Theorem ud4lem2 582

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

Assertion

Ref	Expression
ud4lem2	((a ∪ (a⊥ ∩ b⊥ )) →4 a) = (a ∪ b)

Proof of Theorem ud4lem2

Step	Hyp	Ref	Expression
1		df-i4 47	. 2 ((a ∪ (a⊥ ∩ b⊥ )) →4 a) = ((((a ∪ (a⊥ ∩ b⊥ )) ∩ a) ∪ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a)) ∪ (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) ∩ a⊥ ))
2		ancom 74	. . . . . . 7 ((a ∪ (a⊥ ∩ b⊥ )) ∩ a) = (a ∩ (a ∪ (a⊥ ∩ b⊥ )))
3		anabs 121	. . . . . . 7 (a ∩ (a ∪ (a⊥ ∩ b⊥ ))) = a
4	2, 3	ax-r2 36	. . . . . 6 ((a ∪ (a⊥ ∩ b⊥ )) ∩ a) = a
5		oran 87	. . . . . . . . 9 (a ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∩ (a⊥ ∩ b⊥ )⊥ )⊥
6	5	con2 67	. . . . . . . 8 (a ∪ (a⊥ ∩ b⊥ ))⊥ = (a⊥ ∩ (a⊥ ∩ b⊥ )⊥ )
7	6	ran 78	. . . . . . 7 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a) = ((a⊥ ∩ (a⊥ ∩ b⊥ )⊥ ) ∩ a)
8		ancom 74	. . . . . . . 8 ((a⊥ ∩ (a⊥ ∩ b⊥ )⊥ ) ∩ a) = (a ∩ (a⊥ ∩ (a⊥ ∩ b⊥ )⊥ ))
9		anass 76	. . . . . . . . . 10 ((a ∩ a⊥ ) ∩ (a⊥ ∩ b⊥ )⊥ ) = (a ∩ (a⊥ ∩ (a⊥ ∩ b⊥ )⊥ ))
10	9	ax-r1 35	. . . . . . . . 9 (a ∩ (a⊥ ∩ (a⊥ ∩ b⊥ )⊥ )) = ((a ∩ a⊥ ) ∩ (a⊥ ∩ b⊥ )⊥ )
11		ancom 74	. . . . . . . . . 10 ((a ∩ a⊥ ) ∩ (a⊥ ∩ b⊥ )⊥ ) = ((a⊥ ∩ b⊥ )⊥ ∩ (a ∩ a⊥ ))
12		dff 101	. . . . . . . . . . . . 13 0 = (a ∩ a⊥ )
13	12	lan 77	. . . . . . . . . . . 12 ((a⊥ ∩ b⊥ )⊥ ∩ 0) = ((a⊥ ∩ b⊥ )⊥ ∩ (a ∩ a⊥ ))
14	13	ax-r1 35	. . . . . . . . . . 11 ((a⊥ ∩ b⊥ )⊥ ∩ (a ∩ a⊥ )) = ((a⊥ ∩ b⊥ )⊥ ∩ 0)
15		an0 108	. . . . . . . . . . 11 ((a⊥ ∩ b⊥ )⊥ ∩ 0) = 0
16	14, 15	ax-r2 36	. . . . . . . . . 10 ((a⊥ ∩ b⊥ )⊥ ∩ (a ∩ a⊥ )) = 0
17	11, 16	ax-r2 36	. . . . . . . . 9 ((a ∩ a⊥ ) ∩ (a⊥ ∩ b⊥ )⊥ ) = 0
18	10, 17	ax-r2 36	. . . . . . . 8 (a ∩ (a⊥ ∩ (a⊥ ∩ b⊥ )⊥ )) = 0
19	8, 18	ax-r2 36	. . . . . . 7 ((a⊥ ∩ (a⊥ ∩ b⊥ )⊥ ) ∩ a) = 0
20	7, 19	ax-r2 36	. . . . . 6 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a) = 0
21	4, 20	2or 72	. . . . 5 (((a ∪ (a⊥ ∩ b⊥ )) ∩ a) ∪ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a)) = (a ∪ 0)
22		or0 102	. . . . 5 (a ∪ 0) = a
23	21, 22	ax-r2 36	. . . 4 (((a ∪ (a⊥ ∩ b⊥ )) ∩ a) ∪ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a)) = a
24		ancom 74	. . . . 5 (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) ∩ a⊥ ) = (a⊥ ∩ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a))
25		oran 87	. . . . . . . . . . . . 13 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
26	25	ax-r1 35	. . . . . . . . . . . 12 (a⊥ ∩ b⊥ )⊥ = (a ∪ b)
27	26	con3 68	. . . . . . . . . . 11 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
28	27	lor 70	. . . . . . . . . 10 (a ∪ (a⊥ ∩ b⊥ )) = (a ∪ (a ∪ b)⊥ )
29		anor2 89	. . . . . . . . . . . 12 (a⊥ ∩ (a ∪ b)) = (a ∪ (a ∪ b)⊥ )⊥
30	29	ax-r1 35	. . . . . . . . . . 11 (a ∪ (a ∪ b)⊥ )⊥ = (a⊥ ∩ (a ∪ b))
31	30	con3 68	. . . . . . . . . 10 (a ∪ (a ∪ b)⊥ ) = (a⊥ ∩ (a ∪ b))⊥
32	28, 31	ax-r2 36	. . . . . . . . 9 (a ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∩ (a ∪ b))⊥
33	32	con2 67	. . . . . . . 8 (a ∪ (a⊥ ∩ b⊥ ))⊥ = (a⊥ ∩ (a ∪ b))
34	33	ax-r5 38	. . . . . . 7 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) = ((a⊥ ∩ (a ∪ b)) ∪ a)
35		comid 187	. . . . . . . . . 10 a C a
36	35	comcom2 183	. . . . . . . . 9 a C a⊥
37		comorr 184	. . . . . . . . 9 a C (a ∪ b)
38	36, 37	fh3r 475	. . . . . . . 8 ((a⊥ ∩ (a ∪ b)) ∪ a) = ((a⊥ ∪ a) ∩ ((a ∪ b) ∪ a))
39		ancom 74	. . . . . . . . . 10 ((a⊥ ∪ a) ∩ ((a ∪ b) ∪ a)) = (((a ∪ b) ∪ a) ∩ (a⊥ ∪ a))
40		or32 82	. . . . . . . . . . . 12 ((a ∪ b) ∪ a) = ((a ∪ a) ∪ b)
41		oridm 110	. . . . . . . . . . . . 13 (a ∪ a) = a
42	41	ax-r5 38	. . . . . . . . . . . 12 ((a ∪ a) ∪ b) = (a ∪ b)
43	40, 42	ax-r2 36	. . . . . . . . . . 11 ((a ∪ b) ∪ a) = (a ∪ b)
44		df-t 41	. . . . . . . . . . . . 13 1 = (a ∪ a⊥ )
45		ax-a2 31	. . . . . . . . . . . . 13 (a ∪ a⊥ ) = (a⊥ ∪ a)
46	44, 45	ax-r2 36	. . . . . . . . . . . 12 1 = (a⊥ ∪ a)
47	46	ax-r1 35	. . . . . . . . . . 11 (a⊥ ∪ a) = 1
48	43, 47	2an 79	. . . . . . . . . 10 (((a ∪ b) ∪ a) ∩ (a⊥ ∪ a)) = ((a ∪ b) ∩ 1)
49	39, 48	ax-r2 36	. . . . . . . . 9 ((a⊥ ∪ a) ∩ ((a ∪ b) ∪ a)) = ((a ∪ b) ∩ 1)
50		an1 106	. . . . . . . . 9 ((a ∪ b) ∩ 1) = (a ∪ b)
51	49, 50	ax-r2 36	. . . . . . . 8 ((a⊥ ∪ a) ∩ ((a ∪ b) ∪ a)) = (a ∪ b)
52	38, 51	ax-r2 36	. . . . . . 7 ((a⊥ ∩ (a ∪ b)) ∪ a) = (a ∪ b)
53	34, 52	ax-r2 36	. . . . . 6 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) = (a ∪ b)
54	53	lan 77	. . . . 5 (a⊥ ∩ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a)) = (a⊥ ∩ (a ∪ b))
55	24, 54	ax-r2 36	. . . 4 (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) ∩ a⊥ ) = (a⊥ ∩ (a ∪ b))
56	23, 55	2or 72	. . 3 ((((a ∪ (a⊥ ∩ b⊥ )) ∩ a) ∪ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a)) ∪ (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) ∩ a⊥ )) = (a ∪ (a⊥ ∩ (a ∪ b)))
57		oml 445	. . 3 (a ∪ (a⊥ ∩ (a ∪ b))) = (a ∪ b)
58	56, 57	ax-r2 36	. 2 ((((a ∪ (a⊥ ∩ b⊥ )) ∩ a) ∪ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a)) ∪ (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) ∩ a⊥ )) = (a ∪ b)
59	1, 58	ax-r2 36	1 ((a ∪ (a⊥ ∩ b⊥ )) →4 a) = (a ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →₄ wi4 15

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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  ud4  598