Theorem u3lemanb 617

Description: Lemma for Kalmbach implication study. (Contributed by NM, 14-Dec-1997.)

Assertion

Ref	Expression
u3lemanb	((a →3 b) ∩ b⊥ ) = (a⊥ ∩ b⊥ )

Proof of Theorem u3lemanb

Step	Hyp	Ref	Expression
1		df-i3 46	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2	1	ran 78	. 2 ((a →3 b) ∩ b⊥ ) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ b⊥ )
3		comanr2 465	. . . . . . 7 b C (a⊥ ∩ b)
4	3	comcom3 454	. . . . . 6 b⊥ C (a⊥ ∩ b)
5		comanr2 465	. . . . . 6 b⊥ C (a⊥ ∩ b⊥ )
6	4, 5	com2or 483	. . . . 5 b⊥ C ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
7	6	comcom 453	. . . 4 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) C b⊥
8		coman1 185	. . . . . . . . 9 (a⊥ ∩ b) C a⊥
9	8	comcom7 460	. . . . . . . 8 (a⊥ ∩ b) C a
10		coman2 186	. . . . . . . . 9 (a⊥ ∩ b) C b
11	8, 10	com2or 483	. . . . . . . 8 (a⊥ ∩ b) C (a⊥ ∪ b)
12	9, 11	com2an 484	. . . . . . 7 (a⊥ ∩ b) C (a ∩ (a⊥ ∪ b))
13	12	comcom 453	. . . . . 6 (a ∩ (a⊥ ∪ b)) C (a⊥ ∩ b)
14		coman1 185	. . . . . . . . 9 (a⊥ ∩ b⊥ ) C a⊥
15	14	comcom7 460	. . . . . . . 8 (a⊥ ∩ b⊥ ) C a
16		coman2 186	. . . . . . . . . 10 (a⊥ ∩ b⊥ ) C b⊥
17	16	comcom7 460	. . . . . . . . 9 (a⊥ ∩ b⊥ ) C b
18	14, 17	com2or 483	. . . . . . . 8 (a⊥ ∩ b⊥ ) C (a⊥ ∪ b)
19	15, 18	com2an 484	. . . . . . 7 (a⊥ ∩ b⊥ ) C (a ∩ (a⊥ ∪ b))
20	19	comcom 453	. . . . . 6 (a ∩ (a⊥ ∪ b)) C (a⊥ ∩ b⊥ )
21	13, 20	com2or 483	. . . . 5 (a ∩ (a⊥ ∪ b)) C ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
22	21	comcom 453	. . . 4 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) C (a ∩ (a⊥ ∪ b))
23	7, 22	fh2r 474	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ b⊥ ) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b⊥ ) ∪ ((a ∩ (a⊥ ∪ b)) ∩ b⊥ ))
24	10	comcom2 183	. . . . . . 7 (a⊥ ∩ b) C b⊥
25	8, 24	com2an 484	. . . . . . 7 (a⊥ ∩ b) C (a⊥ ∩ b⊥ )
26	24, 25	fh2r 474	. . . . . 6 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b⊥ ) = (((a⊥ ∩ b) ∩ b⊥ ) ∪ ((a⊥ ∩ b⊥ ) ∩ b⊥ ))
27		ax-a2 31	. . . . . . 7 (((a⊥ ∩ b) ∩ b⊥ ) ∪ ((a⊥ ∩ b⊥ ) ∩ b⊥ )) = (((a⊥ ∩ b⊥ ) ∩ b⊥ ) ∪ ((a⊥ ∩ b) ∩ b⊥ ))
28		anass 76	. . . . . . . . . 10 ((a⊥ ∩ b⊥ ) ∩ b⊥ ) = (a⊥ ∩ (b⊥ ∩ b⊥ ))
29		anidm 111	. . . . . . . . . . 11 (b⊥ ∩ b⊥ ) = b⊥
30	29	lan 77	. . . . . . . . . 10 (a⊥ ∩ (b⊥ ∩ b⊥ )) = (a⊥ ∩ b⊥ )
31	28, 30	ax-r2 36	. . . . . . . . 9 ((a⊥ ∩ b⊥ ) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
32		anass 76	. . . . . . . . . 10 ((a⊥ ∩ b) ∩ b⊥ ) = (a⊥ ∩ (b ∩ b⊥ ))
33		dff 101	. . . . . . . . . . . . 13 0 = (b ∩ b⊥ )
34	33	lan 77	. . . . . . . . . . . 12 (a⊥ ∩ 0) = (a⊥ ∩ (b ∩ b⊥ ))
35	34	ax-r1 35	. . . . . . . . . . 11 (a⊥ ∩ (b ∩ b⊥ )) = (a⊥ ∩ 0)
36		an0 108	. . . . . . . . . . 11 (a⊥ ∩ 0) = 0
37	35, 36	ax-r2 36	. . . . . . . . . 10 (a⊥ ∩ (b ∩ b⊥ )) = 0
38	32, 37	ax-r2 36	. . . . . . . . 9 ((a⊥ ∩ b) ∩ b⊥ ) = 0
39	31, 38	2or 72	. . . . . . . 8 (((a⊥ ∩ b⊥ ) ∩ b⊥ ) ∪ ((a⊥ ∩ b) ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ 0)
40		or0 102	. . . . . . . 8 ((a⊥ ∩ b⊥ ) ∪ 0) = (a⊥ ∩ b⊥ )
41	39, 40	ax-r2 36	. . . . . . 7 (((a⊥ ∩ b⊥ ) ∩ b⊥ ) ∪ ((a⊥ ∩ b) ∩ b⊥ )) = (a⊥ ∩ b⊥ )
42	27, 41	ax-r2 36	. . . . . 6 (((a⊥ ∩ b) ∩ b⊥ ) ∪ ((a⊥ ∩ b⊥ ) ∩ b⊥ )) = (a⊥ ∩ b⊥ )
43	26, 42	ax-r2 36	. . . . 5 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
44		an32 83	. . . . . 6 ((a ∩ (a⊥ ∪ b)) ∩ b⊥ ) = ((a ∩ b⊥ ) ∩ (a⊥ ∪ b))
45		ancom 74	. . . . . . 7 ((a ∩ b⊥ ) ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∩ (a ∩ b⊥ ))
46		anor1 88	. . . . . . . . 9 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
47	46	lan 77	. . . . . . . 8 ((a⊥ ∪ b) ∩ (a ∩ b⊥ )) = ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ )
48		dff 101	. . . . . . . . 9 0 = ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ )
49	48	ax-r1 35	. . . . . . . 8 ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ ) = 0
50	47, 49	ax-r2 36	. . . . . . 7 ((a⊥ ∪ b) ∩ (a ∩ b⊥ )) = 0
51	45, 50	ax-r2 36	. . . . . 6 ((a ∩ b⊥ ) ∩ (a⊥ ∪ b)) = 0
52	44, 51	ax-r2 36	. . . . 5 ((a ∩ (a⊥ ∪ b)) ∩ b⊥ ) = 0
53	43, 52	2or 72	. . . 4 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b⊥ ) ∪ ((a ∩ (a⊥ ∪ b)) ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ 0)
54	53, 40	ax-r2 36	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b⊥ ) ∪ ((a ∩ (a⊥ ∪ b)) ∩ b⊥ )) = (a⊥ ∩ b⊥ )
55	23, 54	ax-r2 36	. 2 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
56	2, 55	ax-r2 36	1 ((a →3 b) ∩ b⊥ ) = (a⊥ ∩ b⊥ )

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →₃ wi3 14

This proof depends on 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 proof 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 used by:  u3lemnob  672  u3lem3  751  u3lem13b  790  neg3antlem2  865