Theorem u1lem11 780

Description: Lemma used in study of orthoarguesian law. (Contributed by NM, 28-Dec-1998.)

Assertion

Ref	Expression
u1lem11	((a⊥ →1 b) →1 b) = (a →1 b)

Proof of Theorem u1lem11

Step	Hyp	Ref	Expression
1		ud1lem0c 277	. . . . 5 (a⊥ →1 b)⊥ = (a⊥ ∩ (a⊥ ⊥ ∪ b⊥ ))
2		ax-a1 30	. . . . . . . 8 a = a⊥ ⊥
3	2	ax-r1 35	. . . . . . 7 a⊥ ⊥ = a
4	3	ax-r5 38	. . . . . 6 (a⊥ ⊥ ∪ b⊥ ) = (a ∪ b⊥ )
5	4	lan 77	. . . . 5 (a⊥ ∩ (a⊥ ⊥ ∪ b⊥ )) = (a⊥ ∩ (a ∪ b⊥ ))
6	1, 5	ax-r2 36	. . . 4 (a⊥ →1 b)⊥ = (a⊥ ∩ (a ∪ b⊥ ))
7		u1lemab 610	. . . . 5 ((a⊥ →1 b) ∩ b) = ((a⊥ ∩ b) ∪ (a⊥ ⊥ ∩ b))
8		ax-a2 31	. . . . 5 ((a⊥ ∩ b) ∪ (a⊥ ⊥ ∩ b)) = ((a⊥ ⊥ ∩ b) ∪ (a⊥ ∩ b))
9	2	ran 78	. . . . . . 7 (a ∩ b) = (a⊥ ⊥ ∩ b)
10	9	ax-r5 38	. . . . . 6 ((a ∩ b) ∪ (a⊥ ∩ b)) = ((a⊥ ⊥ ∩ b) ∪ (a⊥ ∩ b))
11	10	ax-r1 35	. . . . 5 ((a⊥ ⊥ ∩ b) ∪ (a⊥ ∩ b)) = ((a ∩ b) ∪ (a⊥ ∩ b))
12	7, 8, 11	3tr 65	. . . 4 ((a⊥ →1 b) ∩ b) = ((a ∩ b) ∪ (a⊥ ∩ b))
13	6, 12	2or 72	. . 3 ((a⊥ →1 b)⊥ ∪ ((a⊥ →1 b) ∩ b)) = ((a⊥ ∩ (a ∪ b⊥ )) ∪ ((a ∩ b) ∪ (a⊥ ∩ b)))
14		comanr1 464	. . . . . . 7 a C (a ∩ b)
15	14	comcom3 454	. . . . . 6 a⊥ C (a ∩ b)
16		comanr1 464	. . . . . 6 a⊥ C (a⊥ ∩ b)
17	15, 16	com2or 483	. . . . 5 a⊥ C ((a ∩ b) ∪ (a⊥ ∩ b))
18	17	comcom 453	. . . 4 ((a ∩ b) ∪ (a⊥ ∩ b)) C a⊥
19		comor1 461	. . . . . . 7 (a ∪ b⊥ ) C a
20		comor2 462	. . . . . . . 8 (a ∪ b⊥ ) C b⊥
21	20	comcom7 460	. . . . . . 7 (a ∪ b⊥ ) C b
22	19, 21	com2an 484	. . . . . 6 (a ∪ b⊥ ) C (a ∩ b)
23	19	comcom2 183	. . . . . . 7 (a ∪ b⊥ ) C a⊥
24	23, 21	com2an 484	. . . . . 6 (a ∪ b⊥ ) C (a⊥ ∩ b)
25	22, 24	com2or 483	. . . . 5 (a ∪ b⊥ ) C ((a ∩ b) ∪ (a⊥ ∩ b))
26	25	comcom 453	. . . 4 ((a ∩ b) ∪ (a⊥ ∩ b)) C (a ∪ b⊥ )
27	18, 26	fh3r 475	. . 3 ((a⊥ ∩ (a ∪ b⊥ )) ∪ ((a ∩ b) ∪ (a⊥ ∩ b))) = ((a⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b))) ∩ ((a ∪ b⊥ ) ∪ ((a ∩ b) ∪ (a⊥ ∩ b))))
28		or32 82	. . . . . 6 ((a⊥ ∪ (a ∩ b)) ∪ (a⊥ ∩ b)) = ((a⊥ ∪ (a⊥ ∩ b)) ∪ (a ∩ b))
29		ax-a3 32	. . . . . 6 ((a⊥ ∪ (a ∩ b)) ∪ (a⊥ ∩ b)) = (a⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b)))
30		orabs 120	. . . . . . 7 (a⊥ ∪ (a⊥ ∩ b)) = a⊥
31	30	ax-r5 38	. . . . . 6 ((a⊥ ∪ (a⊥ ∩ b)) ∪ (a ∩ b)) = (a⊥ ∪ (a ∩ b))
32	28, 29, 31	3tr2 64	. . . . 5 (a⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b))) = (a⊥ ∪ (a ∩ b))
33		or12 80	. . . . . 6 ((a ∪ b⊥ ) ∪ ((a ∩ b) ∪ (a⊥ ∩ b))) = ((a ∩ b) ∪ ((a ∪ b⊥ ) ∪ (a⊥ ∩ b)))
34		anor2 89	. . . . . . . . 9 (a⊥ ∩ b) = (a ∪ b⊥ )⊥
35	34	lor 70	. . . . . . . 8 ((a ∪ b⊥ ) ∪ (a⊥ ∩ b)) = ((a ∪ b⊥ ) ∪ (a ∪ b⊥ )⊥ )
36		df-t 41	. . . . . . . . 9 1 = ((a ∪ b⊥ ) ∪ (a ∪ b⊥ )⊥ )
37	36	ax-r1 35	. . . . . . . 8 ((a ∪ b⊥ ) ∪ (a ∪ b⊥ )⊥ ) = 1
38	35, 37	ax-r2 36	. . . . . . 7 ((a ∪ b⊥ ) ∪ (a⊥ ∩ b)) = 1
39	38	lor 70	. . . . . 6 ((a ∩ b) ∪ ((a ∪ b⊥ ) ∪ (a⊥ ∩ b))) = ((a ∩ b) ∪ 1)
40		or1 104	. . . . . 6 ((a ∩ b) ∪ 1) = 1
41	33, 39, 40	3tr 65	. . . . 5 ((a ∪ b⊥ ) ∪ ((a ∩ b) ∪ (a⊥ ∩ b))) = 1
42	32, 41	2an 79	. . . 4 ((a⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b))) ∩ ((a ∪ b⊥ ) ∪ ((a ∩ b) ∪ (a⊥ ∩ b)))) = ((a⊥ ∪ (a ∩ b)) ∩ 1)
43		an1 106	. . . 4 ((a⊥ ∪ (a ∩ b)) ∩ 1) = (a⊥ ∪ (a ∩ b))
44	42, 43	ax-r2 36	. . 3 ((a⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b))) ∩ ((a ∪ b⊥ ) ∪ ((a ∩ b) ∪ (a⊥ ∩ b)))) = (a⊥ ∪ (a ∩ b))
45	13, 27, 44	3tr 65	. 2 ((a⊥ →1 b)⊥ ∪ ((a⊥ →1 b) ∩ b)) = (a⊥ ∪ (a ∩ b))
46		df-i1 44	. 2 ((a⊥ →1 b) →1 b) = ((a⊥ →1 b)⊥ ∪ ((a⊥ →1 b) ∩ b))
47		df-i1 44	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
48	45, 46, 47	3tr1 63	1 ((a⊥ →1 b) →1 b) = (a →1 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12

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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u1lem12  781  2oai1u  822  1oath1i1u  828  oa4to4u  973  3oa2  1024