Theorem df2i3 498

Description: Alternate definition for Kalmbach implication. (Contributed by NM, 7-Nov-1997.)

Assertion

Ref	Expression
df2i3	(a →3 b) = ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))

Proof of Theorem df2i3

Step	Hyp	Ref	Expression
1		df-i3 46	. 2 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2		ax-a3 32	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = ((a⊥ ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b))))
3		or12 80	. . . 4 ((a⊥ ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b)))) = ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∩ b) ∪ (a ∩ (a⊥ ∪ b))))
4		coman1 185	. . . . . . . . . 10 (a⊥ ∩ b) C a⊥
5	4	comcom 453	. . . . . . . . 9 a⊥ C (a⊥ ∩ b)
6	5	comcom2 183	. . . . . . . 8 a⊥ C (a⊥ ∩ b)⊥
7	6	comcom5 458	. . . . . . 7 a C (a⊥ ∩ b)
8		comorr 184	. . . . . . . . 9 a⊥ C (a⊥ ∪ b)
9	8	comcom2 183	. . . . . . . 8 a⊥ C (a⊥ ∪ b)⊥
10	9	comcom5 458	. . . . . . 7 a C (a⊥ ∪ b)
11	7, 10	fh4 472	. . . . . 6 ((a⊥ ∩ b) ∪ (a ∩ (a⊥ ∪ b))) = (((a⊥ ∩ b) ∪ a) ∩ ((a⊥ ∩ b) ∪ (a⊥ ∪ b)))
12		lea 160	. . . . . . . . . 10 (a⊥ ∩ b) ≤ a⊥
13		leo 158	. . . . . . . . . 10 a⊥ ≤ (a⊥ ∪ b)
14	12, 13	letr 137	. . . . . . . . 9 (a⊥ ∩ b) ≤ (a⊥ ∪ b)
15	14	df-le2 131	. . . . . . . 8 ((a⊥ ∩ b) ∪ (a⊥ ∪ b)) = (a⊥ ∪ b)
16	15	lan 77	. . . . . . 7 (((a⊥ ∩ b) ∪ a) ∩ ((a⊥ ∩ b) ∪ (a⊥ ∪ b))) = (((a⊥ ∩ b) ∪ a) ∩ (a⊥ ∪ b))
17		ancom 74	. . . . . . . 8 (((a⊥ ∩ b) ∪ a) ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∩ ((a⊥ ∩ b) ∪ a))
18		ax-a2 31	. . . . . . . . 9 ((a⊥ ∩ b) ∪ a) = (a ∪ (a⊥ ∩ b))
19	18	lan 77	. . . . . . . 8 ((a⊥ ∪ b) ∩ ((a⊥ ∩ b) ∪ a)) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))
20	17, 19	ax-r2 36	. . . . . . 7 (((a⊥ ∩ b) ∪ a) ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))
21	16, 20	ax-r2 36	. . . . . 6 (((a⊥ ∩ b) ∪ a) ∩ ((a⊥ ∩ b) ∪ (a⊥ ∪ b))) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))
22	11, 21	ax-r2 36	. . . . 5 ((a⊥ ∩ b) ∪ (a ∩ (a⊥ ∪ b))) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))
23	22	lor 70	. . . 4 ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∩ b) ∪ (a ∩ (a⊥ ∪ b)))) = ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))
24	3, 23	ax-r2 36	. . 3 ((a⊥ ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b)))) = ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))
25	2, 24	ax-r2 36	. 2 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))
26	1, 25	ax-r2 36	1 (a →3 b) = ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 14

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

This theorem is referenced by:  i3n2  501  ni32  502  i3lem1  504  i3th1  543  i3orlem5  556