Theorem ud5lem3 594

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

Assertion

Ref	Expression
ud5lem3	((a →5 b) →5 (a ∪ (a⊥ ∩ b))) = (a ∪ b)

Proof of Theorem ud5lem3

Step	Hyp	Ref	Expression
1		df-i5 48	. 2 ((a →5 b) →5 (a ∪ (a⊥ ∩ b))) = ((((a →5 b) ∩ (a ∪ (a⊥ ∩ b))) ∪ ((a →5 b)⊥ ∩ (a ∪ (a⊥ ∩ b)))) ∪ ((a →5 b)⊥ ∩ (a ∪ (a⊥ ∩ b))⊥ ))
2		ud5lem3a 591	. . . . 5 ((a →5 b) ∩ (a ∪ (a⊥ ∩ b))) = ((a ∩ b) ∪ (a⊥ ∩ b))
3		ud5lem3b 592	. . . . 5 ((a →5 b)⊥ ∩ (a ∪ (a⊥ ∩ b))) = (a ∩ (a⊥ ∪ b⊥ ))
4	2, 3	2or 72	. . . 4 (((a →5 b) ∩ (a ∪ (a⊥ ∩ b))) ∪ ((a →5 b)⊥ ∩ (a ∪ (a⊥ ∩ b)))) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a ∩ (a⊥ ∪ b⊥ )))
5		ud5lem3c 593	. . . 4 ((a →5 b)⊥ ∩ (a ∪ (a⊥ ∩ b))⊥ ) = (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ )
6	4, 5	2or 72	. . 3 ((((a →5 b) ∩ (a ∪ (a⊥ ∩ b))) ∪ ((a →5 b)⊥ ∩ (a ∪ (a⊥ ∩ b)))) ∪ ((a →5 b)⊥ ∩ (a ∪ (a⊥ ∩ b))⊥ )) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a ∩ (a⊥ ∪ b⊥ ))) ∪ (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ ))
7		ax-a3 32	. . . 4 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a ∩ (a⊥ ∪ b⊥ ))) ∪ (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ )) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a ∩ (a⊥ ∪ b⊥ )) ∪ (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ )))
8		or4 84	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a ∩ (a⊥ ∪ b⊥ )) ∪ (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ ))) = (((a ∩ b) ∪ (a ∩ (a⊥ ∪ b⊥ ))) ∪ ((a⊥ ∩ b) ∪ (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ )))
9		comanr1 464	. . . . . . . . 9 a C (a ∩ b)
10		comorr 184	. . . . . . . . . 10 a⊥ C (a⊥ ∪ b⊥ )
11	10	comcom6 459	. . . . . . . . 9 a C (a⊥ ∪ b⊥ )
12	9, 11	fh4 472	. . . . . . . 8 ((a ∩ b) ∪ (a ∩ (a⊥ ∪ b⊥ ))) = (((a ∩ b) ∪ a) ∩ ((a ∩ b) ∪ (a⊥ ∪ b⊥ )))
13		ax-a2 31	. . . . . . . . . . 11 ((a ∩ b) ∪ a) = (a ∪ (a ∩ b))
14		orabs 120	. . . . . . . . . . 11 (a ∪ (a ∩ b)) = a
15	13, 14	ax-r2 36	. . . . . . . . . 10 ((a ∩ b) ∪ a) = a
16		df-a 40	. . . . . . . . . . . . . 14 (a ∩ b) = (a⊥ ∪ b⊥ )⊥
17	16	ax-r1 35	. . . . . . . . . . . . 13 (a⊥ ∪ b⊥ )⊥ = (a ∩ b)
18	17	con3 68	. . . . . . . . . . . 12 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
19	18	lor 70	. . . . . . . . . . 11 ((a ∩ b) ∪ (a⊥ ∪ b⊥ )) = ((a ∩ b) ∪ (a ∩ b)⊥ )
20		df-t 41	. . . . . . . . . . . 12 1 = ((a ∩ b) ∪ (a ∩ b)⊥ )
21	20	ax-r1 35	. . . . . . . . . . 11 ((a ∩ b) ∪ (a ∩ b)⊥ ) = 1
22	19, 21	ax-r2 36	. . . . . . . . . 10 ((a ∩ b) ∪ (a⊥ ∪ b⊥ )) = 1
23	15, 22	2an 79	. . . . . . . . 9 (((a ∩ b) ∪ a) ∩ ((a ∩ b) ∪ (a⊥ ∪ b⊥ ))) = (a ∩ 1)
24		an1 106	. . . . . . . . 9 (a ∩ 1) = a
25	23, 24	ax-r2 36	. . . . . . . 8 (((a ∩ b) ∪ a) ∩ ((a ∩ b) ∪ (a⊥ ∪ b⊥ ))) = a
26	12, 25	ax-r2 36	. . . . . . 7 ((a ∩ b) ∪ (a ∩ (a⊥ ∪ b⊥ ))) = a
27		coman1 185	. . . . . . . . . . . 12 (a⊥ ∩ b) C a⊥
28	27	comcom7 460	. . . . . . . . . . 11 (a⊥ ∩ b) C a
29		coman2 186	. . . . . . . . . . 11 (a⊥ ∩ b) C b
30	28, 29	com2or 483	. . . . . . . . . 10 (a⊥ ∩ b) C (a ∪ b)
31	29	comcom2 183	. . . . . . . . . . 11 (a⊥ ∩ b) C b⊥
32	28, 31	com2or 483	. . . . . . . . . 10 (a⊥ ∩ b) C (a ∪ b⊥ )
33	30, 32	com2an 484	. . . . . . . . 9 (a⊥ ∩ b) C ((a ∪ b) ∩ (a ∪ b⊥ ))
34	33, 27	fh3 471	. . . . . . . 8 ((a⊥ ∩ b) ∪ (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ )) = (((a⊥ ∩ b) ∪ ((a ∪ b) ∩ (a ∪ b⊥ ))) ∩ ((a⊥ ∩ b) ∪ a⊥ ))
35		comor1 461	. . . . . . . . . . . . . 14 (a ∪ b) C a
36	35	comcom2 183	. . . . . . . . . . . . 13 (a ∪ b) C a⊥
37		comor2 462	. . . . . . . . . . . . 13 (a ∪ b) C b
38	36, 37	com2an 484	. . . . . . . . . . . 12 (a ∪ b) C (a⊥ ∩ b)
39	37	comcom2 183	. . . . . . . . . . . . 13 (a ∪ b) C b⊥
40	35, 39	com2or 483	. . . . . . . . . . . 12 (a ∪ b) C (a ∪ b⊥ )
41	38, 40	fh4 472	. . . . . . . . . . 11 ((a⊥ ∩ b) ∪ ((a ∪ b) ∩ (a ∪ b⊥ ))) = (((a⊥ ∩ b) ∪ (a ∪ b)) ∩ ((a⊥ ∩ b) ∪ (a ∪ b⊥ )))
42	36, 37	fh3r 475	. . . . . . . . . . . . . 14 ((a⊥ ∩ b) ∪ (a ∪ b)) = ((a⊥ ∪ (a ∪ b)) ∩ (b ∪ (a ∪ b)))
43		ax-a2 31	. . . . . . . . . . . . . . . 16 (a⊥ ∪ (a ∪ b)) = ((a ∪ b) ∪ a⊥ )
44		or12 80	. . . . . . . . . . . . . . . . 17 (b ∪ (a ∪ b)) = (a ∪ (b ∪ b))
45		oridm 110	. . . . . . . . . . . . . . . . . 18 (b ∪ b) = b
46	45	lor 70	. . . . . . . . . . . . . . . . 17 (a ∪ (b ∪ b)) = (a ∪ b)
47	44, 46	ax-r2 36	. . . . . . . . . . . . . . . 16 (b ∪ (a ∪ b)) = (a ∪ b)
48	43, 47	2an 79	. . . . . . . . . . . . . . 15 ((a⊥ ∪ (a ∪ b)) ∩ (b ∪ (a ∪ b))) = (((a ∪ b) ∪ a⊥ ) ∩ (a ∪ b))
49		ancom 74	. . . . . . . . . . . . . . . 16 (((a ∪ b) ∪ a⊥ ) ∩ (a ∪ b)) = ((a ∪ b) ∩ ((a ∪ b) ∪ a⊥ ))
50		anabs 121	. . . . . . . . . . . . . . . 16 ((a ∪ b) ∩ ((a ∪ b) ∪ a⊥ )) = (a ∪ b)
51	49, 50	ax-r2 36	. . . . . . . . . . . . . . 15 (((a ∪ b) ∪ a⊥ ) ∩ (a ∪ b)) = (a ∪ b)
52	48, 51	ax-r2 36	. . . . . . . . . . . . . 14 ((a⊥ ∪ (a ∪ b)) ∩ (b ∪ (a ∪ b))) = (a ∪ b)
53	42, 52	ax-r2 36	. . . . . . . . . . . . 13 ((a⊥ ∩ b) ∪ (a ∪ b)) = (a ∪ b)
54		anor2 89	. . . . . . . . . . . . . . . . 17 (a⊥ ∩ b) = (a ∪ b⊥ )⊥
55	54	ax-r1 35	. . . . . . . . . . . . . . . 16 (a ∪ b⊥ )⊥ = (a⊥ ∩ b)
56	55	con3 68	. . . . . . . . . . . . . . 15 (a ∪ b⊥ ) = (a⊥ ∩ b)⊥
57	56	lor 70	. . . . . . . . . . . . . 14 ((a⊥ ∩ b) ∪ (a ∪ b⊥ )) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b)⊥ )
58		df-t 41	. . . . . . . . . . . . . . 15 1 = ((a⊥ ∩ b) ∪ (a⊥ ∩ b)⊥ )
59	58	ax-r1 35	. . . . . . . . . . . . . 14 ((a⊥ ∩ b) ∪ (a⊥ ∩ b)⊥ ) = 1
60	57, 59	ax-r2 36	. . . . . . . . . . . . 13 ((a⊥ ∩ b) ∪ (a ∪ b⊥ )) = 1
61	53, 60	2an 79	. . . . . . . . . . . 12 (((a⊥ ∩ b) ∪ (a ∪ b)) ∩ ((a⊥ ∩ b) ∪ (a ∪ b⊥ ))) = ((a ∪ b) ∩ 1)
62		an1 106	. . . . . . . . . . . 12 ((a ∪ b) ∩ 1) = (a ∪ b)
63	61, 62	ax-r2 36	. . . . . . . . . . 11 (((a⊥ ∩ b) ∪ (a ∪ b)) ∩ ((a⊥ ∩ b) ∪ (a ∪ b⊥ ))) = (a ∪ b)
64	41, 63	ax-r2 36	. . . . . . . . . 10 ((a⊥ ∩ b) ∪ ((a ∪ b) ∩ (a ∪ b⊥ ))) = (a ∪ b)
65		ax-a2 31	. . . . . . . . . . 11 ((a⊥ ∩ b) ∪ a⊥ ) = (a⊥ ∪ (a⊥ ∩ b))
66		orabs 120	. . . . . . . . . . 11 (a⊥ ∪ (a⊥ ∩ b)) = a⊥
67	65, 66	ax-r2 36	. . . . . . . . . 10 ((a⊥ ∩ b) ∪ a⊥ ) = a⊥
68	64, 67	2an 79	. . . . . . . . 9 (((a⊥ ∩ b) ∪ ((a ∪ b) ∩ (a ∪ b⊥ ))) ∩ ((a⊥ ∩ b) ∪ a⊥ )) = ((a ∪ b) ∩ a⊥ )
69		ancom 74	. . . . . . . . 9 ((a ∪ b) ∩ a⊥ ) = (a⊥ ∩ (a ∪ b))
70	68, 69	ax-r2 36	. . . . . . . 8 (((a⊥ ∩ b) ∪ ((a ∪ b) ∩ (a ∪ b⊥ ))) ∩ ((a⊥ ∩ b) ∪ a⊥ )) = (a⊥ ∩ (a ∪ b))
71	34, 70	ax-r2 36	. . . . . . 7 ((a⊥ ∩ b) ∪ (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ )) = (a⊥ ∩ (a ∪ b))
72	26, 71	2or 72	. . . . . 6 (((a ∩ b) ∪ (a ∩ (a⊥ ∪ b⊥ ))) ∪ ((a⊥ ∩ b) ∪ (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ ))) = (a ∪ (a⊥ ∩ (a ∪ b)))
73		oml 445	. . . . . 6 (a ∪ (a⊥ ∩ (a ∪ b))) = (a ∪ b)
74	72, 73	ax-r2 36	. . . . 5 (((a ∩ b) ∪ (a ∩ (a⊥ ∪ b⊥ ))) ∪ ((a⊥ ∩ b) ∪ (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ ))) = (a ∪ b)
75	8, 74	ax-r2 36	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a ∩ (a⊥ ∪ b⊥ )) ∪ (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ ))) = (a ∪ b)
76	7, 75	ax-r2 36	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a ∩ (a⊥ ∪ b⊥ ))) ∪ (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ )) = (a ∪ b)
77	6, 76	ax-r2 36	. 2 ((((a →5 b) ∩ (a ∪ (a⊥ ∩ b))) ∪ ((a →5 b)⊥ ∩ (a ∪ (a⊥ ∩ b)))) ∪ ((a →5 b)⊥ ∩ (a ∪ (a⊥ ∩ b))⊥ )) = (a ∪ b)
78	1, 77	ax-r2 36	1 ((a →5 b) →5 (a ∪ (a⊥ ∩ b))) = (a ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₅ wi5 16

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

This theorem is referenced by:  ud5  599