Theorem i3lem1 486

Description: Lemma for Kalmbach implication.

Hypothesis

Ref	Expression
i3lem.1	(a →3 b) = 1

Assertion

Ref	Expression
i3lem1	((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = a⊥

Proof of Theorem i3lem1

Step	Hyp	Ref	Expression
1		coman1 177	. . . . . . 7 (a⊥ ∩ b⊥ ) C a⊥
2	1	comcom 435	. . . . . 6 a⊥ C (a⊥ ∩ b⊥ )
3		comorr 176	. . . . . . 7 a⊥ C (a⊥ ∪ b)
4		comorr 176	. . . . . . . 8 a C (a ∪ (a⊥ ∩ b))
5	4	comcom3 436	. . . . . . 7 a⊥ C (a ∪ (a⊥ ∩ b))
6	3, 5	com2an 466	. . . . . 6 a⊥ C ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))
7	2, 6	fh1 451	. . . . 5 (a⊥ ∩ ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))) = ((a⊥ ∩ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))))
8		anass 69	. . . . . . . . 9 ((a⊥ ∩ a⊥ ) ∩ b⊥ ) = (a⊥ ∩ (a⊥ ∩ b⊥ ))
9	8	ax-r1 34	. . . . . . . 8 (a⊥ ∩ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ a⊥ ) ∩ b⊥ )
10		anidm 103	. . . . . . . . 9 (a⊥ ∩ a⊥ ) = a⊥
11	10	ran 71	. . . . . . . 8 ((a⊥ ∩ a⊥ ) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
12	9, 11	ax-r2 35	. . . . . . 7 (a⊥ ∩ (a⊥ ∩ b⊥ )) = (a⊥ ∩ b⊥ )
13		anass 69	. . . . . . . . 9 ((a⊥ ∩ (a⊥ ∪ b)) ∩ (a ∪ (a⊥ ∩ b))) = (a⊥ ∩ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))
14	13	ax-r1 34	. . . . . . . 8 (a⊥ ∩ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))) = ((a⊥ ∩ (a⊥ ∪ b)) ∩ (a ∪ (a⊥ ∩ b)))
15		a5c 113	. . . . . . . . . 10 (a⊥ ∩ (a⊥ ∪ b)) = a⊥
16	15	ran 71	. . . . . . . . 9 ((a⊥ ∩ (a⊥ ∪ b)) ∩ (a ∪ (a⊥ ∩ b))) = (a⊥ ∩ (a ∪ (a⊥ ∩ b)))
17		omlan 430	. . . . . . . . 9 (a⊥ ∩ (a ∪ (a⊥ ∩ b))) = (a⊥ ∩ b)
18	16, 17	ax-r2 35	. . . . . . . 8 ((a⊥ ∩ (a⊥ ∪ b)) ∩ (a ∪ (a⊥ ∩ b))) = (a⊥ ∩ b)
19	14, 18	ax-r2 35	. . . . . . 7 (a⊥ ∩ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))) = (a⊥ ∩ b)
20	12, 19	2or 67	. . . . . 6 ((a⊥ ∩ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))) = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b))
21		ax-a2 30	. . . . . 6 ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b)) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
22	20, 21	ax-r2 35	. . . . 5 ((a⊥ ∩ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
23	7, 22	ax-r2 35	. . . 4 (a⊥ ∩ ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
24	23	ax-r1 34	. . 3 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∩ ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))))
25		df2i3 480	. . . . . 6 (a →3 b) = ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))
26	25	ax-r1 34	. . . . 5 ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))) = (a →3 b)
27		i3lem.1	. . . . 5 (a →3 b) = 1
28	26, 27	ax-r2 35	. . . 4 ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))) = 1
29	28	lan 70	. . 3 (a⊥ ∩ ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))) = (a⊥ ∩ 1)
30	24, 29	ax-r2 35	. 2 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∩ 1)
31		an1 98	. 2 (a⊥ ∩ 1) = a⊥
32	30, 31	ax-r2 35	1 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = a⊥

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₃ wi3 15

This theorem is referenced by:  i3lem2 487  i3lem3 488  i3lem4 489

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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