Theorem ud5lem3b 592

Description: Lemma for unified disjunction. (Contributed by NM, 26-Nov-1997.)

Assertion

Ref	Expression
ud5lem3b	((a →5 b)⊥ ∩ (a ∪ (a⊥ ∩ b))) = (a ∩ (a⊥ ∪ b⊥ ))

Proof of Theorem ud5lem3b

Step	Hyp	Ref	Expression
1		ud5lem0c 281	. . 3 (a →5 b)⊥ = (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b))
2	1	ran 78	. 2 ((a →5 b)⊥ ∩ (a ∪ (a⊥ ∩ b))) = ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b)) ∩ (a ∪ (a⊥ ∩ b)))
3		comorr 184	. . . . . . 7 a⊥ C (a⊥ ∪ b⊥ )
4	3	comcom6 459	. . . . . 6 a C (a⊥ ∪ b⊥ )
5		comorr 184	. . . . . 6 a C (a ∪ b⊥ )
6	4, 5	com2an 484	. . . . 5 a C ((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ ))
7		comorr 184	. . . . 5 a C (a ∪ b)
8	6, 7	com2an 484	. . . 4 a C (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b))
9		comanr1 464	. . . . 5 a⊥ C (a⊥ ∩ b)
10	9	comcom6 459	. . . 4 a C (a⊥ ∩ b)
11	8, 10	fh2 470	. . 3 ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b)) ∩ (a ∪ (a⊥ ∩ b))) = (((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b)) ∩ a) ∪ ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b)) ∩ (a⊥ ∩ b)))
12		anass 76	. . . . . 6 ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b)) ∩ a) = (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a ∪ b) ∩ a))
13		ancom 74	. . . . . . . . 9 ((a ∪ b) ∩ a) = (a ∩ (a ∪ b))
14		anabs 121	. . . . . . . . 9 (a ∩ (a ∪ b)) = a
15	13, 14	ax-r2 36	. . . . . . . 8 ((a ∪ b) ∩ a) = a
16	15	lan 77	. . . . . . 7 (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a ∪ b) ∩ a)) = (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ a)
17		anass 76	. . . . . . . 8 (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ a) = ((a⊥ ∪ b⊥ ) ∩ ((a ∪ b⊥ ) ∩ a))
18		ancom 74	. . . . . . . . . . 11 ((a ∪ b⊥ ) ∩ a) = (a ∩ (a ∪ b⊥ ))
19		anabs 121	. . . . . . . . . . 11 (a ∩ (a ∪ b⊥ )) = a
20	18, 19	ax-r2 36	. . . . . . . . . 10 ((a ∪ b⊥ ) ∩ a) = a
21	20	lan 77	. . . . . . . . 9 ((a⊥ ∪ b⊥ ) ∩ ((a ∪ b⊥ ) ∩ a)) = ((a⊥ ∪ b⊥ ) ∩ a)
22		ancom 74	. . . . . . . . 9 ((a⊥ ∪ b⊥ ) ∩ a) = (a ∩ (a⊥ ∪ b⊥ ))
23	21, 22	ax-r2 36	. . . . . . . 8 ((a⊥ ∪ b⊥ ) ∩ ((a ∪ b⊥ ) ∩ a)) = (a ∩ (a⊥ ∪ b⊥ ))
24	17, 23	ax-r2 36	. . . . . . 7 (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ a) = (a ∩ (a⊥ ∪ b⊥ ))
25	16, 24	ax-r2 36	. . . . . 6 (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a ∪ b) ∩ a)) = (a ∩ (a⊥ ∪ b⊥ ))
26	12, 25	ax-r2 36	. . . . 5 ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b)) ∩ a) = (a ∩ (a⊥ ∪ b⊥ ))
27		an32 83	. . . . . 6 ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b)) ∩ (a⊥ ∩ b)) = ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a⊥ ∩ b)) ∩ (a ∪ b))
28		anass 76	. . . . . . . . 9 (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a⊥ ∩ b)) = ((a⊥ ∪ b⊥ ) ∩ ((a ∪ b⊥ ) ∩ (a⊥ ∩ b)))
29		anor2 89	. . . . . . . . . . . . 13 (a⊥ ∩ b) = (a ∪ b⊥ )⊥
30	29	lan 77	. . . . . . . . . . . 12 ((a ∪ b⊥ ) ∩ (a⊥ ∩ b)) = ((a ∪ b⊥ ) ∩ (a ∪ b⊥ )⊥ )
31		dff 101	. . . . . . . . . . . . 13 0 = ((a ∪ b⊥ ) ∩ (a ∪ b⊥ )⊥ )
32	31	ax-r1 35	. . . . . . . . . . . 12 ((a ∪ b⊥ ) ∩ (a ∪ b⊥ )⊥ ) = 0
33	30, 32	ax-r2 36	. . . . . . . . . . 11 ((a ∪ b⊥ ) ∩ (a⊥ ∩ b)) = 0
34	33	lan 77	. . . . . . . . . 10 ((a⊥ ∪ b⊥ ) ∩ ((a ∪ b⊥ ) ∩ (a⊥ ∩ b))) = ((a⊥ ∪ b⊥ ) ∩ 0)
35		an0 108	. . . . . . . . . 10 ((a⊥ ∪ b⊥ ) ∩ 0) = 0
36	34, 35	ax-r2 36	. . . . . . . . 9 ((a⊥ ∪ b⊥ ) ∩ ((a ∪ b⊥ ) ∩ (a⊥ ∩ b))) = 0
37	28, 36	ax-r2 36	. . . . . . . 8 (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a⊥ ∩ b)) = 0
38	37	ran 78	. . . . . . 7 ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a⊥ ∩ b)) ∩ (a ∪ b)) = (0 ∩ (a ∪ b))
39		an0r 109	. . . . . . 7 (0 ∩ (a ∪ b)) = 0
40	38, 39	ax-r2 36	. . . . . 6 ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a⊥ ∩ b)) ∩ (a ∪ b)) = 0
41	27, 40	ax-r2 36	. . . . 5 ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b)) ∩ (a⊥ ∩ b)) = 0
42	26, 41	2or 72	. . . 4 (((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b)) ∩ a) ∪ ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b)) ∩ (a⊥ ∩ b))) = ((a ∩ (a⊥ ∪ b⊥ )) ∪ 0)
43		or0 102	. . . 4 ((a ∩ (a⊥ ∪ b⊥ )) ∪ 0) = (a ∩ (a⊥ ∪ b⊥ ))
44	42, 43	ax-r2 36	. . 3 (((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b)) ∩ a) ∪ ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b)) ∩ (a⊥ ∩ b))) = (a ∩ (a⊥ ∪ b⊥ ))
45	11, 44	ax-r2 36	. 2 ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b)) ∩ (a ∪ (a⊥ ∩ b))) = (a ∩ (a⊥ ∪ b⊥ ))
46	2, 45	ax-r2 36	1 ((a →5 b)⊥ ∩ (a ∪ (a⊥ ∩ b))) = (a ∩ (a⊥ ∪ b⊥ ))

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →₅ wi5 16

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-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is used by:  ud5lem3  594