Theorem mlaconjo 886

Description: OML proof of Mladen's conjecture. (Contributed by NM, 10-Mar-2002.)

Assertion

Ref	Expression
mlaconjo	((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) ≤ (a ≡ c)

Proof of Theorem mlaconjo

Step	Hyp	Ref	Expression
1		dfb 94	. . . 4 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
2	1	bile 142	. . 3 (a ≡ b) ≤ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
3		mlaconjolem 885	. . 3 ((a ≡ c) ∪ (b ≡ c)) ≤ ((c ∩ (a ∪ b)) ∪ (c⊥ ∩ (a⊥ ∪ b⊥ )))
4	2, 3	le2an 169	. 2 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) ≤ (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ ((c ∩ (a ∪ b)) ∪ (c⊥ ∩ (a⊥ ∪ b⊥ ))))
5		lea 160	. . . . 5 (a ∩ b) ≤ a
6		lea 160	. . . . 5 (c ∩ (a ∪ b)) ≤ c
7	5, 6	le2an 169	. . . 4 ((a ∩ b) ∩ (c ∩ (a ∪ b))) ≤ (a ∩ c)
8		lea 160	. . . . 5 (a⊥ ∩ b⊥ ) ≤ a⊥
9		lea 160	. . . . 5 (c⊥ ∩ (a⊥ ∪ b⊥ )) ≤ c⊥
10	8, 9	le2an 169	. . . 4 ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ ))) ≤ (a⊥ ∩ c⊥ )
11	7, 10	le2or 168	. . 3 (((a ∩ b) ∩ (c ∩ (a ∪ b))) ∪ ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ )))) ≤ ((a ∩ c) ∪ (a⊥ ∩ c⊥ ))
12		leao1 162	. . . . . . . 8 (a ∩ b) ≤ (a ∪ b)
13		oran 87	. . . . . . . 8 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
14	12, 13	lbtr 139	. . . . . . 7 (a ∩ b) ≤ (a⊥ ∩ b⊥ )⊥
15	14	lecom 180	. . . . . 6 (a ∩ b) C (a⊥ ∩ b⊥ )⊥
16	15	comcom7 460	. . . . 5 (a ∩ b) C (a⊥ ∩ b⊥ )
17		leor 159	. . . . . . . 8 (a ∩ b) ≤ (c ∪ (a ∩ b))
18		df-a 40	. . . . . . . . . 10 (a ∩ b) = (a⊥ ∪ b⊥ )⊥
19	18	lor 70	. . . . . . . . 9 (c ∪ (a ∩ b)) = (c ∪ (a⊥ ∪ b⊥ )⊥ )
20		oran1 91	. . . . . . . . 9 (c ∪ (a⊥ ∪ b⊥ )⊥ ) = (c⊥ ∩ (a⊥ ∪ b⊥ ))⊥
21	19, 20	ax-r2 36	. . . . . . . 8 (c ∪ (a ∩ b)) = (c⊥ ∩ (a⊥ ∪ b⊥ ))⊥
22	17, 21	lbtr 139	. . . . . . 7 (a ∩ b) ≤ (c⊥ ∩ (a⊥ ∪ b⊥ ))⊥
23	22	lecom 180	. . . . . 6 (a ∩ b) C (c⊥ ∩ (a⊥ ∪ b⊥ ))⊥
24	23	comcom7 460	. . . . 5 (a ∩ b) C (c⊥ ∩ (a⊥ ∪ b⊥ ))
25		lear 161	. . . . . . . 8 (c ∩ (a ∪ b)) ≤ (a ∪ b)
26	25, 13	lbtr 139	. . . . . . 7 (c ∩ (a ∪ b)) ≤ (a⊥ ∩ b⊥ )⊥
27	26	lecom 180	. . . . . 6 (c ∩ (a ∪ b)) C (a⊥ ∩ b⊥ )⊥
28	27	comcom7 460	. . . . 5 (c ∩ (a ∪ b)) C (a⊥ ∩ b⊥ )
29		leao1 162	. . . . . . . 8 (c ∩ (a ∪ b)) ≤ (c ∪ (a⊥ ∪ b⊥ )⊥ )
30	29, 20	lbtr 139	. . . . . . 7 (c ∩ (a ∪ b)) ≤ (c⊥ ∩ (a⊥ ∪ b⊥ ))⊥
31	30	lecom 180	. . . . . 6 (c ∩ (a ∪ b)) C (c⊥ ∩ (a⊥ ∪ b⊥ ))⊥
32	31	comcom7 460	. . . . 5 (c ∩ (a ∪ b)) C (c⊥ ∩ (a⊥ ∪ b⊥ ))
33	16, 24, 28, 32	mh 879	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ ((c ∩ (a ∪ b)) ∪ (c⊥ ∩ (a⊥ ∪ b⊥ )))) = ((((a ∩ b) ∩ (c ∩ (a ∪ b))) ∪ ((a ∩ b) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ )))) ∪ (((a⊥ ∩ b⊥ ) ∩ (c ∩ (a ∪ b))) ∪ ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ )))))
34		an12 81	. . . . . . . 8 ((a ∩ b) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ ))) = (c⊥ ∩ ((a ∩ b) ∩ (a⊥ ∪ b⊥ )))
35		oran3 93	. . . . . . . . . . 11 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
36	35	lan 77	. . . . . . . . . 10 ((a ∩ b) ∩ (a⊥ ∪ b⊥ )) = ((a ∩ b) ∩ (a ∩ b)⊥ )
37		dff 101	. . . . . . . . . . 11 0 = ((a ∩ b) ∩ (a ∩ b)⊥ )
38	37	ax-r1 35	. . . . . . . . . 10 ((a ∩ b) ∩ (a ∩ b)⊥ ) = 0
39	36, 38	ax-r2 36	. . . . . . . . 9 ((a ∩ b) ∩ (a⊥ ∪ b⊥ )) = 0
40	39	lan 77	. . . . . . . 8 (c⊥ ∩ ((a ∩ b) ∩ (a⊥ ∪ b⊥ ))) = (c⊥ ∩ 0)
41		an0 108	. . . . . . . 8 (c⊥ ∩ 0) = 0
42	34, 40, 41	3tr 65	. . . . . . 7 ((a ∩ b) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ ))) = 0
43	42	lor 70	. . . . . 6 (((a ∩ b) ∩ (c ∩ (a ∪ b))) ∪ ((a ∩ b) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ )))) = (((a ∩ b) ∩ (c ∩ (a ∪ b))) ∪ 0)
44		or0 102	. . . . . 6 (((a ∩ b) ∩ (c ∩ (a ∪ b))) ∪ 0) = ((a ∩ b) ∩ (c ∩ (a ∪ b)))
45	43, 44	ax-r2 36	. . . . 5 (((a ∩ b) ∩ (c ∩ (a ∪ b))) ∪ ((a ∩ b) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ )))) = ((a ∩ b) ∩ (c ∩ (a ∪ b)))
46		an12 81	. . . . . . . 8 ((a⊥ ∩ b⊥ ) ∩ (c ∩ (a ∪ b))) = (c ∩ ((a⊥ ∩ b⊥ ) ∩ (a ∪ b)))
47	13	lan 77	. . . . . . . . . 10 ((a⊥ ∩ b⊥ ) ∩ (a ∪ b)) = ((a⊥ ∩ b⊥ ) ∩ (a⊥ ∩ b⊥ )⊥ )
48		dff 101	. . . . . . . . . . 11 0 = ((a⊥ ∩ b⊥ ) ∩ (a⊥ ∩ b⊥ )⊥ )
49	48	ax-r1 35	. . . . . . . . . 10 ((a⊥ ∩ b⊥ ) ∩ (a⊥ ∩ b⊥ )⊥ ) = 0
50	47, 49	ax-r2 36	. . . . . . . . 9 ((a⊥ ∩ b⊥ ) ∩ (a ∪ b)) = 0
51	50	lan 77	. . . . . . . 8 (c ∩ ((a⊥ ∩ b⊥ ) ∩ (a ∪ b))) = (c ∩ 0)
52		an0 108	. . . . . . . 8 (c ∩ 0) = 0
53	46, 51, 52	3tr 65	. . . . . . 7 ((a⊥ ∩ b⊥ ) ∩ (c ∩ (a ∪ b))) = 0
54	53	ax-r5 38	. . . . . 6 (((a⊥ ∩ b⊥ ) ∩ (c ∩ (a ∪ b))) ∪ ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ )))) = (0 ∪ ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ ))))
55		or0r 103	. . . . . 6 (0 ∪ ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ )))) = ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ )))
56	54, 55	ax-r2 36	. . . . 5 (((a⊥ ∩ b⊥ ) ∩ (c ∩ (a ∪ b))) ∪ ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ )))) = ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ )))
57	45, 56	2or 72	. . . 4 ((((a ∩ b) ∩ (c ∩ (a ∪ b))) ∪ ((a ∩ b) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ )))) ∪ (((a⊥ ∩ b⊥ ) ∩ (c ∩ (a ∪ b))) ∪ ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ ))))) = (((a ∩ b) ∩ (c ∩ (a ∪ b))) ∪ ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ ))))
58	33, 57	ax-r2 36	. . 3 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ ((c ∩ (a ∪ b)) ∪ (c⊥ ∩ (a⊥ ∪ b⊥ )))) = (((a ∩ b) ∩ (c ∩ (a ∪ b))) ∪ ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ (a⊥ ∪ b⊥ ))))
59		dfb 94	. . 3 (a ≡ c) = ((a ∩ c) ∪ (a⊥ ∩ c⊥ ))
60	11, 58, 59	le3tr1 140	. 2 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ ((c ∩ (a ∪ b)) ∪ (c⊥ ∩ (a⊥ ∪ b⊥ )))) ≤ (a ≡ c)
61	4, 60	letr 137	1 ((a ≡ b) ∩ ((a ≡ c) ∪ (b ≡ c))) ≤ (a ≡ c)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  0wf 9

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-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  distid  887