Theorem u3lemonb 619

Description: Lemma for Kalmbach implication study.

Assertion

Ref	Expression
u3lemonb	((a →3 b) ∪ b⊥ ) = 1

Proof of Theorem u3lemonb

Step	Hyp	Ref	Expression
1		df-i3 45	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2	1	ax-r5 37	. 2 ((a →3 b) ∪ b⊥ ) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∪ b⊥ )
3		or32 75	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∪ b⊥ ) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ ) ∪ (a ∩ (a⊥ ∪ b)))
4		ax-a3 31	. . . . . 6 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ ) = ((a⊥ ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∪ b⊥ ))
5		lear 153	. . . . . . . 8 (a⊥ ∩ b⊥ ) ≤ b⊥
6	5	df-le2 123	. . . . . . 7 ((a⊥ ∩ b⊥ ) ∪ b⊥ ) = b⊥
7	6	lor 66	. . . . . 6 ((a⊥ ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∪ b⊥ )) = ((a⊥ ∩ b) ∪ b⊥ )
8	4, 7	ax-r2 35	. . . . 5 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ ) = ((a⊥ ∩ b) ∪ b⊥ )
9		ancom 68	. . . . 5 (a ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∩ a)
10	8, 9	2or 67	. . . 4 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ ) ∪ (a ∩ (a⊥ ∪ b))) = (((a⊥ ∩ b) ∪ b⊥ ) ∪ ((a⊥ ∪ b) ∩ a))
11		comor1 443	. . . . . . . 8 (a⊥ ∪ b) C a⊥
12		comor2 444	. . . . . . . 8 (a⊥ ∪ b) C b
13	11, 12	com2an 466	. . . . . . 7 (a⊥ ∪ b) C (a⊥ ∩ b)
14	12	comcom2 175	. . . . . . 7 (a⊥ ∪ b) C b⊥
15	13, 14	com2or 465	. . . . . 6 (a⊥ ∪ b) C ((a⊥ ∩ b) ∪ b⊥ )
16	11	comcom7 442	. . . . . 6 (a⊥ ∪ b) C a
17	15, 16	fh4 454	. . . . 5 (((a⊥ ∩ b) ∪ b⊥ ) ∪ ((a⊥ ∪ b) ∩ a)) = ((((a⊥ ∩ b) ∪ b⊥ ) ∪ (a⊥ ∪ b)) ∩ (((a⊥ ∩ b) ∪ b⊥ ) ∪ a))
18		ax-a3 31	. . . . . . . 8 (((a⊥ ∩ b) ∪ b⊥ ) ∪ (a⊥ ∪ b)) = ((a⊥ ∩ b) ∪ (b⊥ ∪ (a⊥ ∪ b)))
19		ax-a2 30	. . . . . . . . . . 11 (b⊥ ∪ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∪ b⊥ )
20		ax-a3 31	. . . . . . . . . . . 12 ((a⊥ ∪ b) ∪ b⊥ ) = (a⊥ ∪ (b ∪ b⊥ ))
21		df-t 40	. . . . . . . . . . . . . . 15 1 = (b ∪ b⊥ )
22	21	ax-r1 34	. . . . . . . . . . . . . 14 (b ∪ b⊥ ) = 1
23	22	lor 66	. . . . . . . . . . . . 13 (a⊥ ∪ (b ∪ b⊥ )) = (a⊥ ∪ 1)
24		or1 96	. . . . . . . . . . . . 13 (a⊥ ∪ 1) = 1
25	23, 24	ax-r2 35	. . . . . . . . . . . 12 (a⊥ ∪ (b ∪ b⊥ )) = 1
26	20, 25	ax-r2 35	. . . . . . . . . . 11 ((a⊥ ∪ b) ∪ b⊥ ) = 1
27	19, 26	ax-r2 35	. . . . . . . . . 10 (b⊥ ∪ (a⊥ ∪ b)) = 1
28	27	lor 66	. . . . . . . . 9 ((a⊥ ∩ b) ∪ (b⊥ ∪ (a⊥ ∪ b))) = ((a⊥ ∩ b) ∪ 1)
29		or1 96	. . . . . . . . 9 ((a⊥ ∩ b) ∪ 1) = 1
30	28, 29	ax-r2 35	. . . . . . . 8 ((a⊥ ∩ b) ∪ (b⊥ ∪ (a⊥ ∪ b))) = 1
31	18, 30	ax-r2 35	. . . . . . 7 (((a⊥ ∩ b) ∪ b⊥ ) ∪ (a⊥ ∪ b)) = 1
32		ax-a3 31	. . . . . . . 8 (((a⊥ ∩ b) ∪ b⊥ ) ∪ a) = ((a⊥ ∩ b) ∪ (b⊥ ∪ a))
33		ancom 68	. . . . . . . . . . . . 13 (a⊥ ∩ b) = (b ∩ a⊥ )
34		anor1 80	. . . . . . . . . . . . 13 (b ∩ a⊥ ) = (b⊥ ∪ a)⊥
35	33, 34	ax-r2 35	. . . . . . . . . . . 12 (a⊥ ∩ b) = (b⊥ ∪ a)⊥
36	35	con2 64	. . . . . . . . . . 11 (a⊥ ∩ b)⊥ = (b⊥ ∪ a)
37	36	ax-r1 34	. . . . . . . . . 10 (b⊥ ∪ a) = (a⊥ ∩ b)⊥
38	37	lor 66	. . . . . . . . 9 ((a⊥ ∩ b) ∪ (b⊥ ∪ a)) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b)⊥ )
39		df-t 40	. . . . . . . . . 10 1 = ((a⊥ ∩ b) ∪ (a⊥ ∩ b)⊥ )
40	39	ax-r1 34	. . . . . . . . 9 ((a⊥ ∩ b) ∪ (a⊥ ∩ b)⊥ ) = 1
41	38, 40	ax-r2 35	. . . . . . . 8 ((a⊥ ∩ b) ∪ (b⊥ ∪ a)) = 1
42	32, 41	ax-r2 35	. . . . . . 7 (((a⊥ ∩ b) ∪ b⊥ ) ∪ a) = 1
43	31, 42	2an 72	. . . . . 6 ((((a⊥ ∩ b) ∪ b⊥ ) ∪ (a⊥ ∪ b)) ∩ (((a⊥ ∩ b) ∪ b⊥ ) ∪ a)) = (1 ∩ 1)
44		an1 98	. . . . . 6 (1 ∩ 1) = 1
45	43, 44	ax-r2 35	. . . . 5 ((((a⊥ ∩ b) ∪ b⊥ ) ∪ (a⊥ ∪ b)) ∩ (((a⊥ ∩ b) ∪ b⊥ ) ∪ a)) = 1
46	17, 45	ax-r2 35	. . . 4 (((a⊥ ∩ b) ∪ b⊥ ) ∪ ((a⊥ ∪ b) ∩ a)) = 1
47	10, 46	ax-r2 35	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ ) ∪ (a ∩ (a⊥ ∪ b))) = 1
48	3, 47	ax-r2 35	. 2 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∪ b⊥ ) = 1
49	2, 48	ax-r2 35	1 ((a →3 b) ∪ b⊥ ) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₃ wi3 15

This theorem is referenced by:  u3lemnab 634  u3lem3 733  u3lem4 740

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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