Theorem i3id 251

Description: Identity for Kalmbach implication.

Assertion

Ref	Expression
i3id	(a →3 a) = 1

Proof of Theorem i3id

Step	Hyp	Ref	Expression
1		ancom 74	. . . . . . . 8 (a⊥ ∩ a) = (a ∩ a⊥ )
2		dff 101	. . . . . . . . 9 0 = (a ∩ a⊥ )
3	2	ax-r1 35	. . . . . . . 8 (a ∩ a⊥ ) = 0
4	1, 3	ax-r2 36	. . . . . . 7 (a⊥ ∩ a) = 0
5		anidm 111	. . . . . . 7 (a⊥ ∩ a⊥ ) = a⊥
6	4, 5	2or 72	. . . . . 6 ((a⊥ ∩ a) ∪ (a⊥ ∩ a⊥ )) = (0 ∪ a⊥ )
7		ax-a2 31	. . . . . 6 (0 ∪ a⊥ ) = (a⊥ ∪ 0)
8	6, 7	ax-r2 36	. . . . 5 ((a⊥ ∩ a) ∪ (a⊥ ∩ a⊥ )) = (a⊥ ∪ 0)
9		or0 102	. . . . 5 (a⊥ ∪ 0) = a⊥
10	8, 9	ax-r2 36	. . . 4 ((a⊥ ∩ a) ∪ (a⊥ ∩ a⊥ )) = a⊥
11		ax-a2 31	. . . . . . 7 (a⊥ ∪ a) = (a ∪ a⊥ )
12		df-t 41	. . . . . . . 8 1 = (a ∪ a⊥ )
13	12	ax-r1 35	. . . . . . 7 (a ∪ a⊥ ) = 1
14	11, 13	ax-r2 36	. . . . . 6 (a⊥ ∪ a) = 1
15	14	lan 77	. . . . 5 (a ∩ (a⊥ ∪ a)) = (a ∩ 1)
16		an1 106	. . . . 5 (a ∩ 1) = a
17	15, 16	ax-r2 36	. . . 4 (a ∩ (a⊥ ∪ a)) = a
18	10, 17	2or 72	. . 3 (((a⊥ ∩ a) ∪ (a⊥ ∩ a⊥ )) ∪ (a ∩ (a⊥ ∪ a))) = (a⊥ ∪ a)
19	18, 11	ax-r2 36	. 2 (((a⊥ ∩ a) ∪ (a⊥ ∩ a⊥ )) ∪ (a ∩ (a⊥ ∪ a))) = (a ∪ a⊥ )
20		df-i3 46	. 2 (a →3 a) = (((a⊥ ∩ a) ∪ (a⊥ ∩ a⊥ )) ∪ (a ∩ (a⊥ ∪ a)))
21	19, 20, 12	3tr1 63	1 (a →3 a) = 1

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

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  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

This theorem is referenced by:  bina1  282  bina2  283  ska14  514  i3orcom  525  i3ancom  526  i3th4  546