Theorem u3lemab 594

Description: Lemma for Kalmbach implication study.

Assertion

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

Proof of Theorem u3lemab

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	. . . . . 6 b C (a⊥ ∩ b)
4		comanr2 447	. . . . . . 7 b⊥ C (a⊥ ∩ b⊥ )
5	4	comcom6 441	. . . . . 6 b C (a⊥ ∩ b⊥ )
6	3, 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	3, 5	fh1r 455	. . . . . 6 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b) = (((a⊥ ∩ b) ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∩ b))
25		anass 69	. . . . . . . . 9 ((a⊥ ∩ b) ∩ b) = (a⊥ ∩ (b ∩ b))
26		anidm 103	. . . . . . . . . 10 (b ∩ b) = b
27	26	lan 70	. . . . . . . . 9 (a⊥ ∩ (b ∩ b)) = (a⊥ ∩ b)
28	25, 27	ax-r2 35	. . . . . . . 8 ((a⊥ ∩ b) ∩ b) = (a⊥ ∩ b)
29		an32 76	. . . . . . . . 9 ((a⊥ ∩ b⊥ ) ∩ b) = ((a⊥ ∩ b) ∩ b⊥ )
30		anass 69	. . . . . . . . . 10 ((a⊥ ∩ b) ∩ b⊥ ) = (a⊥ ∩ (b ∩ b⊥ ))
31		dff 93	. . . . . . . . . . . . 13 0 = (b ∩ b⊥ )
32	31	ax-r1 34	. . . . . . . . . . . 12 (b ∩ b⊥ ) = 0
33	32	lan 70	. . . . . . . . . . 11 (a⊥ ∩ (b ∩ b⊥ )) = (a⊥ ∩ 0)
34		an0 100	. . . . . . . . . . 11 (a⊥ ∩ 0) = 0
35	33, 34	ax-r2 35	. . . . . . . . . 10 (a⊥ ∩ (b ∩ b⊥ )) = 0
36	30, 35	ax-r2 35	. . . . . . . . 9 ((a⊥ ∩ b) ∩ b⊥ ) = 0
37	29, 36	ax-r2 35	. . . . . . . 8 ((a⊥ ∩ b⊥ ) ∩ b) = 0
38	28, 37	2or 67	. . . . . . 7 (((a⊥ ∩ b) ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∩ b)) = ((a⊥ ∩ b) ∪ 0)
39		or0 94	. . . . . . 7 ((a⊥ ∩ b) ∪ 0) = (a⊥ ∩ b)
40	38, 39	ax-r2 35	. . . . . 6 (((a⊥ ∩ b) ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∩ b)) = (a⊥ ∩ b)
41	24, 40	ax-r2 35	. . . . 5 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b) = (a⊥ ∩ b)
42		anass 69	. . . . . 6 ((a ∩ (a⊥ ∪ b)) ∩ b) = (a ∩ ((a⊥ ∪ b) ∩ b))
43		ancom 68	. . . . . . . 8 ((a⊥ ∪ b) ∩ b) = (b ∩ (a⊥ ∪ b))
44		ax-a2 30	. . . . . . . . . 10 (a⊥ ∪ b) = (b ∪ a⊥ )
45	44	lan 70	. . . . . . . . 9 (b ∩ (a⊥ ∪ b)) = (b ∩ (b ∪ a⊥ ))
46		a5c 113	. . . . . . . . 9 (b ∩ (b ∪ a⊥ )) = b
47	45, 46	ax-r2 35	. . . . . . . 8 (b ∩ (a⊥ ∪ b)) = b
48	43, 47	ax-r2 35	. . . . . . 7 ((a⊥ ∪ b) ∩ b) = b
49	48	lan 70	. . . . . 6 (a ∩ ((a⊥ ∪ b) ∩ b)) = (a ∩ b)
50	42, 49	ax-r2 35	. . . . 5 ((a ∩ (a⊥ ∪ b)) ∩ b) = (a ∩ b)
51	41, 50	2or 67	. . . 4 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b) ∪ ((a ∩ (a⊥ ∪ b)) ∩ b)) = ((a⊥ ∩ b) ∪ (a ∩ b))
52		ax-a2 30	. . . 4 ((a⊥ ∩ b) ∪ (a ∩ b)) = ((a ∩ b) ∪ (a⊥ ∩ b))
53	51, 52	ax-r2 35	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ b) ∪ ((a ∩ (a⊥ ∪ b)) ∩ b)) = ((a ∩ b) ∪ (a⊥ ∩ b))
54	23, 53	ax-r2 35	. 2 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ b) = ((a ∩ b) ∪ (a⊥ ∩ b))
55	2, 54	ax-r2 35	1 ((a →3 b) ∩ b) = ((a ∩ b) ∪ (a⊥ ∩ b))

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

This theorem is referenced by:  u3lemnonb 659  neg3antlem1 846  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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