Theorem lem3.3.7i3e1 1066

Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 3, and this is the first part of the equation. (Contributed by Roy F. Longton, 28-Jun-2005.) (Revised by Roy F. Longton, 3-Jul-2005.)

Assertion

Ref	Expression
lem3.3.7i3e1	(a →3 (a ∩ b)) = (a ≡3 (a ∩ b))

Proof of Theorem lem3.3.7i3e1

Step	Hyp	Ref	Expression
1		anass 76	. . . . . 6 ((a⊥ ∩ a) ∩ b) = (a⊥ ∩ (a ∩ b))
2	1	ax-r1 35	. . . . 5 (a⊥ ∩ (a ∩ b)) = ((a⊥ ∩ a) ∩ b)
3	2	ax-r5 38	. . . 4 ((a⊥ ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = (((a⊥ ∩ a) ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ ))
4	3	ax-r5 38	. . 3 (((a⊥ ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = ((((a⊥ ∩ a) ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b))))
5		ancom 74	. . . . . 6 (a⊥ ∩ a) = (a ∩ a⊥ )
6	5	ran 78	. . . . 5 ((a⊥ ∩ a) ∩ b) = ((a ∩ a⊥ ) ∩ b)
7	6	ax-r5 38	. . . 4 (((a⊥ ∩ a) ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = (((a ∩ a⊥ ) ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ ))
8	7	ax-r5 38	. . 3 ((((a⊥ ∩ a) ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = ((((a ∩ a⊥ ) ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b))))
9		dff 101	. . . . . . . 8 0 = (a ∩ a⊥ )
10	9	ax-r1 35	. . . . . . 7 (a ∩ a⊥ ) = 0
11	10	ran 78	. . . . . 6 ((a ∩ a⊥ ) ∩ b) = (0 ∩ b)
12	11	ax-r5 38	. . . . 5 (((a ∩ a⊥ ) ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = ((0 ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ ))
13	12	ax-r5 38	. . . 4 ((((a ∩ a⊥ ) ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = (((0 ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b))))
14		an0r 109	. . . . . 6 (0 ∩ b) = 0
15	14	ax-r5 38	. . . . 5 ((0 ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = (0 ∪ (a⊥ ∩ (a ∩ b)⊥ ))
16	15	ax-r5 38	. . . 4 (((0 ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = ((0 ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b))))
17		or0r 103	. . . . . 6 (0 ∪ (a⊥ ∩ (a ∩ b)⊥ )) = (a⊥ ∩ (a ∩ b)⊥ )
18	17	ax-r5 38	. . . . 5 ((0 ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = ((a⊥ ∩ (a ∩ b)⊥ ) ∪ (a ∩ (a⊥ ∪ (a ∩ b))))
19		anor3 90	. . . . . . 7 (a⊥ ∩ (a ∩ b)⊥ ) = (a ∪ (a ∩ b))⊥
20	19	ax-r5 38	. . . . . 6 ((a⊥ ∩ (a ∩ b)⊥ ) ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = ((a ∪ (a ∩ b))⊥ ∪ (a ∩ (a⊥ ∪ (a ∩ b))))
21		orabs 120	. . . . . . . 8 (a ∪ (a ∩ b)) = a
22	21	ax-r4 37	. . . . . . 7 (a ∪ (a ∩ b))⊥ = a⊥
23	22	ax-r5 38	. . . . . 6 ((a ∪ (a ∩ b))⊥ ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = (a⊥ ∪ (a ∩ (a⊥ ∪ (a ∩ b))))
24		womaa 222	. . . . . . . 8 (a⊥ ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = (a⊥ ∪ (a ∩ b))
25		an1 106	. . . . . . . . 9 ((a⊥ ∪ (a ∩ b)) ∩ 1) = (a⊥ ∪ (a ∩ b))
26	25	ax-r1 35	. . . . . . . 8 (a⊥ ∪ (a ∩ b)) = ((a⊥ ∪ (a ∩ b)) ∩ 1)
27		df-t 41	. . . . . . . . 9 1 = (a ∪ a⊥ )
28	27	lan 77	. . . . . . . 8 ((a⊥ ∪ (a ∩ b)) ∩ 1) = ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ a⊥ ))
29	24, 26, 28	3tr 65	. . . . . . 7 (a⊥ ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ a⊥ ))
30	21	ax-r1 35	. . . . . . . . . 10 a = (a ∪ (a ∩ b))
31	30	ax-r4 37	. . . . . . . . 9 a⊥ = (a ∪ (a ∩ b))⊥
32	31	lor 70	. . . . . . . 8 (a ∪ a⊥ ) = (a ∪ (a ∪ (a ∩ b))⊥ )
33	32	lan 77	. . . . . . 7 ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ a⊥ )) = ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a ∪ (a ∩ b))⊥ ))
34	19	ax-r1 35	. . . . . . . . 9 (a ∪ (a ∩ b))⊥ = (a⊥ ∩ (a ∩ b)⊥ )
35	34	lor 70	. . . . . . . 8 (a ∪ (a ∪ (a ∩ b))⊥ ) = (a ∪ (a⊥ ∩ (a ∩ b)⊥ ))
36	35	lan 77	. . . . . . 7 ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a ∪ (a ∩ b))⊥ )) = ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a⊥ ∩ (a ∩ b)⊥ )))
37	29, 33, 36	3tr 65	. . . . . 6 (a⊥ ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a⊥ ∩ (a ∩ b)⊥ )))
38	20, 23, 37	3tr 65	. . . . 5 ((a⊥ ∩ (a ∩ b)⊥ ) ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a⊥ ∩ (a ∩ b)⊥ )))
39	18, 38	ax-r2 36	. . . 4 ((0 ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a⊥ ∩ (a ∩ b)⊥ )))
40	13, 16, 39	3tr 65	. . 3 ((((a ∩ a⊥ ) ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a⊥ ∩ (a ∩ b)⊥ )))
41	4, 8, 40	3tr 65	. 2 (((a⊥ ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b)))) = ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a⊥ ∩ (a ∩ b)⊥ )))
42		df-i3 46	. 2 (a →3 (a ∩ b)) = (((a⊥ ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ )) ∪ (a ∩ (a⊥ ∪ (a ∩ b))))
43		df-id3 52	. 2 (a ≡3 (a ∩ b)) = ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ (a⊥ ∩ (a ∩ b)⊥ )))
44	41, 42, 43	3tr1 63	1 (a →3 (a ∩ b)) = (a ≡3 (a ∩ b))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →₃ wi3 14   ≡₃ wid3 20

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

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i3 46  df-id3 52  df-le1 130  df-le2 131

This theorem is referenced by: (None)