Theorem u2lemaa 601

Description: Lemma for Dishkant implication study. (Contributed by NM, 14-Dec-1997.)

Assertion

Ref	Expression
u2lemaa	((a →2 b) ∩ a) = (a ∩ b)

Proof of Theorem u2lemaa

Step	Hyp	Ref	Expression
1		df-i2 45	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
2	1	ran 78	. 2 ((a →2 b) ∩ a) = ((b ∪ (a⊥ ∩ b⊥ )) ∩ a)
3		ax-a2 31	. . . 4 (b ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ b)
4	3	ran 78	. . 3 ((b ∪ (a⊥ ∩ b⊥ )) ∩ a) = (((a⊥ ∩ b⊥ ) ∪ b) ∩ a)
5		coman1 185	. . . . . 6 (a⊥ ∩ b⊥ ) C a⊥
6	5	comcom7 460	. . . . 5 (a⊥ ∩ b⊥ ) C a
7		coman2 186	. . . . . 6 (a⊥ ∩ b⊥ ) C b⊥
8	7	comcom7 460	. . . . 5 (a⊥ ∩ b⊥ ) C b
9	6, 8	fh2r 474	. . . 4 (((a⊥ ∩ b⊥ ) ∪ b) ∩ a) = (((a⊥ ∩ b⊥ ) ∩ a) ∪ (b ∩ a))
10		ax-a2 31	. . . . 5 (((a⊥ ∩ b⊥ ) ∩ a) ∪ (b ∩ a)) = ((b ∩ a) ∪ ((a⊥ ∩ b⊥ ) ∩ a))
11		ancom 74	. . . . . . 7 (b ∩ a) = (a ∩ b)
12		ancom 74	. . . . . . . 8 ((a⊥ ∩ b⊥ ) ∩ a) = (a ∩ (a⊥ ∩ b⊥ ))
13		anass 76	. . . . . . . . . 10 ((a ∩ a⊥ ) ∩ b⊥ ) = (a ∩ (a⊥ ∩ b⊥ ))
14	13	ax-r1 35	. . . . . . . . 9 (a ∩ (a⊥ ∩ b⊥ )) = ((a ∩ a⊥ ) ∩ b⊥ )
15		ancom 74	. . . . . . . . . 10 ((a ∩ a⊥ ) ∩ b⊥ ) = (b⊥ ∩ (a ∩ a⊥ ))
16		dff 101	. . . . . . . . . . . . 13 0 = (a ∩ a⊥ )
17	16	ax-r1 35	. . . . . . . . . . . 12 (a ∩ a⊥ ) = 0
18	17	lan 77	. . . . . . . . . . 11 (b⊥ ∩ (a ∩ a⊥ )) = (b⊥ ∩ 0)
19		an0 108	. . . . . . . . . . 11 (b⊥ ∩ 0) = 0
20	18, 19	ax-r2 36	. . . . . . . . . 10 (b⊥ ∩ (a ∩ a⊥ )) = 0
21	15, 20	ax-r2 36	. . . . . . . . 9 ((a ∩ a⊥ ) ∩ b⊥ ) = 0
22	14, 21	ax-r2 36	. . . . . . . 8 (a ∩ (a⊥ ∩ b⊥ )) = 0
23	12, 22	ax-r2 36	. . . . . . 7 ((a⊥ ∩ b⊥ ) ∩ a) = 0
24	11, 23	2or 72	. . . . . 6 ((b ∩ a) ∪ ((a⊥ ∩ b⊥ ) ∩ a)) = ((a ∩ b) ∪ 0)
25		or0 102	. . . . . 6 ((a ∩ b) ∪ 0) = (a ∩ b)
26	24, 25	ax-r2 36	. . . . 5 ((b ∩ a) ∪ ((a⊥ ∩ b⊥ ) ∩ a)) = (a ∩ b)
27	10, 26	ax-r2 36	. . . 4 (((a⊥ ∩ b⊥ ) ∩ a) ∪ (b ∩ a)) = (a ∩ b)
28	9, 27	ax-r2 36	. . 3 (((a⊥ ∩ b⊥ ) ∪ b) ∩ a) = (a ∩ b)
29	4, 28	ax-r2 36	. 2 ((b ∪ (a⊥ ∩ b⊥ )) ∩ a) = (a ∩ b)
30	2, 29	ax-r2 36	1 ((a →2 b) ∩ a) = (a ∩ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →₂ wi2 13

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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u2lemnona  666