Theorem ud1lem3 562

Description: Lemma for unified disjunction. (Contributed by NM, 23-Nov-1997.)

Assertion

Ref	Expression
ud1lem3	((a →1 b) →1 (a ∪ b)) = (a ∪ b)

Proof of Theorem ud1lem3

Step	Hyp	Ref	Expression
1		df-i1 44	. 2 ((a →1 b) →1 (a ∪ b)) = ((a →1 b)⊥ ∪ ((a →1 b) ∩ (a ∪ b)))
2		ud1lem0c 277	. . . 4 (a →1 b)⊥ = (a ∩ (a⊥ ∪ b⊥ ))
3	2	con3 68	. . . . 5 (a →1 b) = (a ∩ (a⊥ ∪ b⊥ ))⊥
4	3	ran 78	. . . 4 ((a →1 b) ∩ (a ∪ b)) = ((a ∩ (a⊥ ∪ b⊥ ))⊥ ∩ (a ∪ b))
5	2, 4	2or 72	. . 3 ((a →1 b)⊥ ∪ ((a →1 b) ∩ (a ∪ b))) = ((a ∩ (a⊥ ∪ b⊥ )) ∪ ((a ∩ (a⊥ ∪ b⊥ ))⊥ ∩ (a ∪ b)))
6		comid 187	. . . . . 6 (a ∩ (a⊥ ∪ b⊥ )) C (a ∩ (a⊥ ∪ b⊥ ))
7	6	comcom2 183	. . . . 5 (a ∩ (a⊥ ∪ b⊥ )) C (a ∩ (a⊥ ∪ b⊥ ))⊥
8		comor1 461	. . . . . . 7 (a ∪ b) C a
9	8	comcom2 183	. . . . . . . 8 (a ∪ b) C a⊥
10		comor2 462	. . . . . . . . 9 (a ∪ b) C b
11	10	comcom2 183	. . . . . . . 8 (a ∪ b) C b⊥
12	9, 11	com2or 483	. . . . . . 7 (a ∪ b) C (a⊥ ∪ b⊥ )
13	8, 12	com2an 484	. . . . . 6 (a ∪ b) C (a ∩ (a⊥ ∪ b⊥ ))
14	13	comcom 453	. . . . 5 (a ∩ (a⊥ ∪ b⊥ )) C (a ∪ b)
15	7, 14	fh3 471	. . . 4 ((a ∩ (a⊥ ∪ b⊥ )) ∪ ((a ∩ (a⊥ ∪ b⊥ ))⊥ ∩ (a ∪ b))) = (((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∩ (a⊥ ∪ b⊥ ))⊥ ) ∩ ((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∪ b)))
16		ancom 74	. . . . 5 (((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∩ (a⊥ ∪ b⊥ ))⊥ ) ∩ ((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∪ b))) = (((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∪ b)) ∩ ((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∩ (a⊥ ∪ b⊥ ))⊥ ))
17		df-t 41	. . . . . . . 8 1 = ((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∩ (a⊥ ∪ b⊥ ))⊥ )
18	17	ax-r1 35	. . . . . . 7 ((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∩ (a⊥ ∪ b⊥ ))⊥ ) = 1
19	18	lan 77	. . . . . 6 (((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∪ b)) ∩ ((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∩ (a⊥ ∪ b⊥ ))⊥ )) = (((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∪ b)) ∩ 1)
20		an1 106	. . . . . . 7 (((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∪ b)) ∩ 1) = ((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∪ b))
21		comorr 184	. . . . . . . . 9 a C (a ∪ b)
22		comorr 184	. . . . . . . . . . 11 a⊥ C (a⊥ ∪ b⊥ )
23	22	comcom2 183	. . . . . . . . . 10 a⊥ C (a⊥ ∪ b⊥ )⊥
24	23	comcom5 458	. . . . . . . . 9 a C (a⊥ ∪ b⊥ )
25	21, 24	fh4r 476	. . . . . . . 8 ((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∪ b)) = ((a ∪ (a ∪ b)) ∩ ((a⊥ ∪ b⊥ ) ∪ (a ∪ b)))
26		ax-a2 31	. . . . . . . . . . 11 ((a⊥ ∪ b⊥ ) ∪ (a ∪ b)) = ((a ∪ b) ∪ (a⊥ ∪ b⊥ ))
27		or4 84	. . . . . . . . . . . 12 ((a ∪ b) ∪ (a⊥ ∪ b⊥ )) = ((a ∪ a⊥ ) ∪ (b ∪ b⊥ ))
28		df-t 41	. . . . . . . . . . . . . . 15 1 = (b ∪ b⊥ )
29	28	ax-r1 35	. . . . . . . . . . . . . 14 (b ∪ b⊥ ) = 1
30	29	lor 70	. . . . . . . . . . . . 13 ((a ∪ a⊥ ) ∪ (b ∪ b⊥ )) = ((a ∪ a⊥ ) ∪ 1)
31		or1 104	. . . . . . . . . . . . 13 ((a ∪ a⊥ ) ∪ 1) = 1
32	30, 31	ax-r2 36	. . . . . . . . . . . 12 ((a ∪ a⊥ ) ∪ (b ∪ b⊥ )) = 1
33	27, 32	ax-r2 36	. . . . . . . . . . 11 ((a ∪ b) ∪ (a⊥ ∪ b⊥ )) = 1
34	26, 33	ax-r2 36	. . . . . . . . . 10 ((a⊥ ∪ b⊥ ) ∪ (a ∪ b)) = 1
35	34	lan 77	. . . . . . . . 9 ((a ∪ (a ∪ b)) ∩ ((a⊥ ∪ b⊥ ) ∪ (a ∪ b))) = ((a ∪ (a ∪ b)) ∩ 1)
36		an1 106	. . . . . . . . . 10 ((a ∪ (a ∪ b)) ∩ 1) = (a ∪ (a ∪ b))
37		ax-a3 32	. . . . . . . . . . . 12 ((a ∪ a) ∪ b) = (a ∪ (a ∪ b))
38	37	ax-r1 35	. . . . . . . . . . 11 (a ∪ (a ∪ b)) = ((a ∪ a) ∪ b)
39		oridm 110	. . . . . . . . . . . 12 (a ∪ a) = a
40	39	ax-r5 38	. . . . . . . . . . 11 ((a ∪ a) ∪ b) = (a ∪ b)
41	38, 40	ax-r2 36	. . . . . . . . . 10 (a ∪ (a ∪ b)) = (a ∪ b)
42	36, 41	ax-r2 36	. . . . . . . . 9 ((a ∪ (a ∪ b)) ∩ 1) = (a ∪ b)
43	35, 42	ax-r2 36	. . . . . . . 8 ((a ∪ (a ∪ b)) ∩ ((a⊥ ∪ b⊥ ) ∪ (a ∪ b))) = (a ∪ b)
44	25, 43	ax-r2 36	. . . . . . 7 ((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∪ b)) = (a ∪ b)
45	20, 44	ax-r2 36	. . . . . 6 (((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∪ b)) ∩ 1) = (a ∪ b)
46	19, 45	ax-r2 36	. . . . 5 (((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∪ b)) ∩ ((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∩ (a⊥ ∪ b⊥ ))⊥ )) = (a ∪ b)
47	16, 46	ax-r2 36	. . . 4 (((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∩ (a⊥ ∪ b⊥ ))⊥ ) ∩ ((a ∩ (a⊥ ∪ b⊥ )) ∪ (a ∪ b))) = (a ∪ b)
48	15, 47	ax-r2 36	. . 3 ((a ∩ (a⊥ ∪ b⊥ )) ∪ ((a ∩ (a⊥ ∪ b⊥ ))⊥ ∩ (a ∪ b))) = (a ∪ b)
49	5, 48	ax-r2 36	. 2 ((a →1 b)⊥ ∪ ((a →1 b) ∩ (a ∪ b))) = (a ∪ b)
50	1, 49	ax-r2 36	1 ((a →1 b) →1 (a ∪ b)) = (a ∪ 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:  ud1  595