Theorem u4lemana 608

Description: Lemma for non-tollens implication study. (Contributed by NM, 14-Dec-1997.)

Assertion

Ref	Expression
u4lemana	((a →4 b) ∩ a⊥ ) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))

Proof of Theorem u4lemana

Step	Hyp	Ref	Expression
1		df-i4 47	. . 3 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
2	1	ran 78	. 2 ((a →4 b) ∩ a⊥ ) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∩ a⊥ )
3		comanr1 464	. . . . . . 7 a C (a ∩ b)
4	3	comcom3 454	. . . . . 6 a⊥ C (a ∩ b)
5		comanr1 464	. . . . . 6 a⊥ C (a⊥ ∩ b)
6	4, 5	com2or 483	. . . . 5 a⊥ C ((a ∩ b) ∪ (a⊥ ∩ b))
7	6	comcom 453	. . . 4 ((a ∩ b) ∪ (a⊥ ∩ b)) C a⊥
8		comor1 461	. . . . . . . . 9 (a⊥ ∪ b) C a⊥
9	8	comcom7 460	. . . . . . . 8 (a⊥ ∪ b) C a
10		comor2 462	. . . . . . . 8 (a⊥ ∪ b) C b
11	9, 10	com2an 484	. . . . . . 7 (a⊥ ∪ b) C (a ∩ b)
12	8, 10	com2an 484	. . . . . . 7 (a⊥ ∪ b) C (a⊥ ∩ b)
13	11, 12	com2or 483	. . . . . 6 (a⊥ ∪ b) C ((a ∩ b) ∪ (a⊥ ∩ b))
14	13	comcom 453	. . . . 5 ((a ∩ b) ∪ (a⊥ ∩ b)) C (a⊥ ∪ b)
15		comanr2 465	. . . . . . . 8 b C (a ∩ b)
16	15	comcom3 454	. . . . . . 7 b⊥ C (a ∩ b)
17		comanr2 465	. . . . . . . 8 b C (a⊥ ∩ b)
18	17	comcom3 454	. . . . . . 7 b⊥ C (a⊥ ∩ b)
19	16, 18	com2or 483	. . . . . 6 b⊥ C ((a ∩ b) ∪ (a⊥ ∩ b))
20	19	comcom 453	. . . . 5 ((a ∩ b) ∪ (a⊥ ∩ b)) C b⊥
21	14, 20	com2an 484	. . . 4 ((a ∩ b) ∪ (a⊥ ∩ b)) C ((a⊥ ∪ b) ∩ b⊥ )
22	7, 21	fh2r 474	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∩ a⊥ ) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∩ a⊥ ) ∪ (((a⊥ ∪ b) ∩ b⊥ ) ∩ a⊥ ))
23	4, 5	fh1r 473	. . . . . 6 (((a ∩ b) ∪ (a⊥ ∩ b)) ∩ a⊥ ) = (((a ∩ b) ∩ a⊥ ) ∪ ((a⊥ ∩ b) ∩ a⊥ ))
24		an32 83	. . . . . . . . 9 ((a ∩ b) ∩ a⊥ ) = ((a ∩ a⊥ ) ∩ b)
25		ancom 74	. . . . . . . . . 10 ((a ∩ a⊥ ) ∩ b) = (b ∩ (a ∩ a⊥ ))
26		dff 101	. . . . . . . . . . . . 13 0 = (a ∩ a⊥ )
27	26	ax-r1 35	. . . . . . . . . . . 12 (a ∩ a⊥ ) = 0
28	27	lan 77	. . . . . . . . . . 11 (b ∩ (a ∩ a⊥ )) = (b ∩ 0)
29		an0 108	. . . . . . . . . . 11 (b ∩ 0) = 0
30	28, 29	ax-r2 36	. . . . . . . . . 10 (b ∩ (a ∩ a⊥ )) = 0
31	25, 30	ax-r2 36	. . . . . . . . 9 ((a ∩ a⊥ ) ∩ b) = 0
32	24, 31	ax-r2 36	. . . . . . . 8 ((a ∩ b) ∩ a⊥ ) = 0
33		an32 83	. . . . . . . . 9 ((a⊥ ∩ b) ∩ a⊥ ) = ((a⊥ ∩ a⊥ ) ∩ b)
34		anidm 111	. . . . . . . . . 10 (a⊥ ∩ a⊥ ) = a⊥
35	34	ran 78	. . . . . . . . 9 ((a⊥ ∩ a⊥ ) ∩ b) = (a⊥ ∩ b)
36	33, 35	ax-r2 36	. . . . . . . 8 ((a⊥ ∩ b) ∩ a⊥ ) = (a⊥ ∩ b)
37	32, 36	2or 72	. . . . . . 7 (((a ∩ b) ∩ a⊥ ) ∪ ((a⊥ ∩ b) ∩ a⊥ )) = (0 ∪ (a⊥ ∩ b))
38		ax-a2 31	. . . . . . . 8 (0 ∪ (a⊥ ∩ b)) = ((a⊥ ∩ b) ∪ 0)
39		or0 102	. . . . . . . 8 ((a⊥ ∩ b) ∪ 0) = (a⊥ ∩ b)
40	38, 39	ax-r2 36	. . . . . . 7 (0 ∪ (a⊥ ∩ b)) = (a⊥ ∩ b)
41	37, 40	ax-r2 36	. . . . . 6 (((a ∩ b) ∩ a⊥ ) ∪ ((a⊥ ∩ b) ∩ a⊥ )) = (a⊥ ∩ b)
42	23, 41	ax-r2 36	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b)) ∩ a⊥ ) = (a⊥ ∩ b)
43		an32 83	. . . . . 6 (((a⊥ ∪ b) ∩ b⊥ ) ∩ a⊥ ) = (((a⊥ ∪ b) ∩ a⊥ ) ∩ b⊥ )
44		ancom 74	. . . . . . . 8 ((a⊥ ∪ b) ∩ a⊥ ) = (a⊥ ∩ (a⊥ ∪ b))
45		leo 158	. . . . . . . . 9 a⊥ ≤ (a⊥ ∪ b)
46	45	df2le2 136	. . . . . . . 8 (a⊥ ∩ (a⊥ ∪ b)) = a⊥
47	44, 46	ax-r2 36	. . . . . . 7 ((a⊥ ∪ b) ∩ a⊥ ) = a⊥
48	47	ran 78	. . . . . 6 (((a⊥ ∪ b) ∩ a⊥ ) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
49	43, 48	ax-r2 36	. . . . 5 (((a⊥ ∪ b) ∩ b⊥ ) ∩ a⊥ ) = (a⊥ ∩ b⊥ )
50	42, 49	2or 72	. . . 4 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∩ a⊥ ) ∪ (((a⊥ ∪ b) ∩ b⊥ ) ∩ a⊥ )) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
51		id 59	. . . 4 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
52	50, 51	ax-r2 36	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∩ a⊥ ) ∪ (((a⊥ ∪ b) ∩ b⊥ ) ∩ a⊥ )) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
53	22, 52	ax-r2 36	. 2 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∩ a⊥ ) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
54	2, 53	ax-r2 36	1 ((a →4 b) ∩ a⊥ ) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →₄ wi4 15

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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u4lemnoa  663  u4lem5  764