Theorem ud3lem1d 551

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud3lem1d	((a →3 b) ∩ ((a →3 b)⊥ ∪ (b →3 a))) = ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b)))

Proof of Theorem ud3lem1d

Step	Hyp	Ref	Expression
1		df-i3 45	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2		ud3lem1c 550	. . 3 ((a →3 b)⊥ ∪ (b →3 a)) = (a ∪ b⊥ )
3	1, 2	2an 72	. 2 ((a →3 b) ∩ ((a →3 b)⊥ ∪ (b →3 a))) = ((((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	. . . . . . 7 (a ∪ b⊥ ) C b⊥
7	6	comcom7 442	. . . . . 6 (a ∪ b⊥ ) C b
8	5, 7	com2an 466	. . . . 5 (a ∪ b⊥ ) C (a⊥ ∩ b)
9	5, 6	com2an 466	. . . . 5 (a ∪ b⊥ ) C (a⊥ ∩ b⊥ )
10	8, 9	com2or 465	. . . 4 (a ∪ b⊥ ) C ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
11	5, 7	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	8, 9	fh1r 455	. . . . 5 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ b⊥ )) = (((a⊥ ∩ b) ∩ (a ∪ b⊥ )) ∪ ((a⊥ ∩ b⊥ ) ∩ (a ∪ b⊥ )))
15		an32 76	. . . . . 6 ((a ∩ (a⊥ ∪ b)) ∩ (a ∪ b⊥ )) = ((a ∩ (a ∪ b⊥ )) ∩ (a⊥ ∪ b))
16		a5c 113	. . . . . . 7 (a ∩ (a ∪ b⊥ )) = a
17	16	ran 71	. . . . . 6 ((a ∩ (a ∪ b⊥ )) ∩ (a⊥ ∪ b)) = (a ∩ (a⊥ ∪ b))
18	15, 17	ax-r2 35	. . . . 5 ((a ∩ (a⊥ ∪ b)) ∩ (a ∪ b⊥ )) = (a ∩ (a⊥ ∪ b))
19	14, 18	2or 67	. . . 4 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ b⊥ )) ∪ ((a ∩ (a⊥ ∪ b)) ∩ (a ∪ b⊥ ))) = ((((a⊥ ∩ b) ∩ (a ∪ b⊥ )) ∪ ((a⊥ ∩ b⊥ ) ∩ (a ∪ b⊥ ))) ∪ (a ∩ (a⊥ ∪ b)))
20		ancom 68	. . . . . . . 8 ((a⊥ ∩ b) ∩ (a ∪ b⊥ )) = ((a ∪ b⊥ ) ∩ (a⊥ ∩ b))
21		anor2 81	. . . . . . . . . 10 (a⊥ ∩ b) = (a ∪ b⊥ )⊥
22	21	lan 70	. . . . . . . . 9 ((a ∪ b⊥ ) ∩ (a⊥ ∩ b)) = ((a ∪ b⊥ ) ∩ (a ∪ b⊥ )⊥ )
23		dff 93	. . . . . . . . . 10 0 = ((a ∪ b⊥ ) ∩ (a ∪ b⊥ )⊥ )
24	23	ax-r1 34	. . . . . . . . 9 ((a ∪ b⊥ ) ∩ (a ∪ b⊥ )⊥ ) = 0
25	22, 24	ax-r2 35	. . . . . . . 8 ((a ∪ b⊥ ) ∩ (a⊥ ∩ b)) = 0
26	20, 25	ax-r2 35	. . . . . . 7 ((a⊥ ∩ b) ∩ (a ∪ b⊥ )) = 0
27		lear 153	. . . . . . . . 9 (a⊥ ∩ b⊥ ) ≤ b⊥
28		leor 151	. . . . . . . . 9 b⊥ ≤ (a ∪ b⊥ )
29	27, 28	letr 129	. . . . . . . 8 (a⊥ ∩ b⊥ ) ≤ (a ∪ b⊥ )
30	29	df2le2 128	. . . . . . 7 ((a⊥ ∩ b⊥ ) ∩ (a ∪ b⊥ )) = (a⊥ ∩ b⊥ )
31	26, 30	2or 67	. . . . . 6 (((a⊥ ∩ b) ∩ (a ∪ b⊥ )) ∪ ((a⊥ ∩ b⊥ ) ∩ (a ∪ b⊥ ))) = (0 ∪ (a⊥ ∩ b⊥ ))
32		or0r 95	. . . . . 6 (0 ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∩ b⊥ )
33	31, 32	ax-r2 35	. . . . 5 (((a⊥ ∩ b) ∩ (a ∪ b⊥ )) ∪ ((a⊥ ∩ b⊥ ) ∩ (a ∪ b⊥ ))) = (a⊥ ∩ b⊥ )
34	33	ax-r5 37	. . . 4 ((((a⊥ ∩ b) ∩ (a ∪ b⊥ )) ∪ ((a⊥ ∩ b⊥ ) ∩ (a ∪ b⊥ ))) ∪ (a ∩ (a⊥ ∪ b))) = ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b)))
35	19, 34	ax-r2 35	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ b⊥ )) ∪ ((a ∩ (a⊥ ∪ b)) ∩ (a ∪ b⊥ ))) = ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b)))
36	13, 35	ax-r2 35	. 2 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ (a ∪ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b)))
37	3, 36	ax-r2 35	1 ((a →3 b) ∩ ((a →3 b)⊥ ∪ (b →3 a))) = ((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:  ud3lem1 552

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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