Theorem u3lem15 795

Description: Lemma for Kalmbach implication. (Contributed by NM, 7-Aug-2001.)

Assertion

Ref	Expression
u3lem15	((a →3 b) ∩ (a ∪ b)) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))

Proof of Theorem u3lem15

Step	Hyp	Ref	Expression
1		dfi3b 499	. . 3 (a →3 b) = ((a⊥ ∪ b) ∩ ((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b)))
2	1	ran 78	. 2 ((a →3 b) ∩ (a ∪ b)) = (((a⊥ ∪ b) ∩ ((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b))) ∩ (a ∪ b))
3		anass 76	. 2 (((a⊥ ∪ b) ∩ ((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b))) ∩ (a ∪ b)) = ((a⊥ ∪ b) ∩ (((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b)) ∩ (a ∪ b)))
4		comor1 461	. . . . . 6 (a ∪ b) C a
5	4	comcom2 183	. . . . . . 7 (a ∪ b) C a⊥
6		comor2 462	. . . . . . . 8 (a ∪ b) C b
7	6	comcom2 183	. . . . . . 7 (a ∪ b) C b⊥
8	5, 7	com2an 484	. . . . . 6 (a ∪ b) C (a⊥ ∩ b⊥ )
9	4, 8	com2or 483	. . . . 5 (a ∪ b) C (a ∪ (a⊥ ∩ b⊥ ))
10		leao4 165	. . . . . . 7 (a⊥ ∩ b) ≤ (a ∪ b)
11	10	lecom 180	. . . . . 6 (a⊥ ∩ b) C (a ∪ b)
12	11	comcom 453	. . . . 5 (a ∪ b) C (a⊥ ∩ b)
13	9, 12	fh1r 473	. . . 4 (((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b)) ∩ (a ∪ b)) = (((a ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ b)) ∪ ((a⊥ ∩ b) ∩ (a ∪ b)))
14	4, 8	fh1r 473	. . . . . 6 ((a ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ b)) = ((a ∩ (a ∪ b)) ∪ ((a⊥ ∩ b⊥ ) ∩ (a ∪ b)))
15		anabs 121	. . . . . . 7 (a ∩ (a ∪ b)) = a
16		oran 87	. . . . . . . . 9 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
17	16	lan 77	. . . . . . . 8 ((a⊥ ∩ b⊥ ) ∩ (a ∪ b)) = ((a⊥ ∩ b⊥ ) ∩ (a⊥ ∩ b⊥ )⊥ )
18		dff 101	. . . . . . . . 9 0 = ((a⊥ ∩ b⊥ ) ∩ (a⊥ ∩ b⊥ )⊥ )
19	18	ax-r1 35	. . . . . . . 8 ((a⊥ ∩ b⊥ ) ∩ (a⊥ ∩ b⊥ )⊥ ) = 0
20	17, 19	ax-r2 36	. . . . . . 7 ((a⊥ ∩ b⊥ ) ∩ (a ∪ b)) = 0
21	15, 20	2or 72	. . . . . 6 ((a ∩ (a ∪ b)) ∪ ((a⊥ ∩ b⊥ ) ∩ (a ∪ b))) = (a ∪ 0)
22		or0 102	. . . . . 6 (a ∪ 0) = a
23	14, 21, 22	3tr 65	. . . . 5 ((a ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ b)) = a
24	10	df2le2 136	. . . . 5 ((a⊥ ∩ b) ∩ (a ∪ b)) = (a⊥ ∩ b)
25	23, 24	2or 72	. . . 4 (((a ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ b)) ∪ ((a⊥ ∩ b) ∩ (a ∪ b))) = (a ∪ (a⊥ ∩ b))
26	13, 25	ax-r2 36	. . 3 (((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b)) ∩ (a ∪ b)) = (a ∪ (a⊥ ∩ b))
27	26	lan 77	. 2 ((a⊥ ∪ b) ∩ (((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b)) ∩ (a ∪ b))) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))
28	2, 3, 27	3tr 65	1 ((a →3 b) ∩ (a ∪ b)) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →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:  neg3antlem2  865