Theorem lem4 511

Description: Lemma 4 of Kalmbach p. 240. (Contributed by NM, 5-Nov-1997.)

Assertion

Ref	Expression
lem4	(a →3 (a →3 b)) = (a⊥ ∪ b)

Proof of Theorem lem4

Step	Hyp	Ref	Expression
1		df-i3 46	. 2 (a →3 (a →3 b)) = (((a⊥ ∩ (a →3 b)) ∪ (a⊥ ∩ (a →3 b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a →3 b))))
2		df-i3 46	. . . . . . . 8 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
3	2	lan 77	. . . . . . 7 (a⊥ ∩ (a →3 b)) = (a⊥ ∩ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))))
4		lea 160	. . . . . . . . . . . . 13 (a⊥ ∩ b) ≤ a⊥
5		lea 160	. . . . . . . . . . . . 13 (a⊥ ∩ b⊥ ) ≤ a⊥
6	4, 5	le2or 168	. . . . . . . . . . . 12 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ (a⊥ ∪ a⊥ )
7		oridm 110	. . . . . . . . . . . 12 (a⊥ ∪ a⊥ ) = a⊥
8	6, 7	lbtr 139	. . . . . . . . . . 11 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ a⊥
9	8	lecom 180	. . . . . . . . . 10 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) C a⊥
10	9	comcom 453	. . . . . . . . 9 a⊥ C ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
11		lea 160	. . . . . . . . . . . 12 (a ∩ (a⊥ ∪ b)) ≤ a
12	11	lecom 180	. . . . . . . . . . 11 (a ∩ (a⊥ ∪ b)) C a
13	12	comcom 453	. . . . . . . . . 10 a C (a ∩ (a⊥ ∪ b))
14	13	comcom3 454	. . . . . . . . 9 a⊥ C (a ∩ (a⊥ ∪ b))
15	10, 14	fh1 469	. . . . . . . 8 (a⊥ ∩ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))) = ((a⊥ ∩ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∩ (a ∩ (a⊥ ∪ b))))
16		ancom 74	. . . . . . . . . . 11 ((a⊥ ∩ a) ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∩ (a⊥ ∩ a))
17		anass 76	. . . . . . . . . . 11 ((a⊥ ∩ a) ∩ (a⊥ ∪ b)) = (a⊥ ∩ (a ∩ (a⊥ ∪ b)))
18		dff 101	. . . . . . . . . . . . . . 15 0 = (a ∩ a⊥ )
19		ancom 74	. . . . . . . . . . . . . . 15 (a ∩ a⊥ ) = (a⊥ ∩ a)
20	18, 19	ax-r2 36	. . . . . . . . . . . . . 14 0 = (a⊥ ∩ a)
21	20	lan 77	. . . . . . . . . . . . 13 ((a⊥ ∪ b) ∩ 0) = ((a⊥ ∪ b) ∩ (a⊥ ∩ a))
22	21	ax-r1 35	. . . . . . . . . . . 12 ((a⊥ ∪ b) ∩ (a⊥ ∩ a)) = ((a⊥ ∪ b) ∩ 0)
23		an0 108	. . . . . . . . . . . 12 ((a⊥ ∪ b) ∩ 0) = 0
24	22, 23	ax-r2 36	. . . . . . . . . . 11 ((a⊥ ∪ b) ∩ (a⊥ ∩ a)) = 0
25	16, 17, 24	3tr2 64	. . . . . . . . . 10 (a⊥ ∩ (a ∩ (a⊥ ∪ b))) = 0
26	25	lor 70	. . . . . . . . 9 ((a⊥ ∩ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∩ (a ∩ (a⊥ ∪ b)))) = ((a⊥ ∩ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ 0)
27		or0 102	. . . . . . . . . 10 ((a⊥ ∩ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ 0) = (a⊥ ∩ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
28		ancom 74	. . . . . . . . . . 11 (a⊥ ∩ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ a⊥ )
29	8	df2le2 136	. . . . . . . . . . 11 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ a⊥ ) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
30	28, 29	ax-r2 36	. . . . . . . . . 10 (a⊥ ∩ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
31	27, 30	ax-r2 36	. . . . . . . . 9 ((a⊥ ∩ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ 0) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
32	26, 31	ax-r2 36	. . . . . . . 8 ((a⊥ ∩ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a⊥ ∩ (a ∩ (a⊥ ∪ b)))) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
33	15, 32	ax-r2 36	. . . . . . 7 (a⊥ ∩ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
34	3, 33	ax-r2 36	. . . . . 6 (a⊥ ∩ (a →3 b)) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
35	2	lor 70	. . . . . . . . 9 (a ∪ (a →3 b)) = (a ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))))
36		orordi 112	. . . . . . . . . 10 (a ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))) = ((a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a ∪ (a ∩ (a⊥ ∪ b))))
37		orabs 120	. . . . . . . . . . . 12 (a ∪ (a ∩ (a⊥ ∪ b))) = a
38	37	lor 70	. . . . . . . . . . 11 ((a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a ∪ (a ∩ (a⊥ ∪ b)))) = ((a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a)
39		or32 82	. . . . . . . . . . . 12 ((a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a) = ((a ∪ a) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
40		oridm 110	. . . . . . . . . . . . 13 (a ∪ a) = a
41	40	ax-r5 38	. . . . . . . . . . . 12 ((a ∪ a) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
42	39, 41	ax-r2 36	. . . . . . . . . . 11 ((a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
43	38, 42	ax-r2 36	. . . . . . . . . 10 ((a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a ∪ (a ∩ (a⊥ ∪ b)))) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
44	36, 43	ax-r2 36	. . . . . . . . 9 (a ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
45	35, 44	ax-r2 36	. . . . . . . 8 (a ∪ (a →3 b)) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
46	45	ax-r4 37	. . . . . . 7 (a ∪ (a →3 b))⊥ = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))⊥
47		oran 87	. . . . . . . 8 (a ∪ (a →3 b)) = (a⊥ ∩ (a →3 b)⊥ )⊥
48	47	con2 67	. . . . . . 7 (a ∪ (a →3 b))⊥ = (a⊥ ∩ (a →3 b)⊥ )
49		oran 87	. . . . . . . . 9 (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (a⊥ ∩ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ )⊥
50	49	con2 67	. . . . . . . 8 (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))⊥ = (a⊥ ∩ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ )
51		ancom 74	. . . . . . . 8 (a⊥ ∩ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a⊥ )
52	50, 51	ax-r2 36	. . . . . . 7 (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))⊥ = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a⊥ )
53	46, 48, 52	3tr2 64	. . . . . 6 (a⊥ ∩ (a →3 b)⊥ ) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a⊥ )
54	34, 53	2or 72	. . . . 5 ((a⊥ ∩ (a →3 b)) ∪ (a⊥ ∩ (a →3 b)⊥ )) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a⊥ ))
55	8	oml2 451	. . . . 5 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a⊥ )) = a⊥
56	54, 55	ax-r2 36	. . . 4 ((a⊥ ∩ (a →3 b)) ∪ (a⊥ ∩ (a →3 b)⊥ )) = a⊥
57	2	lor 70	. . . . . . 7 (a⊥ ∪ (a →3 b)) = (a⊥ ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))))
58		ax-a3 32	. . . . . . . . 9 ((a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))))
59	58	ax-r1 35	. . . . . . . 8 (a⊥ ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))) = ((a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a ∩ (a⊥ ∪ b)))
60		orordi 112	. . . . . . . . . 10 (a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = ((a⊥ ∪ (a⊥ ∩ b)) ∪ (a⊥ ∪ (a⊥ ∩ b⊥ )))
61		orabs 120	. . . . . . . . . . . 12 (a⊥ ∪ (a⊥ ∩ b)) = a⊥
62		orabs 120	. . . . . . . . . . . 12 (a⊥ ∪ (a⊥ ∩ b⊥ )) = a⊥
63	61, 62	2or 72	. . . . . . . . . . 11 ((a⊥ ∪ (a⊥ ∩ b)) ∪ (a⊥ ∪ (a⊥ ∩ b⊥ ))) = (a⊥ ∪ a⊥ )
64	63, 7	ax-r2 36	. . . . . . . . . 10 ((a⊥ ∪ (a⊥ ∩ b)) ∪ (a⊥ ∪ (a⊥ ∩ b⊥ ))) = a⊥
65	60, 64	ax-r2 36	. . . . . . . . 9 (a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = a⊥
66	65	ax-r5 38	. . . . . . . 8 ((a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ (a ∩ (a⊥ ∪ b)))
67	59, 66	ax-r2 36	. . . . . . 7 (a⊥ ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))) = (a⊥ ∪ (a ∩ (a⊥ ∪ b)))
68	57, 67	ax-r2 36	. . . . . 6 (a⊥ ∪ (a →3 b)) = (a⊥ ∪ (a ∩ (a⊥ ∪ b)))
69		omln 446	. . . . . 6 (a⊥ ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ b)
70	68, 69	ax-r2 36	. . . . 5 (a⊥ ∪ (a →3 b)) = (a⊥ ∪ b)
71	70	lan 77	. . . 4 (a ∩ (a⊥ ∪ (a →3 b))) = (a ∩ (a⊥ ∪ b))
72	56, 71	2or 72	. . 3 (((a⊥ ∩ (a →3 b)) ∪ (a⊥ ∩ (a →3 b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a →3 b)))) = (a⊥ ∪ (a ∩ (a⊥ ∪ b)))
73	72, 69	ax-r2 36	. 2 (((a⊥ ∩ (a →3 b)) ∪ (a⊥ ∩ (a →3 b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a →3 b)))) = (a⊥ ∪ b)
74	1, 73	ax-r2 36	1 (a →3 (a →3 b)) = (a⊥ ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  i0i3  512  i3i0  513  ska14  514  i3abs1  522  i3abs3  524  i3th1  543  i3th2  544  i3th3  545  i3th5  547  i3th7  549  i3th8  550  u3lem4  758  u3lem12  788  u3lemax4  796  u3lemax5  797