Theorem u3lemob 632

Description: Lemma for Kalmbach implication study. (Contributed by NM, 15-Dec-1997.)

Assertion

Ref	Expression
u3lemob	((a →3 b) ∪ b) = (a⊥ ∪ b)

Proof of Theorem u3lemob

Step	Hyp	Ref	Expression
1		df-i3 46	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2	1	ax-r5 38	. 2 ((a →3 b) ∪ b) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∪ b)
3		or32 82	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∪ b) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ b) ∪ (a ∩ (a⊥ ∪ b)))
4		or32 82	. . . . . 6 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ b) = (((a⊥ ∩ b) ∪ b) ∪ (a⊥ ∩ b⊥ ))
5		lear 161	. . . . . . . 8 (a⊥ ∩ b) ≤ b
6	5	df-le2 131	. . . . . . 7 ((a⊥ ∩ b) ∪ b) = b
7	6	ax-r5 38	. . . . . 6 (((a⊥ ∩ b) ∪ b) ∪ (a⊥ ∩ b⊥ )) = (b ∪ (a⊥ ∩ b⊥ ))
8	4, 7	ax-r2 36	. . . . 5 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ b) = (b ∪ (a⊥ ∩ b⊥ ))
9		ancom 74	. . . . 5 (a ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∩ a)
10	8, 9	2or 72	. . . 4 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ b) ∪ (a ∩ (a⊥ ∪ b))) = ((b ∪ (a⊥ ∩ b⊥ )) ∪ ((a⊥ ∪ b) ∩ a))
11		comor2 462	. . . . . . 7 (a⊥ ∪ b) C b
12		comor1 461	. . . . . . . 8 (a⊥ ∪ b) C a⊥
13	11	comcom2 183	. . . . . . . 8 (a⊥ ∪ b) C b⊥
14	12, 13	com2an 484	. . . . . . 7 (a⊥ ∪ b) C (a⊥ ∩ b⊥ )
15	11, 14	com2or 483	. . . . . 6 (a⊥ ∪ b) C (b ∪ (a⊥ ∩ b⊥ ))
16	12	comcom7 460	. . . . . 6 (a⊥ ∪ b) C a
17	15, 16	fh4 472	. . . . 5 ((b ∪ (a⊥ ∩ b⊥ )) ∪ ((a⊥ ∪ b) ∩ a)) = (((b ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ b)) ∩ ((b ∪ (a⊥ ∩ b⊥ )) ∪ a))
18		or32 82	. . . . . . . 8 ((b ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ b)) = ((b ∪ (a⊥ ∪ b)) ∪ (a⊥ ∩ b⊥ ))
19		or12 80	. . . . . . . . . . 11 (b ∪ (a⊥ ∪ b)) = (a⊥ ∪ (b ∪ b))
20		oridm 110	. . . . . . . . . . . 12 (b ∪ b) = b
21	20	lor 70	. . . . . . . . . . 11 (a⊥ ∪ (b ∪ b)) = (a⊥ ∪ b)
22	19, 21	ax-r2 36	. . . . . . . . . 10 (b ∪ (a⊥ ∪ b)) = (a⊥ ∪ b)
23	22	ax-r5 38	. . . . . . . . 9 ((b ∪ (a⊥ ∪ b)) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∪ b) ∪ (a⊥ ∩ b⊥ ))
24		ax-a2 31	. . . . . . . . . 10 ((a⊥ ∪ b) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∪ b))
25		lea 160	. . . . . . . . . . . 12 (a⊥ ∩ b⊥ ) ≤ a⊥
26		leo 158	. . . . . . . . . . . 12 a⊥ ≤ (a⊥ ∪ b)
27	25, 26	letr 137	. . . . . . . . . . 11 (a⊥ ∩ b⊥ ) ≤ (a⊥ ∪ b)
28	27	df-le2 131	. . . . . . . . . 10 ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∪ b)) = (a⊥ ∪ b)
29	24, 28	ax-r2 36	. . . . . . . . 9 ((a⊥ ∪ b) ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∪ b)
30	23, 29	ax-r2 36	. . . . . . . 8 ((b ∪ (a⊥ ∪ b)) ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∪ b)
31	18, 30	ax-r2 36	. . . . . . 7 ((b ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ b)) = (a⊥ ∪ b)
32		or32 82	. . . . . . . 8 ((b ∪ (a⊥ ∩ b⊥ )) ∪ a) = ((b ∪ a) ∪ (a⊥ ∩ b⊥ ))
33		ancom 74	. . . . . . . . . . 11 (a⊥ ∩ b⊥ ) = (b⊥ ∩ a⊥ )
34		oran 87	. . . . . . . . . . . . 13 (b ∪ a) = (b⊥ ∩ a⊥ )⊥
35	34	con2 67	. . . . . . . . . . . 12 (b ∪ a)⊥ = (b⊥ ∩ a⊥ )
36	35	ax-r1 35	. . . . . . . . . . 11 (b⊥ ∩ a⊥ ) = (b ∪ a)⊥
37	33, 36	ax-r2 36	. . . . . . . . . 10 (a⊥ ∩ b⊥ ) = (b ∪ a)⊥
38	37	lor 70	. . . . . . . . 9 ((b ∪ a) ∪ (a⊥ ∩ b⊥ )) = ((b ∪ a) ∪ (b ∪ a)⊥ )
39		df-t 41	. . . . . . . . . 10 1 = ((b ∪ a) ∪ (b ∪ a)⊥ )
40	39	ax-r1 35	. . . . . . . . 9 ((b ∪ a) ∪ (b ∪ a)⊥ ) = 1
41	38, 40	ax-r2 36	. . . . . . . 8 ((b ∪ a) ∪ (a⊥ ∩ b⊥ )) = 1
42	32, 41	ax-r2 36	. . . . . . 7 ((b ∪ (a⊥ ∩ b⊥ )) ∪ a) = 1
43	31, 42	2an 79	. . . . . 6 (((b ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ b)) ∩ ((b ∪ (a⊥ ∩ b⊥ )) ∪ a)) = ((a⊥ ∪ b) ∩ 1)
44		an1 106	. . . . . 6 ((a⊥ ∪ b) ∩ 1) = (a⊥ ∪ b)
45	43, 44	ax-r2 36	. . . . 5 (((b ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∪ b)) ∩ ((b ∪ (a⊥ ∩ b⊥ )) ∪ a)) = (a⊥ ∪ b)
46	17, 45	ax-r2 36	. . . 4 ((b ∪ (a⊥ ∩ b⊥ )) ∪ ((a⊥ ∪ b) ∩ a)) = (a⊥ ∪ b)
47	10, 46	ax-r2 36	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ b) ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ b)
48	3, 47	ax-r2 36	. 2 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∪ b) = (a⊥ ∪ b)
49	2, 48	ax-r2 36	1 ((a →3 b) ∪ b) = (a⊥ ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →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:  u3lemnanb  657  neg3antlem2  865