. . . . . . . 8 (a⊥ ∩ b) ≤ a⊥
9		lea 160	. . . . . . . 8 (a⊥ ∩ b⊥ ) ≤ a⊥
10	8, 9	lel2or 170	. . . . . . 7 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ a⊥
11	10	ler 149	. . . . . 6 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ (a⊥ ∪ b)
12	7, 11	lel2or 170	. . . . 5 ((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ≤ (a⊥ ∪ b)
13	12	lecom 180	. . . 4 ((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) C (a⊥ ∪ b)
14	5, 13	fh3 471	. . 3 (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a ∩ (a⊥ ∪ b))) = ((((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a) ∩ (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∪ b)))
15		ax-a3 32	. . 3 (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a ∩ (a⊥ ∪ b))) = ((a⊥ ∪ (a ∩ b)) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))))
16		ax-a2 31	. . . . 5 (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a) = (a ∪ ((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))))
17	16	ran 78	. . . 4 ((((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a) ∩ (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∪ b))) = ((a ∪ ((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))) ∩ (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∪ b)))
18		ax-a3 32	. . . . . 6 ((a ∪ (a⊥ ∪ (a ∩ b))) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (a ∪ ((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))))
19	18	ax-r1 35	. . . . 5 (a ∪ ((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))) = ((a ∪ (a⊥ ∪ (a ∩ b))) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
20	19	ran 78	. . . 4 ((a ∪ ((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))) ∩ (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∪ b))) = (((a ∪ (a⊥ ∪ (a ∩ b))) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∩ (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∪ b)))
21		ax-a3 32	. . . . . . . 8 ((a ∪ a⊥ ) ∪ (a ∩ b)) = (a ∪ (a⊥ ∪ (a ∩ b)))
22	21	ax-r1 35	. . . . . . 7 (a ∪ (a⊥ ∪ (a ∩ b))) = ((a ∪ a⊥ ) ∪ (a ∩ b))
23	22	ax-r5 38	. . . . . 6 ((a ∪ (a⊥ ∪ (a ∩ b))) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (((a ∪ a⊥ ) ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
24	23	ran 78	. . . . 5 (((a ∪ (a⊥ ∪ (a ∩ b))) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∩ (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∪ b))) = ((((a ∪ a⊥ ) ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∩ (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∪ b)))
25		ax-a4 33	. . . . . . . . . 10 (1 ∪ (a ∪ a⊥ )) = (a ∪ a⊥ )
26	25	df-le1 130	. . . . . . . . 9 1 ≤ (a ∪ a⊥ )
27	26	ler 149	. . . . . . . 8 1 ≤ ((a ∪ a⊥ ) ∪ (a ∩ b))
28	27	ler 149	. . . . . . 7 1 ≤ (((a ∪ a⊥ ) ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
29	28	lem3.3.5lem 1054	. . . . . 6 (((a ∪ a⊥ ) ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = 1
30	29	ran 78	. . . . 5 ((((a ∪ a⊥ ) ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∩ (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∪ b))) = (1 ∩ (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∪ b)))
31		an1r 107	. . . . . 6 (1 ∩ (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∪ b))) = (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∪ b))
32		ax-a2 31	. . . . . . 7 ((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ (a ∩ b)))
33	32	ax-r5 38	. . . . . 6 (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∪ b)) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ (a ∩ b))) ∪ (a⊥ ∪ b))
34		ax-a3 32	. . . . . . 7 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ (a ∩ b))) ∪ (a⊥ ∪ b)) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ ((a⊥ ∪ (a ∩ b)) ∪ (a⊥ ∪ b)))
35		orordi 112	. . . . . . . . . 10 (a⊥ ∪ ((a ∩ b) ∪ b)) = ((a⊥ ∪ (a ∩ b)) ∪ (a⊥ ∪ b))
36	35	ax-r1 35	. . . . . . . . 9 ((a⊥ ∪ (a ∩ b)) ∪ (a⊥ ∪ b)) = (a⊥ ∪ ((a ∩ b) ∪ b))
37	36	lor 70	. . . . . . . 8 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ ((a⊥ ∪ (a ∩ b)) ∪ (a⊥ ∪ b))) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ ((a ∩ b) ∪ b)))
38	6	df-le2 131	. . . . . . . . . 10 ((a ∩ b) ∪ b) = b
39	38	lor 70	. . . . . . . . 9 (a⊥ ∪ ((a ∩ b) ∪ b)) = (a⊥ ∪ b)
40	39	lor 70	. . . . . . . 8 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ ((a ∩ b) ∪ b))) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ b))
41	11	df-le2 131	. . . . . . . 8 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ b)) = (a⊥ ∪ b)
42	37, 40, 41	3tr 65	. . . . . . 7 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ ((a⊥ ∪ (a ∩ b)) ∪ (a⊥ ∪ b))) = (a⊥ ∪ b)
43	34, 42	ax-r2 36	. . . . . 6 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ (a ∩ b))) ∪ (a⊥ ∪ b)) = (a⊥ ∪ b)
44	31, 33, 43	3tr 65	. . . . 5 (1 ∩ (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∪ b))) = (a⊥ ∪ b)
45	24, 30, 44	3tr 65	. . . 4 (((a ∪ (a⊥ ∪ (a ∩ b))) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∩ (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∪ b))) = (a⊥ ∪ b)
46	17, 20, 45	3tr 65	. . 3 ((((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a) ∩ (((a⊥ ∪ (a ∩ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∪ b))) = (a⊥ ∪ b)
47	14, 15, 46	3tr2 64	. 2 ((a⊥ ∪ (a ∩ b)) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))) = (a⊥ ∪ b)
48		df-i1 44	. . 3 (a →1 b) = (a⊥ ∪ (a ∩ b))
49		df-i3 46	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
50	48, 49	2or 72	. 2 ((a →1 b) ∪ (a →3 b)) = ((a⊥ ∪ (a ∩ b)) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))))
51		df-i0 43	. 2 (a →0 b) = (a⊥ ∪ b)
52	47, 50, 51	3tr1 63	1 ((a →1 b) ∪ (a →3 b)) = (a →0 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₀ wi0 11   →₁ wi1 12   →₃ wi3 14

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  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-i1 44  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)