i3abs3 - Quantum Logic Explorer

	Quantum Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > QLE Home > Th. List > i3abs3		GIF version

Theorem i3abs3 524

Description: Antecedent absorption. (Contributed by NM, 19-Nov-1997.)

**Assertion**
Ref	Expression
i3abs3	((a →₃b) →₃((a →₃b) →₃a)) = ((a →₃b) →₃a)

**Proof of Theorem i3abs3**
Step	Hyp	Ref	Expression
1		df-t 41	. . . . . . . 8 1 = (a ∪ a^⊥ )
2	1	lan 77	. . . . . . 7 ((a →₃b)^⊥ ∩ 1) = ((a →₃b)^⊥ ∩ (a ∪ a^⊥ ))
3		an1 106	. . . . . . 7 ((a →₃b)^⊥ ∩ 1) = (a →₃b)^⊥
4		comi31 508	. . . . . . . . . 10 a C (a →₃b)
5	4	comcom 453	. . . . . . . . 9 (a →₃b) C a
6	5	comcom3 454	. . . . . . . 8 (a →₃b)^⊥ C a
7	5	comcom4 455	. . . . . . . 8 (a →₃b)^⊥ C a^⊥
8	6, 7	fh1 469	. . . . . . 7 ((a →₃b)^⊥ ∩ (a ∪ a^⊥ )) = (((a →₃b)^⊥ ∩ a) ∪ ((a →₃b)^⊥ ∩ a^⊥ ))
9	2, 3, 8	3tr2 64	. . . . . 6 (a →₃b)^⊥ = (((a →₃b)^⊥ ∩ a) ∪ ((a →₃b)^⊥ ∩ a^⊥ ))
10	9	ax-r1 35	. . . . 5 (((a →₃b)^⊥ ∩ a) ∪ ((a →₃b)^⊥ ∩ a^⊥ )) = (a →₃b)^⊥
11		comid 187	. . . . . . . 8 (a →₃b) C (a →₃b)
12	11	comcom2 183	. . . . . . 7 (a →₃b) C (a →₃b)^⊥
13	12, 5	fh1 469	. . . . . 6 ((a →₃b) ∩ ((a →₃b)^⊥ ∪ a)) = (((a →₃b) ∩ (a →₃b)^⊥ ) ∪ ((a →₃b) ∩ a))
14		ax-a2 31	. . . . . . 7 (0 ∪ ((a →₃b) ∩ a)) = (((a →₃b) ∩ a) ∪ 0)
15		dff 101	. . . . . . . 8 0 = ((a →₃b) ∩ (a →₃b)^⊥ )
16	15	ax-r5 38	. . . . . . 7 (0 ∪ ((a →₃b) ∩ a)) = (((a →₃b) ∩ (a →₃b)^⊥ ) ∪ ((a →₃b) ∩ a))
17		or0 102	. . . . . . 7 (((a →₃b) ∩ a) ∪ 0) = ((a →₃b) ∩ a)
18	14, 16, 17	3tr2 64	. . . . . 6 (((a →₃b) ∩ (a →₃b)^⊥ ) ∪ ((a →₃b) ∩ a)) = ((a →₃b) ∩ a)
19	13, 18	ax-r2 36	. . . . 5 ((a →₃b) ∩ ((a →₃b)^⊥ ∪ a)) = ((a →₃b) ∩ a)
20	10, 19	2or 72	. . . 4 ((((a →₃b)^⊥ ∩ a) ∪ ((a →₃b)^⊥ ∩ a^⊥ )) ∪ ((a →₃b) ∩ ((a →₃b)^⊥ ∪ a))) = ((a →₃b)^⊥ ∪ ((a →₃b) ∩ a))
21	12, 5	fh4 472	. . . . 5 ((a →₃b)^⊥ ∪ ((a →₃b) ∩ a)) = (((a →₃b)^⊥ ∪ (a →₃b)) ∩ ((a →₃b)^⊥ ∪ a))
22		ax-a2 31	. . . . . . . . 9 ((a →₃b)^⊥ ∪ (a →₃b)) = ((a →₃b) ∪ (a →₃b)^⊥ )
23		df-t 41	. . . . . . . . . 10 1 = ((a →₃b) ∪ (a →₃b)^⊥ )
24	23	ax-r1 35	. . . . . . . . 9 ((a →₃b) ∪ (a →₃b)^⊥ ) = 1
25	22, 24	ax-r2 36	. . . . . . . 8 ((a →₃b)^⊥ ∪ (a →₃b)) = 1
26	25	ran 78	. . . . . . 7 (((a →₃b)^⊥ ∪ (a →₃b)) ∩ ((a →₃b)^⊥ ∪ a)) = (1 ∩ ((a →₃b)^⊥ ∪ a))
27		ancom 74	. . . . . . 7 (1 ∩ ((a →₃b)^⊥ ∪ a)) = (((a →₃b)^⊥ ∪ a) ∩ 1)
28	26, 27	ax-r2 36	. . . . . 6 (((a →₃b)^⊥ ∪ (a →₃b)) ∩ ((a →₃b)^⊥ ∪ a)) = (((a →₃b)^⊥ ∪ a) ∩ 1)
29		an1 106	. . . . . 6 (((a →₃b)^⊥ ∪ a) ∩ 1) = ((a →₃b)^⊥ ∪ a)
30	28, 29	ax-r2 36	. . . . 5 (((a →₃b)^⊥ ∪ (a →₃b)) ∩ ((a →₃b)^⊥ ∪ a)) = ((a →₃b)^⊥ ∪ a)
31	21, 30	ax-r2 36	. . . 4 ((a →₃b)^⊥ ∪ ((a →₃b) ∩ a)) = ((a →₃b)^⊥ ∪ a)
32	20, 31	ax-r2 36	. . 3 ((((a →₃b)^⊥ ∩ a) ∪ ((a →₃b)^⊥ ∩ a^⊥ )) ∪ ((a →₃b) ∩ ((a →₃b)^⊥ ∪ a))) = ((a →₃b)^⊥ ∪ a)
33	32	ax-r1 35	. 2 ((a →₃b)^⊥ ∪ a) = ((((a →₃b)^⊥ ∩ a) ∪ ((a →₃b)^⊥ ∩ a^⊥ )) ∪ ((a →₃b) ∩ ((a →₃b)^⊥ ∪ a)))
34		lem4 511	. 2 ((a →₃b) →₃((a →₃b) →₃a)) = ((a →₃b)^⊥ ∪ a)
35		df-i3 46	. 2 ((a →₃b) →₃a) = ((((a →₃b)^⊥ ∩ a) ∪ ((a →₃b)^⊥ ∩ a^⊥ )) ∪ ((a →₃b) ∩ ((a →₃b)^⊥ ∪ a)))
36	33, 34, 35	3tr1 63	1 ((a →₃b) →₃((a →₃b) →₃a)) = ((a →₃b) →₃a)

Colors of variables: term

This proof depends on syntax axioms: = wb 1 ^⊥ wn 4 ∪ wo 6 ∩ wa 7 1wt 8 0wf 9 →₃wi3 14

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-i3 46 df-le1 130 df-le2 131 df-c1 132 df-c2 133

This theorem is used by: i3th7 549 i3th8 550

W3C validator