Theorem u3lemab 612

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

Assertion

Ref	Expression
u3lemab	((a →3 b) ∩ b) = ((a ∩ b) ∪ (a⊥ ∩ b))

Proof of Theorem u3lemab

Step	Hyp	Ref	Expression
1		df-i3 46	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2	1	ran 78	. 2 ((a →3 b) ∩ b) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ b)
3		comanr2 465	. . . . . 6 b C (a⊥ ∩ b)
4		comanr2 465	. . . . . . 7 b⊥ C (a⊥ ∩ b⊥ )
5	4	comcom6 459	. . . . . 6 b C (a⊥ ∩ b⊥ )
6	3, 5	com2or 483	. . . . 5 b C ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
7	6	comcom 453	. . . 4 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) C b
8		coman1 185	. . . . . . . . 9 (a⊥ ∩ b) C a⊥
9	8	comcom7 460	. . . . . . . 8 (a⊥ ∩ b) C a
10		coman2 186	. . . . . . . . 9 (a⊥ ∩ b) C b
11	8, 10	com2or 483	. . . . . . . 8 (a⊥ ∩ b) C (a⊥ ∪ b)
12	9, 11	com2an 484	. . . . . . 7 (a⊥ ∩ b) C (a ∩ (a⊥ ∪ b))
13	12	comcom 453	. . . . . 6 (a ∩ (a⊥ ∪ b)) C (a⊥ ∩ b)
14		coman1 185	. . . . . . . . 9 (a⊥ ∩ b⊥ ) C a⊥
15	14	comcom7 460	. . . . . . . 8 (a⊥ ∩ b⊥ ) C a
16		coman2 186	. . . . . . . . . 10 (a⊥ ∩ b⊥ ) C b⊥
17	16	comcom7 460	. . . . . . . . 9 (a⊥ ∩ b⊥ ) C b
18	14, 17	com2or 483	. . . . . . . 8 (a⊥ ∩ b⊥ ) C (a⊥ ∪ b)
19	15, 18	com2an 484	. . . . . . 7 (a⊥ ∩ b⊥ ) C (a ∩ (a⊥ ∪ b))
20	19	comcom 453	. . . . . 6 (a ∩ (a⊥ ∪ b)) C (a⊥ ∩ b⊥ )
21	13, 20	com2or 483	. . . . 5 (a ∩ (a⊥ ∪ b)) C ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
22	21	comcom 453	. . . 4 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) C (a ∩ (a⊥ ∪ b))
23	7, 22	fh2r 474	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ b) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b) ∪ ((a ∩ (a⊥ ∪ b)) ∩ b))
24	3, 5	fh1r 473	. . . . . 6 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b) = (((a⊥ ∩ b) ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∩ b))
25		anass 76	. . . . . . . . 9 ((a⊥ ∩ b) ∩ b) = (a⊥ ∩ (b ∩ b))
26		anidm 111	. . . . . . . . . 10 (b ∩ b) = b
27	26	lan 77	. . . . . . . . 9 (a⊥ ∩ (b ∩ b)) = (a⊥ ∩ b)
28	25, 27	ax-r2 36	. . . . . . . 8 ((a⊥ ∩ b) ∩ b) = (a⊥ ∩ b)
29		an32 83	. . . . . . . . 9 ((a⊥ ∩ b⊥ ) ∩ b) = ((a⊥ ∩ b) ∩ b⊥ )
30		anass 76	. . . . . . . . . 10 ((a⊥ ∩ b) ∩ b⊥ ) = (a⊥ ∩ (b ∩ b⊥ ))
31		dff 101	. . . . . . . . . . . . 13 0 = (b ∩ b⊥ )
32	31	ax-r1 35	. . . . . . . . . . . 12 (b ∩ b⊥ ) = 0
33	32	lan 77	. . . . . . . . . . 11 (a⊥ ∩ (b ∩ b⊥ )) = (a⊥ ∩ 0)
34		an0 108	. . . . . . . . . . 11 (a⊥ ∩ 0) = 0
35	33, 34	ax-r2 36	. . . . . . . . . 10 (a⊥ ∩ (b ∩ b⊥ )) = 0
36	30, 35	ax-r2 36	. . . . . . . . 9 ((a⊥ ∩ b) ∩ b⊥ ) = 0
37	29, 36	ax-r2 36	. . . . . . . 8 ((a⊥ ∩ b⊥ ) ∩ b) = 0
38	28, 37	2or 72	. . . . . . 7 (((a⊥ ∩ b) ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∩ b)) = ((a⊥ ∩ b) ∪ 0)
39		or0 102	. . . . . . 7 ((a⊥ ∩ b) ∪ 0) = (a⊥ ∩ b)
40	38, 39	ax-r2 36	. . . . . 6 (((a⊥ ∩ b) ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∩ b)) = (a⊥ ∩ b)
41	24, 40	ax-r2 36	. . . . 5 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b) = (a⊥ ∩ b)
42		anass 76	. . . . . 6 ((a ∩ (a⊥ ∪ b)) ∩ b) = (a ∩ ((a⊥ ∪ b) ∩ b))
43		ancom 74	. . . . . . . 8 ((a⊥ ∪ b) ∩ b) = (b ∩ (a⊥ ∪ b))
44		ax-a2 31	. . . . . . . . . 10 (a⊥ ∪ b) = (b ∪ a⊥ )
45	44	lan 77	. . . . . . . . 9 (b ∩ (a⊥ ∪ b)) = (b ∩ (b ∪ a⊥ ))
46		anabs 121	. . . . . . . . 9 (b ∩ (b ∪ a⊥ )) = b
47	45, 46	ax-r2 36	. . . . . . . 8 (b ∩ (a⊥ ∪ b)) = b
48	43, 47	ax-r2 36	. . . . . . 7 ((a⊥ ∪ b) ∩ b) = b
49	48	lan 77	. . . . . 6 (a ∩ ((a⊥ ∪ b) ∩ b)) = (a ∩ b)
50	42, 49	ax-r2 36	. . . . 5 ((a ∩ (a⊥ ∪ b)) ∩ b) = (a ∩ b)
51	41, 50	2or 72	. . . 4 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b) ∪ ((a ∩ (a⊥ ∪ b)) ∩ b)) = ((a⊥ ∩ b) ∪ (a ∩ b))
52		ax-a2 31	. . . 4 ((a⊥ ∩ b) ∪ (a ∩ b)) = ((a ∩ b) ∪ (a⊥ ∩ b))
53	51, 52	ax-r2 36	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b) ∪ ((a ∩ (a⊥ ∪ b)) ∩ b)) = ((a ∩ b) ∪ (a⊥ ∩ b))
54	23, 53	ax-r2 36	. 2 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ b) = ((a ∩ b) ∪ (a⊥ ∩ b))
55	2, 54	ax-r2 36	1 ((a →3 b) ∩ b) = ((a ∩ b) ∪ (a⊥ ∩ b))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →₃ wi3 14

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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u3lemnonb  677  neg3antlem1  864  neg3antlem2  865