Theorem u3lemanb 599

Description: Lemma for Kalmbach implication study.

Assertion

Ref	Expression
u3lemanb	((a →3 b) ∩ b⊥ ) = (a⊥ ∩ b⊥ )

Proof of Theorem u3lemanb

Step	Hyp	Ref	Expression
1		df-i3 45	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2	1	ran 71	. 2 ((a →3 b) ∩ b⊥ ) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ b⊥ )
3		comanr2 447	. . . . . . 7 b C (a⊥ ∩ b)
4	3	comcom3 436	. . . . . 6 b⊥ C (a⊥ ∩ b)
5		comanr2 447	. . . . . 6 b⊥ C (a⊥ ∩ b⊥ )
6	4, 5	com2or 465	. . . . 5 b⊥ C ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
7	6	comcom 435	. . . 4 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) C b⊥
8		coman1 177	. . . . . . . . 9 (a⊥ ∩ b) C a⊥
9	8	comcom7 442	. . . . . . . 8 (a⊥ ∩ b) C a
10		coman2 178	. . . . . . . . 9 (a⊥ ∩ b) C b
11	8, 10	com2or 465	. . . . . . . 8 (a⊥ ∩ b) C (a⊥ ∪ b)
12	9, 11	com2an 466	. . . . . . 7 (a⊥ ∩ b) C (a ∩ (a⊥ ∪ b))
13	12	comcom 435	. . . . . 6 (a ∩ (a⊥ ∪ b)) C (a⊥ ∩ b)
14		coman1 177	. . . . . . . . 9 (a⊥ ∩ b⊥ ) C a⊥
15	14	comcom7 442	. . . . . . . 8 (a⊥ ∩ b⊥ ) C a
16		coman2 178	. . . . . . . . . 10 (a⊥ ∩ b⊥ ) C b⊥
17	16	comcom7 442	. . . . . . . . 9 (a⊥ ∩ b⊥ ) C b
18	14, 17	com2or 465	. . . . . . . 8 (a⊥ ∩ b⊥ ) C (a⊥ ∪ b)
19	15, 18	com2an 466	. . . . . . 7 (a⊥ ∩ b⊥ ) C (a ∩ (a⊥ ∪ b))
20	19	comcom 435	. . . . . 6 (a ∩ (a⊥ ∪ b)) C (a⊥ ∩ b⊥ )
21	13, 20	com2or 465	. . . . 5 (a ∩ (a⊥ ∪ b)) C ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
22	21	comcom 435	. . . 4 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) C (a ∩ (a⊥ ∪ b))
23	7, 22	fh2r 456	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ b⊥ ) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b⊥ ) ∪ ((a ∩ (a⊥ ∪ b)) ∩ b⊥ ))
24	10	comcom2 175	. . . . . . 7 (a⊥ ∩ b) C b⊥
25	8, 24	com2an 466	. . . . . . 7 (a⊥ ∩ b) C (a⊥ ∩ b⊥ )
26	24, 25	fh2r 456	. . . . . 6 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b⊥ ) = (((a⊥ ∩ b) ∩ b⊥ ) ∪ ((a⊥ ∩ b⊥ ) ∩ b⊥ ))
27		ax-a2 30	. . . . . . 7 (((a⊥ ∩ b) ∩ b⊥ ) ∪ ((a⊥ ∩ b⊥ ) ∩ b⊥ )) = (((a⊥ ∩ b⊥ ) ∩ b⊥ ) ∪ ((a⊥ ∩ b) ∩ b⊥ ))
28		anass 69	. . . . . . . . . 10 ((a⊥ ∩ b⊥ ) ∩ b⊥ ) = (a⊥ ∩ (b⊥ ∩ b⊥ ))
29		anidm 103	. . . . . . . . . . 11 (b⊥ ∩ b⊥ ) = b⊥
30	29	lan 70	. . . . . . . . . 10 (a⊥ ∩ (b⊥ ∩ b⊥ )) = (a⊥ ∩ b⊥ )
31	28, 30	ax-r2 35	. . . . . . . . 9 ((a⊥ ∩ b⊥ ) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
32		anass 69	. . . . . . . . . 10 ((a⊥ ∩ b) ∩ b⊥ ) = (a⊥ ∩ (b ∩ b⊥ ))
33		dff 93	. . . . . . . . . . . . 13 0 = (b ∩ b⊥ )
34	33	lan 70	. . . . . . . . . . . 12 (a⊥ ∩ 0) = (a⊥ ∩ (b ∩ b⊥ ))
35	34	ax-r1 34	. . . . . . . . . . 11 (a⊥ ∩ (b ∩ b⊥ )) = (a⊥ ∩ 0)
36		an0 100	. . . . . . . . . . 11 (a⊥ ∩ 0) = 0
37	35, 36	ax-r2 35	. . . . . . . . . 10 (a⊥ ∩ (b ∩ b⊥ )) = 0
38	32, 37	ax-r2 35	. . . . . . . . 9 ((a⊥ ∩ b) ∩ b⊥ ) = 0
39	31, 38	2or 67	. . . . . . . 8 (((a⊥ ∩ b⊥ ) ∩ b⊥ ) ∪ ((a⊥ ∩ b) ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ 0)
40		or0 94	. . . . . . . 8 ((a⊥ ∩ b⊥ ) ∪ 0) = (a⊥ ∩ b⊥ )
41	39, 40	ax-r2 35	. . . . . . 7 (((a⊥ ∩ b⊥ ) ∩ b⊥ ) ∪ ((a⊥ ∩ b) ∩ b⊥ )) = (a⊥ ∩ b⊥ )
42	27, 41	ax-r2 35	. . . . . 6 (((a⊥ ∩ b) ∩ b⊥ ) ∪ ((a⊥ ∩ b⊥ ) ∩ b⊥ )) = (a⊥ ∩ b⊥ )
43	26, 42	ax-r2 35	. . . . 5 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
44		an32 76	. . . . . 6 ((a ∩ (a⊥ ∪ b)) ∩ b⊥ ) = ((a ∩ b⊥ ) ∩ (a⊥ ∪ b))
45		ancom 68	. . . . . . 7 ((a ∩ b⊥ ) ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∩ (a ∩ b⊥ ))
46		anor1 80	. . . . . . . . 9 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
47	46	lan 70	. . . . . . . 8 ((a⊥ ∪ b) ∩ (a ∩ b⊥ )) = ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ )
48		dff 93	. . . . . . . . 9 0 = ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ )
49	48	ax-r1 34	. . . . . . . 8 ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ ) = 0
50	47, 49	ax-r2 35	. . . . . . 7 ((a⊥ ∪ b) ∩ (a ∩ b⊥ )) = 0
51	45, 50	ax-r2 35	. . . . . 6 ((a ∩ b⊥ ) ∩ (a⊥ ∪ b)) = 0
52	44, 51	ax-r2 35	. . . . 5 ((a ∩ (a⊥ ∪ b)) ∩ b⊥ ) = 0
53	43, 52	2or 67	. . . 4 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b⊥ ) ∪ ((a ∩ (a⊥ ∪ b)) ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ 0)
54	53, 40	ax-r2 35	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b⊥ ) ∪ ((a ∩ (a⊥ ∪ b)) ∩ b⊥ )) = (a⊥ ∩ b⊥ )
55	23, 54	ax-r2 35	. 2 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
56	2, 55	ax-r2 35	1 ((a →3 b) ∩ b⊥ ) = (a⊥ ∩ b⊥ )

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

This theorem is referenced by:  u3lemnob 654  u3lem3 733  u3lem13b 772  neg3antlem2 847

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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