Theorem ud3lem2 571

Description: Lemma for unified disjunction. (Contributed by NM, 23-Nov-1997.)

Assertion

Ref	Expression
ud3lem2	((a ∪ (a⊥ ∩ b⊥ )) →3 a) = (a ∪ b)

Proof of Theorem ud3lem2

Step	Hyp	Ref	Expression
1		oran 87	. . . . . . 7 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
2	1	ax-r1 35	. . . . . 6 (a⊥ ∩ b⊥ )⊥ = (a ∪ b)
3	2	con3 68	. . . . 5 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
4	3	lor 70	. . . 4 (a ∪ (a⊥ ∩ b⊥ )) = (a ∪ (a ∪ b)⊥ )
5		anor2 89	. . . . . 6 (a⊥ ∩ (a ∪ b)) = (a ∪ (a ∪ b)⊥ )⊥
6	5	ax-r1 35	. . . . 5 (a ∪ (a ∪ b)⊥ )⊥ = (a⊥ ∩ (a ∪ b))
7	6	con3 68	. . . 4 (a ∪ (a ∪ b)⊥ ) = (a⊥ ∩ (a ∪ b))⊥
8	4, 7	ax-r2 36	. . 3 (a ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∩ (a ∪ b))⊥
9	8	ud3lem0b 261	. 2 ((a ∪ (a⊥ ∩ b⊥ )) →3 a) = ((a⊥ ∩ (a ∪ b))⊥ →3 a)
10		df-i3 46	. . 3 ((a⊥ ∩ (a ∪ b))⊥ →3 a) = ((((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a) ∪ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a⊥ )) ∪ ((a⊥ ∩ (a ∪ b))⊥ ∩ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∪ a)))
11		ax-a3 32	. . . 4 ((((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a) ∪ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a⊥ )) ∪ ((a⊥ ∩ (a ∪ b))⊥ ∩ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∪ a))) = (((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a) ∪ (((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a⊥ ) ∪ ((a⊥ ∩ (a ∪ b))⊥ ∩ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∪ a))))
12		ax-a2 31	. . . . 5 (((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a) ∪ (((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a⊥ ) ∪ ((a⊥ ∩ (a ∪ b))⊥ ∩ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∪ a)))) = ((((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a⊥ ) ∪ ((a⊥ ∩ (a ∪ b))⊥ ∩ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∪ a))) ∪ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a))
13		ax-a1 30	. . . . . . . . . . . . 13 (a⊥ ∩ (a ∪ b)) = (a⊥ ∩ (a ∪ b))⊥ ⊥
14	13	ran 78	. . . . . . . . . . . 12 ((a⊥ ∩ (a ∪ b)) ∩ a⊥ ) = ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a⊥ )
15	14	ax-r1 35	. . . . . . . . . . 11 ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a⊥ ) = ((a⊥ ∩ (a ∪ b)) ∩ a⊥ )
16		an32 83	. . . . . . . . . . . 12 ((a⊥ ∩ (a ∪ b)) ∩ a⊥ ) = ((a⊥ ∩ a⊥ ) ∩ (a ∪ b))
17		anidm 111	. . . . . . . . . . . . 13 (a⊥ ∩ a⊥ ) = a⊥
18	17	ran 78	. . . . . . . . . . . 12 ((a⊥ ∩ a⊥ ) ∩ (a ∪ b)) = (a⊥ ∩ (a ∪ b))
19	16, 18	ax-r2 36	. . . . . . . . . . 11 ((a⊥ ∩ (a ∪ b)) ∩ a⊥ ) = (a⊥ ∩ (a ∪ b))
20	15, 19	ax-r2 36	. . . . . . . . . 10 ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a⊥ ) = (a⊥ ∩ (a ∪ b))
21	13	ax-r5 38	. . . . . . . . . . . . 13 ((a⊥ ∩ (a ∪ b)) ∪ a) = ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∪ a)
22	7, 21	2an 79	. . . . . . . . . . . 12 ((a ∪ (a ∪ b)⊥ ) ∩ ((a⊥ ∩ (a ∪ b)) ∪ a)) = ((a⊥ ∩ (a ∪ b))⊥ ∩ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∪ a))
23	22	ax-r1 35	. . . . . . . . . . 11 ((a⊥ ∩ (a ∪ b))⊥ ∩ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∪ a)) = ((a ∪ (a ∪ b)⊥ ) ∩ ((a⊥ ∩ (a ∪ b)) ∪ a))
24		ax-a2 31	. . . . . . . . . . . . . 14 ((a⊥ ∩ (a ∪ b)) ∪ a) = (a ∪ (a⊥ ∩ (a ∪ b)))
25		oml 445	. . . . . . . . . . . . . 14 (a ∪ (a⊥ ∩ (a ∪ b))) = (a ∪ b)
26	24, 25	ax-r2 36	. . . . . . . . . . . . 13 ((a⊥ ∩ (a ∪ b)) ∪ a) = (a ∪ b)
27	26	lan 77	. . . . . . . . . . . 12 ((a ∪ (a ∪ b)⊥ ) ∩ ((a⊥ ∩ (a ∪ b)) ∪ a)) = ((a ∪ (a ∪ b)⊥ ) ∩ (a ∪ b))
28		comorr 184	. . . . . . . . . . . . . 14 a C (a ∪ b)
29	28	comcom2 183	. . . . . . . . . . . . . 14 a C (a ∪ b)⊥
30	28, 29	fh2r 474	. . . . . . . . . . . . 13 ((a ∪ (a ∪ b)⊥ ) ∩ (a ∪ b)) = ((a ∩ (a ∪ b)) ∪ ((a ∪ b)⊥ ∩ (a ∪ b)))
31		anabs 121	. . . . . . . . . . . . . . 15 (a ∩ (a ∪ b)) = a
32		ancom 74	. . . . . . . . . . . . . . . 16 ((a ∪ b)⊥ ∩ (a ∪ b)) = ((a ∪ b) ∩ (a ∪ b)⊥ )
33		dff 101	. . . . . . . . . . . . . . . . 17 0 = ((a ∪ b) ∩ (a ∪ b)⊥ )
34	33	ax-r1 35	. . . . . . . . . . . . . . . 16 ((a ∪ b) ∩ (a ∪ b)⊥ ) = 0
35	32, 34	ax-r2 36	. . . . . . . . . . . . . . 15 ((a ∪ b)⊥ ∩ (a ∪ b)) = 0
36	31, 35	2or 72	. . . . . . . . . . . . . 14 ((a ∩ (a ∪ b)) ∪ ((a ∪ b)⊥ ∩ (a ∪ b))) = (a ∪ 0)
37		or0 102	. . . . . . . . . . . . . 14 (a ∪ 0) = a
38	36, 37	ax-r2 36	. . . . . . . . . . . . 13 ((a ∩ (a ∪ b)) ∪ ((a ∪ b)⊥ ∩ (a ∪ b))) = a
39	30, 38	ax-r2 36	. . . . . . . . . . . 12 ((a ∪ (a ∪ b)⊥ ) ∩ (a ∪ b)) = a
40	27, 39	ax-r2 36	. . . . . . . . . . 11 ((a ∪ (a ∪ b)⊥ ) ∩ ((a⊥ ∩ (a ∪ b)) ∪ a)) = a
41	23, 40	ax-r2 36	. . . . . . . . . 10 ((a⊥ ∩ (a ∪ b))⊥ ∩ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∪ a)) = a
42	20, 41	2or 72	. . . . . . . . 9 (((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a⊥ ) ∪ ((a⊥ ∩ (a ∪ b))⊥ ∩ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∪ a))) = ((a⊥ ∩ (a ∪ b)) ∪ a)
43	42, 24	ax-r2 36	. . . . . . . 8 (((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a⊥ ) ∪ ((a⊥ ∩ (a ∪ b))⊥ ∩ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∪ a))) = (a ∪ (a⊥ ∩ (a ∪ b)))
44	43, 25	ax-r2 36	. . . . . . 7 (((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a⊥ ) ∪ ((a⊥ ∩ (a ∪ b))⊥ ∩ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∪ a))) = (a ∪ b)
45		ancom 74	. . . . . . . 8 ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a) = (a ∩ (a⊥ ∩ (a ∪ b))⊥ ⊥ )
46	13	lan 77	. . . . . . . . . 10 (a ∩ (a⊥ ∩ (a ∪ b))) = (a ∩ (a⊥ ∩ (a ∪ b))⊥ ⊥ )
47	46	ax-r1 35	. . . . . . . . 9 (a ∩ (a⊥ ∩ (a ∪ b))⊥ ⊥ ) = (a ∩ (a⊥ ∩ (a ∪ b)))
48		anass 76	. . . . . . . . . . 11 ((a ∩ a⊥ ) ∩ (a ∪ b)) = (a ∩ (a⊥ ∩ (a ∪ b)))
49	48	ax-r1 35	. . . . . . . . . 10 (a ∩ (a⊥ ∩ (a ∪ b))) = ((a ∩ a⊥ ) ∩ (a ∪ b))
50		ancom 74	. . . . . . . . . . 11 ((a ∩ a⊥ ) ∩ (a ∪ b)) = ((a ∪ b) ∩ (a ∩ a⊥ ))
51		dff 101	. . . . . . . . . . . . . 14 0 = (a ∩ a⊥ )
52	51	lan 77	. . . . . . . . . . . . 13 ((a ∪ b) ∩ 0) = ((a ∪ b) ∩ (a ∩ a⊥ ))
53	52	ax-r1 35	. . . . . . . . . . . 12 ((a ∪ b) ∩ (a ∩ a⊥ )) = ((a ∪ b) ∩ 0)
54		an0 108	. . . . . . . . . . . 12 ((a ∪ b) ∩ 0) = 0
55	53, 54	ax-r2 36	. . . . . . . . . . 11 ((a ∪ b) ∩ (a ∩ a⊥ )) = 0
56	50, 55	ax-r2 36	. . . . . . . . . 10 ((a ∩ a⊥ ) ∩ (a ∪ b)) = 0
57	49, 56	ax-r2 36	. . . . . . . . 9 (a ∩ (a⊥ ∩ (a ∪ b))) = 0
58	47, 57	ax-r2 36	. . . . . . . 8 (a ∩ (a⊥ ∩ (a ∪ b))⊥ ⊥ ) = 0
59	45, 58	ax-r2 36	. . . . . . 7 ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a) = 0
60	44, 59	2or 72	. . . . . 6 ((((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a⊥ ) ∪ ((a⊥ ∩ (a ∪ b))⊥ ∩ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∪ a))) ∪ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a)) = ((a ∪ b) ∪ 0)
61		or0 102	. . . . . 6 ((a ∪ b) ∪ 0) = (a ∪ b)
62	60, 61	ax-r2 36	. . . . 5 ((((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a⊥ ) ∪ ((a⊥ ∩ (a ∪ b))⊥ ∩ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∪ a))) ∪ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a)) = (a ∪ b)
63	12, 62	ax-r2 36	. . . 4 (((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a) ∪ (((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a⊥ ) ∪ ((a⊥ ∩ (a ∪ b))⊥ ∩ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∪ a)))) = (a ∪ b)
64	11, 63	ax-r2 36	. . 3 ((((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a) ∪ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∩ a⊥ )) ∪ ((a⊥ ∩ (a ∪ b))⊥ ∩ ((a⊥ ∩ (a ∪ b))⊥ ⊥ ∪ a))) = (a ∪ b)
65	10, 64	ax-r2 36	. 2 ((a⊥ ∩ (a ∪ b))⊥ →3 a) = (a ∪ b)
66	9, 65	ax-r2 36	1 ((a ∪ (a⊥ ∩ b⊥ )) →3 a) = (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:  ud3  597