Theorem u3lem15 777

Description: Lemma for Kalmbach implication.

Assertion

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

Proof of Theorem u3lem15

Step	Hyp	Ref	Expression
1		dfi3b 481	. . 3 (a →3 b) = ((a⊥ ∪ b) ∩ ((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b)))
2	1	ran 71	. 2 ((a →3 b) ∩ (a ∪ b)) = (((a⊥ ∪ b) ∩ ((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b))) ∩ (a ∪ b))
3		anass 69	. 2 (((a⊥ ∪ b) ∩ ((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b))) ∩ (a ∪ b)) = ((a⊥ ∪ b) ∩ (((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b)) ∩ (a ∪ b)))
4		comor1 443	. . . . . 6 (a ∪ b) C a
5	4	comcom2 175	. . . . . . 7 (a ∪ b) C a⊥
6		comor2 444	. . . . . . . 8 (a ∪ b) C b
7	6	comcom2 175	. . . . . . 7 (a ∪ b) C b⊥
8	5, 7	com2an 466	. . . . . 6 (a ∪ b) C (a⊥ ∩ b⊥ )
9	4, 8	com2or 465	. . . . 5 (a ∪ b) C (a ∪ (a⊥ ∩ b⊥ ))
10		leao4 157	. . . . . . 7 (a⊥ ∩ b) ≤ (a ∪ b)
11	10	lecom 172	. . . . . 6 (a⊥ ∩ b) C (a ∪ b)
12	11	comcom 435	. . . . 5 (a ∪ b) C (a⊥ ∩ b)
13	9, 12	fh1r 455	. . . 4 (((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b)) ∩ (a ∪ b)) = (((a ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ b)) ∪ ((a⊥ ∩ b) ∩ (a ∪ b)))
14	4, 8	fh1r 455	. . . . . 6 ((a ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ b)) = ((a ∩ (a ∪ b)) ∪ ((a⊥ ∩ b⊥ ) ∩ (a ∪ b)))
15		a5c 113	. . . . . . 7 (a ∩ (a ∪ b)) = a
16		oran 79	. . . . . . . . 9 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
17	16	lan 70	. . . . . . . 8 ((a⊥ ∩ b⊥ ) ∩ (a ∪ b)) = ((a⊥ ∩ b⊥ ) ∩ (a⊥ ∩ b⊥ )⊥ )
18		dff 93	. . . . . . . . 9 0 = ((a⊥ ∩ b⊥ ) ∩ (a⊥ ∩ b⊥ )⊥ )
19	18	ax-r1 34	. . . . . . . 8 ((a⊥ ∩ b⊥ ) ∩ (a⊥ ∩ b⊥ )⊥ ) = 0
20	17, 19	ax-r2 35	. . . . . . 7 ((a⊥ ∩ b⊥ ) ∩ (a ∪ b)) = 0
21	15, 20	2or 67	. . . . . 6 ((a ∩ (a ∪ b)) ∪ ((a⊥ ∩ b⊥ ) ∩ (a ∪ b))) = (a ∪ 0)
22		or0 94	. . . . . 6 (a ∪ 0) = a
23	14, 21, 22	3tr 62	. . . . 5 ((a ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ b)) = a
24	10	df2le2 128	. . . . 5 ((a⊥ ∩ b) ∩ (a ∪ b)) = (a⊥ ∩ b)
25	23, 24	2or 67	. . . 4 (((a ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ b)) ∪ ((a⊥ ∩ b) ∩ (a ∪ b))) = (a ∪ (a⊥ ∩ b))
26	13, 25	ax-r2 35	. . 3 (((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b)) ∩ (a ∪ b)) = (a ∪ (a⊥ ∩ b))
27	26	lan 70	. 2 ((a⊥ ∪ b) ∩ (((a ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ b)) ∩ (a ∪ b))) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))
28	2, 3, 27	3tr 62	1 ((a →3 b) ∩ (a ∪ b)) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 10   →₃ wi3 15

This theorem is referenced by:  neg3antlem2 847

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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