Theorem ud3lem3d 557

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud3lem3d	((a →3 b) ∩ ((a →3 b)⊥ ∪ (a ∪ b))) = ((a⊥ ∩ b) ∪ (a ∩ (a⊥ ∪ b)))

Proof of Theorem ud3lem3d

Step	Hyp	Ref	Expression
1		df-i3 45	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2		ud3lem3c 556	. . 3 ((a →3 b)⊥ ∪ (a ∪ b)) = (a ∪ b)
3	1, 2	2an 72	. 2 ((a →3 b) ∩ ((a →3 b)⊥ ∪ (a ∪ b))) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ (a ∪ b))
4		comor1 443	. . . . . . 7 (a ∪ b) C a
5	4	comcom2 175	. . . . . 6 (a ∪ b) C a⊥
6		comor2 444	. . . . . 6 (a ∪ b) C b
7	5, 6	com2an 466	. . . . 5 (a ∪ b) C (a⊥ ∩ b)
8	6	comcom2 175	. . . . . 6 (a ∪ b) C b⊥
9	5, 8	com2an 466	. . . . 5 (a ∪ b) C (a⊥ ∩ b⊥ )
10	7, 9	com2or 465	. . . 4 (a ∪ b) C ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
11	5, 6	com2or 465	. . . . 5 (a ∪ b) C (a⊥ ∪ b)
12	4, 11	com2an 466	. . . 4 (a ∪ b) C (a ∩ (a⊥ ∪ b))
13	10, 12	fh1r 455	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ (a ∪ b)) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ b)) ∪ ((a ∩ (a⊥ ∪ b)) ∩ (a ∪ b)))
14		coman1 177	. . . . . . . . 9 (a⊥ ∩ b) C a⊥
15	14	comcom7 442	. . . . . . . 8 (a⊥ ∩ b) C a
16		coman2 178	. . . . . . . 8 (a⊥ ∩ b) C b
17	15, 16	com2or 465	. . . . . . 7 (a⊥ ∩ b) C (a ∪ b)
18	16	comcom2 175	. . . . . . . 8 (a⊥ ∩ b) C b⊥
19	14, 18	com2an 466	. . . . . . 7 (a⊥ ∩ b) C (a⊥ ∩ b⊥ )
20	17, 19	fh2r 456	. . . . . 6 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ b)) = (((a⊥ ∩ b) ∩ (a ∪ b)) ∪ ((a⊥ ∩ b⊥ ) ∩ (a ∪ b)))
21		lear 153	. . . . . . . . . 10 (a⊥ ∩ b) ≤ b
22		leor 151	. . . . . . . . . 10 b ≤ (a ∪ b)
23	21, 22	letr 129	. . . . . . . . 9 (a⊥ ∩ b) ≤ (a ∪ b)
24	23	df2le2 128	. . . . . . . 8 ((a⊥ ∩ b) ∩ (a ∪ b)) = (a⊥ ∩ b)
25		oran 79	. . . . . . . . . 10 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
26	25	lan 70	. . . . . . . . 9 ((a⊥ ∩ b⊥ ) ∩ (a ∪ b)) = ((a⊥ ∩ b⊥ ) ∩ (a⊥ ∩ b⊥ )⊥ )
27		dff 93	. . . . . . . . . 10 0 = ((a⊥ ∩ b⊥ ) ∩ (a⊥ ∩ b⊥ )⊥ )
28	27	ax-r1 34	. . . . . . . . 9 ((a⊥ ∩ b⊥ ) ∩ (a⊥ ∩ b⊥ )⊥ ) = 0
29	26, 28	ax-r2 35	. . . . . . . 8 ((a⊥ ∩ b⊥ ) ∩ (a ∪ b)) = 0
30	24, 29	2or 67	. . . . . . 7 (((a⊥ ∩ b) ∩ (a ∪ b)) ∪ ((a⊥ ∩ b⊥ ) ∩ (a ∪ b))) = ((a⊥ ∩ b) ∪ 0)
31		or0 94	. . . . . . 7 ((a⊥ ∩ b) ∪ 0) = (a⊥ ∩ b)
32	30, 31	ax-r2 35	. . . . . 6 (((a⊥ ∩ b) ∩ (a ∪ b)) ∪ ((a⊥ ∩ b⊥ ) ∩ (a ∪ b))) = (a⊥ ∩ b)
33	20, 32	ax-r2 35	. . . . 5 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ b)) = (a⊥ ∩ b)
34	33	ax-r5 37	. . . 4 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ b)) ∪ ((a ∩ (a⊥ ∪ b)) ∩ (a ∪ b))) = ((a⊥ ∩ b) ∪ ((a ∩ (a⊥ ∪ b)) ∩ (a ∪ b)))
35		lea 152	. . . . . . 7 (a ∩ (a⊥ ∪ b)) ≤ a
36		leo 150	. . . . . . 7 a ≤ (a ∪ b)
37	35, 36	letr 129	. . . . . 6 (a ∩ (a⊥ ∪ b)) ≤ (a ∪ b)
38	37	df2le2 128	. . . . 5 ((a ∩ (a⊥ ∪ b)) ∩ (a ∪ b)) = (a ∩ (a⊥ ∪ b))
39	38	lor 66	. . . 4 ((a⊥ ∩ b) ∪ ((a ∩ (a⊥ ∪ b)) ∩ (a ∪ b))) = ((a⊥ ∩ b) ∪ (a ∩ (a⊥ ∪ b)))
40	34, 39	ax-r2 35	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ b)) ∪ ((a ∩ (a⊥ ∪ b)) ∩ (a ∪ b))) = ((a⊥ ∩ b) ∪ (a ∩ (a⊥ ∪ b)))
41	13, 40	ax-r2 35	. 2 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ (a ∪ b)) = ((a⊥ ∩ b) ∪ (a ∩ (a⊥ ∪ b)))
42	3, 41	ax-r2 35	1 ((a →3 b) ∩ ((a →3 b)⊥ ∪ (a ∪ b))) = ((a⊥ ∩ b) ∪ (a ∩ (a⊥ ∪ b)))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 10   →₃ wi3 15

This theorem is referenced by:  ud3lem3 558

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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