Theorem lem3.3.7i0e1 1057

Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 0, and this is the first part of the equation. (Contributed by Roy F. Longton, 3-Jul-05.)

Assertion

Ref	Expression
lem3.3.7i0e1	(a →0 (a ∩ b)) = (a ≡0 (a ∩ b))

Proof of Theorem lem3.3.7i0e1

Step	Hyp	Ref	Expression
1		or1 104	. . . . . . 7 (b⊥ ∪ 1) = 1
2	1	ax-r1 35	. . . . . 6 1 = (b⊥ ∪ 1)
3	2	lan 77	. . . . 5 ((a⊥ ∪ (a ∩ b)) ∩ 1) = ((a⊥ ∪ (a ∩ b)) ∩ (b⊥ ∪ 1))
4		an1 106	. . . . 5 ((a⊥ ∪ (a ∩ b)) ∩ 1) = (a⊥ ∪ (a ∩ b))
5		df-t 41	. . . . . . 7 1 = (a ∪ a⊥ )
6	5	lor 70	. . . . . 6 (b⊥ ∪ 1) = (b⊥ ∪ (a ∪ a⊥ ))
7	6	lan 77	. . . . 5 ((a⊥ ∪ (a ∩ b)) ∩ (b⊥ ∪ 1)) = ((a⊥ ∪ (a ∩ b)) ∩ (b⊥ ∪ (a ∪ a⊥ )))
8	3, 4, 7	3tr2 64	. . . 4 (a⊥ ∪ (a ∩ b)) = ((a⊥ ∪ (a ∩ b)) ∩ (b⊥ ∪ (a ∪ a⊥ )))
9		ax-a2 31	. . . . . 6 (a ∪ a⊥ ) = (a⊥ ∪ a)
10	9	lor 70	. . . . 5 (b⊥ ∪ (a ∪ a⊥ )) = (b⊥ ∪ (a⊥ ∪ a))
11	10	lan 77	. . . 4 ((a⊥ ∪ (a ∩ b)) ∩ (b⊥ ∪ (a ∪ a⊥ ))) = ((a⊥ ∪ (a ∩ b)) ∩ (b⊥ ∪ (a⊥ ∪ a)))
12		ax-a3 32	. . . . . 6 ((b⊥ ∪ a⊥ ) ∪ a) = (b⊥ ∪ (a⊥ ∪ a))
13	12	ax-r1 35	. . . . 5 (b⊥ ∪ (a⊥ ∪ a)) = ((b⊥ ∪ a⊥ ) ∪ a)
14	13	lan 77	. . . 4 ((a⊥ ∪ (a ∩ b)) ∩ (b⊥ ∪ (a⊥ ∪ a))) = ((a⊥ ∪ (a ∩ b)) ∩ ((b⊥ ∪ a⊥ ) ∪ a))
15	8, 11, 14	3tr 65	. . 3 (a⊥ ∪ (a ∩ b)) = ((a⊥ ∪ (a ∩ b)) ∩ ((b⊥ ∪ a⊥ ) ∪ a))
16		ax-a2 31	. . . . 5 (b⊥ ∪ a⊥ ) = (a⊥ ∪ b⊥ )
17	16	ax-r5 38	. . . 4 ((b⊥ ∪ a⊥ ) ∪ a) = ((a⊥ ∪ b⊥ ) ∪ a)
18	17	lan 77	. . 3 ((a⊥ ∪ (a ∩ b)) ∩ ((b⊥ ∪ a⊥ ) ∪ a)) = ((a⊥ ∪ (a ∩ b)) ∩ ((a⊥ ∪ b⊥ ) ∪ a))
19		oran3 93	. . . . 5 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
20	19	ax-r5 38	. . . 4 ((a⊥ ∪ b⊥ ) ∪ a) = ((a ∩ b)⊥ ∪ a)
21	20	lan 77	. . 3 ((a⊥ ∪ (a ∩ b)) ∩ ((a⊥ ∪ b⊥ ) ∪ a)) = ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ a))
22	15, 18, 21	3tr 65	. 2 (a⊥ ∪ (a ∩ b)) = ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ a))
23		df-i0 43	. 2 (a →0 (a ∩ b)) = (a⊥ ∪ (a ∩ b))
24		df-id0 49	. 2 (a ≡0 (a ∩ b)) = ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ a))
25	22, 23, 24	3tr1 63	1 (a →0 (a ∩ b)) = (a ≡0 (a ∩ b))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₀ wi0 11   ≡₀ wid0 17

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

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i0 43  df-id0 49

This theorem is referenced by: (None)