Theorem bi3 839

Description: Chained biconditional. (Contributed by NM, 2-Mar-2000.)

Assertion

Ref	Expression
bi3	((a ≡ b) ∩ (b ≡ c)) = (((a ∩ b) ∩ c) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ ))

Proof of Theorem bi3

Step	Hyp	Ref	Expression
1		dfb 94	. . 3 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
2		u12lembi 726	. . . 4 ((b →1 c) ∩ (c →2 b)) = (b ≡ c)
3	2	ax-r1 35	. . 3 (b ≡ c) = ((b →1 c) ∩ (c →2 b))
4	1, 3	2an 79	. 2 ((a ≡ b) ∩ (b ≡ c)) = (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ ((b →1 c) ∩ (c →2 b)))
5		df-i1 44	. . . . . 6 (b →1 c) = (b⊥ ∪ (b ∩ c))
6	5	lan 77	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ (b →1 c)) = (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ (b⊥ ∪ (b ∩ c)))
7		lear 161	. . . . . . . 8 (a⊥ ∩ b⊥ ) ≤ b⊥
8		leo 158	. . . . . . . 8 b⊥ ≤ (b⊥ ∪ (b ∩ c))
9	7, 8	letr 137	. . . . . . 7 (a⊥ ∩ b⊥ ) ≤ (b⊥ ∪ (b ∩ c))
10	9	lecom 180	. . . . . 6 (a⊥ ∩ b⊥ ) C (b⊥ ∪ (b ∩ c))
11		coman1 185	. . . . . . . 8 (a⊥ ∩ b⊥ ) C a⊥
12	11	comcom7 460	. . . . . . 7 (a⊥ ∩ b⊥ ) C a
13		coman2 186	. . . . . . . 8 (a⊥ ∩ b⊥ ) C b⊥
14	13	comcom7 460	. . . . . . 7 (a⊥ ∩ b⊥ ) C b
15	12, 14	com2an 484	. . . . . 6 (a⊥ ∩ b⊥ ) C (a ∩ b)
16	10, 15	fh2rc 480	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ (b⊥ ∪ (b ∩ c))) = (((a ∩ b) ∩ (b⊥ ∪ (b ∩ c))) ∪ ((a⊥ ∩ b⊥ ) ∩ (b⊥ ∪ (b ∩ c))))
17		comanr2 465	. . . . . . . . 9 b C (a ∩ b)
18	17	comcom3 454	. . . . . . . 8 b⊥ C (a ∩ b)
19		comanr1 464	. . . . . . . . 9 b C (b ∩ c)
20	19	comcom3 454	. . . . . . . 8 b⊥ C (b ∩ c)
21	18, 20	fh2 470	. . . . . . 7 ((a ∩ b) ∩ (b⊥ ∪ (b ∩ c))) = (((a ∩ b) ∩ b⊥ ) ∪ ((a ∩ b) ∩ (b ∩ c)))
22		anass 76	. . . . . . . . 9 ((a ∩ b) ∩ b⊥ ) = (a ∩ (b ∩ b⊥ ))
23		dff 101	. . . . . . . . . . 11 0 = (b ∩ b⊥ )
24	23	lan 77	. . . . . . . . . 10 (a ∩ 0) = (a ∩ (b ∩ b⊥ ))
25	24	ax-r1 35	. . . . . . . . 9 (a ∩ (b ∩ b⊥ )) = (a ∩ 0)
26		an0 108	. . . . . . . . 9 (a ∩ 0) = 0
27	22, 25, 26	3tr 65	. . . . . . . 8 ((a ∩ b) ∩ b⊥ ) = 0
28		anass 76	. . . . . . . . . 10 (((a ∩ b) ∩ b) ∩ c) = ((a ∩ b) ∩ (b ∩ c))
29	28	ax-r1 35	. . . . . . . . 9 ((a ∩ b) ∩ (b ∩ c)) = (((a ∩ b) ∩ b) ∩ c)
30		anass 76	. . . . . . . . . . 11 ((a ∩ b) ∩ b) = (a ∩ (b ∩ b))
31		anidm 111	. . . . . . . . . . . 12 (b ∩ b) = b
32	31	lan 77	. . . . . . . . . . 11 (a ∩ (b ∩ b)) = (a ∩ b)
33	30, 32	ax-r2 36	. . . . . . . . . 10 ((a ∩ b) ∩ b) = (a ∩ b)
34	33	ran 78	. . . . . . . . 9 (((a ∩ b) ∩ b) ∩ c) = ((a ∩ b) ∩ c)
35	29, 34	ax-r2 36	. . . . . . . 8 ((a ∩ b) ∩ (b ∩ c)) = ((a ∩ b) ∩ c)
36	27, 35	2or 72	. . . . . . 7 (((a ∩ b) ∩ b⊥ ) ∪ ((a ∩ b) ∩ (b ∩ c))) = (0 ∪ ((a ∩ b) ∩ c))
37		or0r 103	. . . . . . 7 (0 ∪ ((a ∩ b) ∩ c)) = ((a ∩ b) ∩ c)
38	21, 36, 37	3tr 65	. . . . . 6 ((a ∩ b) ∩ (b⊥ ∪ (b ∩ c))) = ((a ∩ b) ∩ c)
39	13	comcom 453	. . . . . . . 8 b⊥ C (a⊥ ∩ b⊥ )
40	39, 20	fh2 470	. . . . . . 7 ((a⊥ ∩ b⊥ ) ∩ (b⊥ ∪ (b ∩ c))) = (((a⊥ ∩ b⊥ ) ∩ b⊥ ) ∪ ((a⊥ ∩ b⊥ ) ∩ (b ∩ c)))
41		anass 76	. . . . . . . . 9 ((a⊥ ∩ b⊥ ) ∩ b⊥ ) = (a⊥ ∩ (b⊥ ∩ b⊥ ))
42		anidm 111	. . . . . . . . . 10 (b⊥ ∩ b⊥ ) = b⊥
43	42	lan 77	. . . . . . . . 9 (a⊥ ∩ (b⊥ ∩ b⊥ )) = (a⊥ ∩ b⊥ )
44	41, 43	ax-r2 36	. . . . . . . 8 ((a⊥ ∩ b⊥ ) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
45		an4 86	. . . . . . . . 9 ((a⊥ ∩ b⊥ ) ∩ (b ∩ c)) = ((a⊥ ∩ b) ∩ (b⊥ ∩ c))
46		anass 76	. . . . . . . . 9 ((a⊥ ∩ b) ∩ (b⊥ ∩ c)) = (a⊥ ∩ (b ∩ (b⊥ ∩ c)))
47	23	ran 78	. . . . . . . . . . . . 13 (0 ∩ c) = ((b ∩ b⊥ ) ∩ c)
48	47	ax-r1 35	. . . . . . . . . . . 12 ((b ∩ b⊥ ) ∩ c) = (0 ∩ c)
49		anass 76	. . . . . . . . . . . 12 ((b ∩ b⊥ ) ∩ c) = (b ∩ (b⊥ ∩ c))
50		an0r 109	. . . . . . . . . . . 12 (0 ∩ c) = 0
51	48, 49, 50	3tr2 64	. . . . . . . . . . 11 (b ∩ (b⊥ ∩ c)) = 0
52	51	lan 77	. . . . . . . . . 10 (a⊥ ∩ (b ∩ (b⊥ ∩ c))) = (a⊥ ∩ 0)
53		an0 108	. . . . . . . . . 10 (a⊥ ∩ 0) = 0
54	52, 53	ax-r2 36	. . . . . . . . 9 (a⊥ ∩ (b ∩ (b⊥ ∩ c))) = 0
55	45, 46, 54	3tr 65	. . . . . . . 8 ((a⊥ ∩ b⊥ ) ∩ (b ∩ c)) = 0
56	44, 55	2or 72	. . . . . . 7 (((a⊥ ∩ b⊥ ) ∩ b⊥ ) ∪ ((a⊥ ∩ b⊥ ) ∩ (b ∩ c))) = ((a⊥ ∩ b⊥ ) ∪ 0)
57		or0 102	. . . . . . 7 ((a⊥ ∩ b⊥ ) ∪ 0) = (a⊥ ∩ b⊥ )
58	40, 56, 57	3tr 65	. . . . . 6 ((a⊥ ∩ b⊥ ) ∩ (b⊥ ∪ (b ∩ c))) = (a⊥ ∩ b⊥ )
59	38, 58	2or 72	. . . . 5 (((a ∩ b) ∩ (b⊥ ∪ (b ∩ c))) ∪ ((a⊥ ∩ b⊥ ) ∩ (b⊥ ∪ (b ∩ c)))) = (((a ∩ b) ∩ c) ∪ (a⊥ ∩ b⊥ ))
60	6, 16, 59	3tr 65	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ (b →1 c)) = (((a ∩ b) ∩ c) ∪ (a⊥ ∩ b⊥ ))
61	60	ran 78	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ (b →1 c)) ∩ (c →2 b)) = ((((a ∩ b) ∩ c) ∪ (a⊥ ∩ b⊥ )) ∩ (c →2 b))
62		anass 76	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ (b →1 c)) ∩ (c →2 b)) = (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ ((b →1 c) ∩ (c →2 b)))
63		lear 161	. . . . . . . 8 ((a ∩ c) ∩ b) ≤ b
64		leo 158	. . . . . . . 8 b ≤ (b ∪ (c⊥ ∩ b⊥ ))
65	63, 64	letr 137	. . . . . . 7 ((a ∩ c) ∩ b) ≤ (b ∪ (c⊥ ∩ b⊥ ))
66		an32 83	. . . . . . 7 ((a ∩ b) ∩ c) = ((a ∩ c) ∩ b)
67		df-i2 45	. . . . . . 7 (c →2 b) = (b ∪ (c⊥ ∩ b⊥ ))
68	65, 66, 67	le3tr1 140	. . . . . 6 ((a ∩ b) ∩ c) ≤ (c →2 b)
69	68	lecom 180	. . . . 5 ((a ∩ b) ∩ c) C (c →2 b)
70		anass 76	. . . . . . . . . 10 ((a ∩ b) ∩ c) = (a ∩ (b ∩ c))
71		lea 160	. . . . . . . . . 10 (a ∩ (b ∩ c)) ≤ a
72	70, 71	bltr 138	. . . . . . . . 9 ((a ∩ b) ∩ c) ≤ a
73		leo 158	. . . . . . . . 9 a ≤ (a ∪ b)
74	72, 73	letr 137	. . . . . . . 8 ((a ∩ b) ∩ c) ≤ (a ∪ b)
75		oran 87	. . . . . . . 8 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
76	74, 75	lbtr 139	. . . . . . 7 ((a ∩ b) ∩ c) ≤ (a⊥ ∩ b⊥ )⊥
77	76	lecom 180	. . . . . 6 ((a ∩ b) ∩ c) C (a⊥ ∩ b⊥ )⊥
78	77	comcom7 460	. . . . 5 ((a ∩ b) ∩ c) C (a⊥ ∩ b⊥ )
79	69, 78	fh2r 474	. . . 4 ((((a ∩ b) ∩ c) ∪ (a⊥ ∩ b⊥ )) ∩ (c →2 b)) = ((((a ∩ b) ∩ c) ∩ (c →2 b)) ∪ ((a⊥ ∩ b⊥ ) ∩ (c →2 b)))
80		anass 76	. . . . . 6 (((a ∩ b) ∩ c) ∩ (c →2 b)) = ((a ∩ b) ∩ (c ∩ (c →2 b)))
81		an4 86	. . . . . 6 ((a ∩ b) ∩ (c ∩ (c →2 b))) = ((a ∩ c) ∩ (b ∩ (c →2 b)))
82		ancom 74	. . . . . . . . 9 (b ∩ (c →2 b)) = ((c →2 b) ∩ b)
83		u2lemab 611	. . . . . . . . 9 ((c →2 b) ∩ b) = b
84	82, 83	ax-r2 36	. . . . . . . 8 (b ∩ (c →2 b)) = b
85	84	lan 77	. . . . . . 7 ((a ∩ c) ∩ (b ∩ (c →2 b))) = ((a ∩ c) ∩ b)
86		an32 83	. . . . . . 7 ((a ∩ c) ∩ b) = ((a ∩ b) ∩ c)
87	85, 86	ax-r2 36	. . . . . 6 ((a ∩ c) ∩ (b ∩ (c →2 b))) = ((a ∩ b) ∩ c)
88	80, 81, 87	3tr 65	. . . . 5 (((a ∩ b) ∩ c) ∩ (c →2 b)) = ((a ∩ b) ∩ c)
89		anass 76	. . . . . 6 ((a⊥ ∩ b⊥ ) ∩ (c →2 b)) = (a⊥ ∩ (b⊥ ∩ (c →2 b)))
90		ancom 74	. . . . . . . 8 (b⊥ ∩ (c →2 b)) = ((c →2 b) ∩ b⊥ )
91		u2lemanb 616	. . . . . . . 8 ((c →2 b) ∩ b⊥ ) = (c⊥ ∩ b⊥ )
92	90, 91	ax-r2 36	. . . . . . 7 (b⊥ ∩ (c →2 b)) = (c⊥ ∩ b⊥ )
93	92	lan 77	. . . . . 6 (a⊥ ∩ (b⊥ ∩ (c →2 b))) = (a⊥ ∩ (c⊥ ∩ b⊥ ))
94		an12 81	. . . . . . 7 (a⊥ ∩ (c⊥ ∩ b⊥ )) = (c⊥ ∩ (a⊥ ∩ b⊥ ))
95		ancom 74	. . . . . . 7 (c⊥ ∩ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∩ c⊥ )
96	94, 95	ax-r2 36	. . . . . 6 (a⊥ ∩ (c⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∩ c⊥ )
97	89, 93, 96	3tr 65	. . . . 5 ((a⊥ ∩ b⊥ ) ∩ (c →2 b)) = ((a⊥ ∩ b⊥ ) ∩ c⊥ )
98	88, 97	2or 72	. . . 4 ((((a ∩ b) ∩ c) ∩ (c →2 b)) ∪ ((a⊥ ∩ b⊥ ) ∩ (c →2 b))) = (((a ∩ b) ∩ c) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ ))
99	79, 98	ax-r2 36	. . 3 ((((a ∩ b) ∩ c) ∪ (a⊥ ∩ b⊥ )) ∩ (c →2 b)) = (((a ∩ b) ∩ c) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ ))
100	61, 62, 99	3tr2 64	. 2 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ ((b →1 c) ∩ (c →2 b))) = (((a ∩ b) ∩ c) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ ))
101	4, 100	ax-r2 36	1 ((a ≡ b) ∩ (b ≡ c)) = (((a ∩ b) ∩ c) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ ))

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  0wf 9   →₁ wi1 12   →₂ wi2 13

This proof depends on 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 proof 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 used by:  bi4  840  mlaconj4  844