Theorem df2i3 480

Description: Alternate definition for Kalmbach implication.

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 45	. 2 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2		ax-a3 31	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = ((a⊥ ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b))))
3		or12 73	. . . 4 ((a⊥ ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b)))) = ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∩ b) ∪ (a ∩ (a⊥ ∪ b))))
4		coman1 177	. . . . . . . . . 10 (a⊥ ∩ b) C a⊥
5	4	comcom 435	. . . . . . . . 9 a⊥ C (a⊥ ∩ b)
6	5	comcom2 175	. . . . . . . 8 a⊥ C (a⊥ ∩ b)⊥
7	6	comcom5 440	. . . . . . 7 a C (a⊥ ∩ b)
8		comorr 176	. . . . . . . . 9 a⊥ C (a⊥ ∪ b)
9	8	comcom2 175	. . . . . . . 8 a⊥ C (a⊥ ∪ b)⊥
10	9	comcom5 440	. . . . . . 7 a C (a⊥ ∪ b)
11	7, 10	fh4 454	. . . . . 6 ((a⊥ ∩ b) ∪ (a ∩ (a⊥ ∪ b))) = (((a⊥ ∩ b) ∪ a) ∩ ((a⊥ ∩ b) ∪ (a⊥ ∪ b)))
12		lea 152	. . . . . . . . . 10 (a⊥ ∩ b) ≤ a⊥
13		leo 150	. . . . . . . . . 10 a⊥ ≤ (a⊥ ∪ b)
14	12, 13	letr 129	. . . . . . . . 9 (a⊥ ∩ b) ≤ (a⊥ ∪ b)
15	14	df-le2 123	. . . . . . . 8 ((a⊥ ∩ b) ∪ (a⊥ ∪ b)) = (a⊥ ∪ b)
16	15	lan 70	. . . . . . 7 (((a⊥ ∩ b) ∪ a) ∩ ((a⊥ ∩ b) ∪ (a⊥ ∪ b))) = (((a⊥ ∩ b) ∪ a) ∩ (a⊥ ∪ b))
17		ancom 68	. . . . . . . 8 (((a⊥ ∩ b) ∪ a) ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∩ ((a⊥ ∩ b) ∪ a))
18		ax-a2 30	. . . . . . . . 9 ((a⊥ ∩ b) ∪ a) = (a ∪ (a⊥ ∩ b))
19	18	lan 70	. . . . . . . 8 ((a⊥ ∪ b) ∩ ((a⊥ ∩ b) ∪ a)) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))
20	17, 19	ax-r2 35	. . . . . . 7 (((a⊥ ∩ b) ∪ a) ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))
21	16, 20	ax-r2 35	. . . . . 6 (((a⊥ ∩ b) ∪ a) ∩ ((a⊥ ∩ b) ∪ (a⊥ ∪ b))) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))
22	11, 21	ax-r2 35	. . . . 5 ((a⊥ ∩ b) ∪ (a ∩ (a⊥ ∪ b))) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))
23	22	lor 66	. . . 4 ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∩ b) ∪ (a ∩ (a⊥ ∪ b)))) = ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))
24	3, 23	ax-r2 35	. . 3 ((a⊥ ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b)))) = ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))
25	2, 24	ax-r2 35	. 2 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))
26	1, 25	ax-r2 35	1 (a →3 b) = ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 15

This theorem is referenced by:  i3n2 483  ni32 484  i3lem1 486  i3th1 525  i3orlem5 538

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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