Theorem bi4 840

Description: Chained biconditional. (Contributed by NM, 25-Jun-2003.)

Assertion

Ref	Expression
bi4	(((a ≡ b) ∩ (b ≡ c)) ∩ (c ≡ d)) = ((((a ∩ b) ∩ c) ∩ d) ∪ (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ d⊥ ))

Proof of Theorem bi4

Step	Hyp	Ref	Expression
1		bi3 839	. . 3 ((a ≡ b) ∩ (b ≡ c)) = (((a ∩ b) ∩ c) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ ))
2		u12lembi 726	. . . 4 ((c →1 d) ∩ (d →2 c)) = (c ≡ d)
3	2	ax-r1 35	. . 3 (c ≡ d) = ((c →1 d) ∩ (d →2 c))
4	1, 3	2an 79	. 2 (((a ≡ b) ∩ (b ≡ c)) ∩ (c ≡ d)) = ((((a ∩ b) ∩ c) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ )) ∩ ((c →1 d) ∩ (d →2 c)))
5		df-i1 44	. . . . . 6 (c →1 d) = (c⊥ ∪ (c ∩ d))
6	5	lan 77	. . . . 5 ((((a ∩ b) ∩ c) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ )) ∩ (c →1 d)) = ((((a ∩ b) ∩ c) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ )) ∩ (c⊥ ∪ (c ∩ d)))
7		leao2 163	. . . . . . 7 ((a⊥ ∩ b⊥ ) ∩ c⊥ ) ≤ (c⊥ ∪ (c ∩ d))
8	7	lecom 180	. . . . . 6 ((a⊥ ∩ b⊥ ) ∩ c⊥ ) C (c⊥ ∪ (c ∩ d))
9		leao4 165	. . . . . . . . . 10 ((a ∩ b) ∩ c) ≤ ((a⊥ ∩ b⊥ )⊥ ∪ c)
10		oran2 92	. . . . . . . . . 10 ((a⊥ ∩ b⊥ )⊥ ∪ c) = ((a⊥ ∩ b⊥ ) ∩ c⊥ )⊥
11	9, 10	lbtr 139	. . . . . . . . 9 ((a ∩ b) ∩ c) ≤ ((a⊥ ∩ b⊥ ) ∩ c⊥ )⊥
12	11	lecom 180	. . . . . . . 8 ((a ∩ b) ∩ c) C ((a⊥ ∩ b⊥ ) ∩ c⊥ )⊥
13	12	comcom 453	. . . . . . 7 ((a⊥ ∩ b⊥ ) ∩ c⊥ )⊥ C ((a ∩ b) ∩ c)
14	13	comcom6 459	. . . . . 6 ((a⊥ ∩ b⊥ ) ∩ c⊥ ) C ((a ∩ b) ∩ c)
15	8, 14	fh2rc 480	. . . . 5 ((((a ∩ b) ∩ c) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ )) ∩ (c⊥ ∪ (c ∩ d))) = ((((a ∩ b) ∩ c) ∩ (c⊥ ∪ (c ∩ d))) ∪ (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ (c⊥ ∪ (c ∩ d))))
16		comanr2 465	. . . . . . . . 9 c C ((a ∩ b) ∩ c)
17	16	comcom3 454	. . . . . . . 8 c⊥ C ((a ∩ b) ∩ c)
18		comanr1 464	. . . . . . . . 9 c C (c ∩ d)
19	18	comcom3 454	. . . . . . . 8 c⊥ C (c ∩ d)
20	17, 19	fh2 470	. . . . . . 7 (((a ∩ b) ∩ c) ∩ (c⊥ ∪ (c ∩ d))) = ((((a ∩ b) ∩ c) ∩ c⊥ ) ∪ (((a ∩ b) ∩ c) ∩ (c ∩ d)))
21		anass 76	. . . . . . . . 9 (((a ∩ b) ∩ c) ∩ c⊥ ) = ((a ∩ b) ∩ (c ∩ c⊥ ))
22		dff 101	. . . . . . . . . . 11 0 = (c ∩ c⊥ )
23	22	lan 77	. . . . . . . . . 10 ((a ∩ b) ∩ 0) = ((a ∩ b) ∩ (c ∩ c⊥ ))
24	23	ax-r1 35	. . . . . . . . 9 ((a ∩ b) ∩ (c ∩ c⊥ )) = ((a ∩ b) ∩ 0)
25		an0 108	. . . . . . . . 9 ((a ∩ b) ∩ 0) = 0
26	21, 24, 25	3tr 65	. . . . . . . 8 (((a ∩ b) ∩ c) ∩ c⊥ ) = 0
27		anass 76	. . . . . . . . . 10 ((((a ∩ b) ∩ c) ∩ c) ∩ d) = (((a ∩ b) ∩ c) ∩ (c ∩ d))
28	27	ax-r1 35	. . . . . . . . 9 (((a ∩ b) ∩ c) ∩ (c ∩ d)) = ((((a ∩ b) ∩ c) ∩ c) ∩ d)
29		anass 76	. . . . . . . . . . 11 (((a ∩ b) ∩ c) ∩ c) = ((a ∩ b) ∩ (c ∩ c))
30		anidm 111	. . . . . . . . . . . 12 (c ∩ c) = c
31	30	lan 77	. . . . . . . . . . 11 ((a ∩ b) ∩ (c ∩ c)) = ((a ∩ b) ∩ c)
32	29, 31	ax-r2 36	. . . . . . . . . 10 (((a ∩ b) ∩ c) ∩ c) = ((a ∩ b) ∩ c)
33	32	ran 78	. . . . . . . . 9 ((((a ∩ b) ∩ c) ∩ c) ∩ d) = (((a ∩ b) ∩ c) ∩ d)
34	28, 33	ax-r2 36	. . . . . . . 8 (((a ∩ b) ∩ c) ∩ (c ∩ d)) = (((a ∩ b) ∩ c) ∩ d)
35	26, 34	2or 72	. . . . . . 7 ((((a ∩ b) ∩ c) ∩ c⊥ ) ∪ (((a ∩ b) ∩ c) ∩ (c ∩ d))) = (0 ∪ (((a ∩ b) ∩ c) ∩ d))
36		or0r 103	. . . . . . 7 (0 ∪ (((a ∩ b) ∩ c) ∩ d)) = (((a ∩ b) ∩ c) ∩ d)
37	20, 35, 36	3tr 65	. . . . . 6 (((a ∩ b) ∩ c) ∩ (c⊥ ∪ (c ∩ d))) = (((a ∩ b) ∩ c) ∩ d)
38		comanr2 465	. . . . . . . 8 c⊥ C ((a⊥ ∩ b⊥ ) ∩ c⊥ )
39	38, 19	fh2 470	. . . . . . 7 (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ (c⊥ ∪ (c ∩ d))) = ((((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ c⊥ ) ∪ (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ (c ∩ d)))
40		anass 76	. . . . . . . . 9 (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ c⊥ ) = ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ c⊥ ))
41		anidm 111	. . . . . . . . . 10 (c⊥ ∩ c⊥ ) = c⊥
42	41	lan 77	. . . . . . . . 9 ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ c⊥ )) = ((a⊥ ∩ b⊥ ) ∩ c⊥ )
43	40, 42	ax-r2 36	. . . . . . . 8 (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ c⊥ ) = ((a⊥ ∩ b⊥ ) ∩ c⊥ )
44		an4 86	. . . . . . . . 9 (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ (c ∩ d)) = (((a⊥ ∩ b⊥ ) ∩ c) ∩ (c⊥ ∩ d))
45		anass 76	. . . . . . . . 9 (((a⊥ ∩ b⊥ ) ∩ c) ∩ (c⊥ ∩ d)) = ((a⊥ ∩ b⊥ ) ∩ (c ∩ (c⊥ ∩ d)))
46	22	ran 78	. . . . . . . . . . . . 13 (0 ∩ d) = ((c ∩ c⊥ ) ∩ d)
47	46	ax-r1 35	. . . . . . . . . . . 12 ((c ∩ c⊥ ) ∩ d) = (0 ∩ d)
48		anass 76	. . . . . . . . . . . 12 ((c ∩ c⊥ ) ∩ d) = (c ∩ (c⊥ ∩ d))
49		an0r 109	. . . . . . . . . . . 12 (0 ∩ d) = 0
50	47, 48, 49	3tr2 64	. . . . . . . . . . 11 (c ∩ (c⊥ ∩ d)) = 0
51	50	lan 77	. . . . . . . . . 10 ((a⊥ ∩ b⊥ ) ∩ (c ∩ (c⊥ ∩ d))) = ((a⊥ ∩ b⊥ ) ∩ 0)
52		an0 108	. . . . . . . . . 10 ((a⊥ ∩ b⊥ ) ∩ 0) = 0
53	51, 52	ax-r2 36	. . . . . . . . 9 ((a⊥ ∩ b⊥ ) ∩ (c ∩ (c⊥ ∩ d))) = 0
54	44, 45, 53	3tr 65	. . . . . . . 8 (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ (c ∩ d)) = 0
55	43, 54	2or 72	. . . . . . 7 ((((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ c⊥ ) ∪ (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ (c ∩ d))) = (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∪ 0)
56		or0 102	. . . . . . 7 (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∪ 0) = ((a⊥ ∩ b⊥ ) ∩ c⊥ )
57	39, 55, 56	3tr 65	. . . . . 6 (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ (c⊥ ∪ (c ∩ d))) = ((a⊥ ∩ b⊥ ) ∩ c⊥ )
58	37, 57	2or 72	. . . . 5 ((((a ∩ b) ∩ c) ∩ (c⊥ ∪ (c ∩ d))) ∪ (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ (c⊥ ∪ (c ∩ d)))) = ((((a ∩ b) ∩ c) ∩ d) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ ))
59	6, 15, 58	3tr 65	. . . 4 ((((a ∩ b) ∩ c) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ )) ∩ (c →1 d)) = ((((a ∩ b) ∩ c) ∩ d) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ ))
60	59	ran 78	. . 3 (((((a ∩ b) ∩ c) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ )) ∩ (c →1 d)) ∩ (d →2 c)) = (((((a ∩ b) ∩ c) ∩ d) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ )) ∩ (d →2 c))
61		anass 76	. . 3 (((((a ∩ b) ∩ c) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ )) ∩ (c →1 d)) ∩ (d →2 c)) = ((((a ∩ b) ∩ c) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ )) ∩ ((c →1 d) ∩ (d →2 c)))
62		anass 76	. . . . . . . 8 ((((a ∩ b) ∩ c) ∩ d) ∩ (d →2 c)) = (((a ∩ b) ∩ c) ∩ (d ∩ (d →2 c)))
63		an4 86	. . . . . . . 8 (((a ∩ b) ∩ c) ∩ (d ∩ (d →2 c))) = (((a ∩ b) ∩ d) ∩ (c ∩ (d →2 c)))
64		ancom 74	. . . . . . . . . . 11 (c ∩ (d →2 c)) = ((d →2 c) ∩ c)
65		u2lemab 611	. . . . . . . . . . 11 ((d →2 c) ∩ c) = c
66	64, 65	ax-r2 36	. . . . . . . . . 10 (c ∩ (d →2 c)) = c
67	66	lan 77	. . . . . . . . 9 (((a ∩ b) ∩ d) ∩ (c ∩ (d →2 c))) = (((a ∩ b) ∩ d) ∩ c)
68		an32 83	. . . . . . . . 9 (((a ∩ b) ∩ d) ∩ c) = (((a ∩ b) ∩ c) ∩ d)
69	67, 68	ax-r2 36	. . . . . . . 8 (((a ∩ b) ∩ d) ∩ (c ∩ (d →2 c))) = (((a ∩ b) ∩ c) ∩ d)
70	62, 63, 69	3tr 65	. . . . . . 7 ((((a ∩ b) ∩ c) ∩ d) ∩ (d →2 c)) = (((a ∩ b) ∩ c) ∩ d)
71	70	df2le1 135	. . . . . 6 (((a ∩ b) ∩ c) ∩ d) ≤ (d →2 c)
72	71	lecom 180	. . . . 5 (((a ∩ b) ∩ c) ∩ d) C (d →2 c)
73		an32 83	. . . . . . . . 9 (((a ∩ b) ∩ c) ∩ d) = (((a ∩ b) ∩ d) ∩ c)
74		leao4 165	. . . . . . . . 9 (((a ∩ b) ∩ d) ∩ c) ≤ ((a⊥ ∩ b⊥ )⊥ ∪ c)
75	73, 74	bltr 138	. . . . . . . 8 (((a ∩ b) ∩ c) ∩ d) ≤ ((a⊥ ∩ b⊥ )⊥ ∪ c)
76	75, 10	lbtr 139	. . . . . . 7 (((a ∩ b) ∩ c) ∩ d) ≤ ((a⊥ ∩ b⊥ ) ∩ c⊥ )⊥
77	76	lecom 180	. . . . . 6 (((a ∩ b) ∩ c) ∩ d) C ((a⊥ ∩ b⊥ ) ∩ c⊥ )⊥
78	77	comcom7 460	. . . . 5 (((a ∩ b) ∩ c) ∩ d) C ((a⊥ ∩ b⊥ ) ∩ c⊥ )
79	72, 78	fh2r 474	. . . 4 (((((a ∩ b) ∩ c) ∩ d) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ )) ∩ (d →2 c)) = (((((a ∩ b) ∩ c) ∩ d) ∩ (d →2 c)) ∪ (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ (d →2 c)))
80		anass 76	. . . . . 6 (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ (d →2 c)) = ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ (d →2 c)))
81		ancom 74	. . . . . . . 8 (c⊥ ∩ (d →2 c)) = ((d →2 c) ∩ c⊥ )
82		u2lemanb 616	. . . . . . . 8 ((d →2 c) ∩ c⊥ ) = (d⊥ ∩ c⊥ )
83	81, 82	ax-r2 36	. . . . . . 7 (c⊥ ∩ (d →2 c)) = (d⊥ ∩ c⊥ )
84	83	lan 77	. . . . . 6 ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ (d →2 c))) = ((a⊥ ∩ b⊥ ) ∩ (d⊥ ∩ c⊥ ))
85		an12 81	. . . . . . 7 ((a⊥ ∩ b⊥ ) ∩ (d⊥ ∩ c⊥ )) = (d⊥ ∩ ((a⊥ ∩ b⊥ ) ∩ c⊥ ))
86		ancom 74	. . . . . . 7 (d⊥ ∩ ((a⊥ ∩ b⊥ ) ∩ c⊥ )) = (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ d⊥ )
87	85, 86	ax-r2 36	. . . . . 6 ((a⊥ ∩ b⊥ ) ∩ (d⊥ ∩ c⊥ )) = (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ d⊥ )
88	80, 84, 87	3tr 65	. . . . 5 (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ (d →2 c)) = (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ d⊥ )
89	70, 88	2or 72	. . . 4 (((((a ∩ b) ∩ c) ∩ d) ∩ (d →2 c)) ∪ (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ (d →2 c))) = ((((a ∩ b) ∩ c) ∩ d) ∪ (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ d⊥ ))
90	79, 89	ax-r2 36	. . 3 (((((a ∩ b) ∩ c) ∩ d) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ )) ∩ (d →2 c)) = ((((a ∩ b) ∩ c) ∩ d) ∪ (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ d⊥ ))
91	60, 61, 90	3tr2 64	. 2 ((((a ∩ b) ∩ c) ∪ ((a⊥ ∩ b⊥ ) ∩ c⊥ )) ∩ ((c →1 d) ∩ (d →2 c))) = ((((a ∩ b) ∩ c) ∩ d) ∪ (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ d⊥ ))
92	4, 91	ax-r2 36	1 (((a ≡ b) ∩ (b ≡ c)) ∩ (c ≡ d)) = ((((a ∩ b) ∩ c) ∩ d) ∪ (((a⊥ ∩ b⊥ ) ∩ c⊥ ) ∩ d⊥ ))

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  0wf 9   →₁ wi1 12   →₂ wi2 13

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:  mhcor1  888