Theorem lem3.3.7i5e1 1072

Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 5, and this is the first part of the equation. (Contributed by Roy F. Longton, 28-Jun-2005.) (Revised by Roy F. Longton, 3-Jul-2005.)

Assertion

Ref	Expression
lem3.3.7i5e1	(a →5 (a ∩ b)) = (a ≡5 (a ∩ b))

Proof of Theorem lem3.3.7i5e1

Step	Hyp	Ref	Expression
1		lear 161	. . . . . 6 (a ∩ (a ∩ b)) ≤ (a ∩ b)
2		lea 160	. . . . . . 7 (a ∩ b) ≤ a
3		leid 148	. . . . . . 7 (a ∩ b) ≤ (a ∩ b)
4	2, 3	ler2an 173	. . . . . 6 (a ∩ b) ≤ (a ∩ (a ∩ b))
5	1, 4	lebi 145	. . . . 5 (a ∩ (a ∩ b)) = (a ∩ b)
6	2	lecon 154	. . . . . 6 a⊥ ≤ (a ∩ b)⊥
7	6	ortha 438	. . . . 5 (a⊥ ∩ (a ∩ b)) = 0
8	5, 7	2or 72	. . . 4 ((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) = ((a ∩ b) ∪ 0)
9	8	ax-r5 38	. . 3 (((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = (((a ∩ b) ∪ 0) ∪ (a⊥ ∩ (a ∩ b)⊥ ))
10		or0 102	. . . 4 ((a ∩ b) ∪ 0) = (a ∩ b)
11	6	df2le2 136	. . . 4 (a⊥ ∩ (a ∩ b)⊥ ) = a⊥
12	10, 11	2or 72	. . 3 (((a ∩ b) ∪ 0) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = ((a ∩ b) ∪ a⊥ )
13	4, 1	lebi 145	. . . 4 (a ∩ b) = (a ∩ (a ∩ b))
14	11	ax-r1 35	. . . 4 a⊥ = (a⊥ ∩ (a ∩ b)⊥ )
15	13, 14	2or 72	. . 3 ((a ∩ b) ∪ a⊥ ) = ((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ ))
16	9, 12, 15	3tr 65	. 2 (((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = ((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ ))
17		df-i5 48	. 2 (a →5 (a ∩ b)) = (((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) ∪ (a⊥ ∩ (a ∩ b)⊥ ))
18		df-id5 1047	. 2 (a ≡5 (a ∩ b)) = ((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ ))
19	16, 17, 18	3tr1 63	1 (a →5 (a ∩ b)) = (a ≡5 (a ∩ b))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →₅ wi5 16   ≡₅ wid5 22

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  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-i5 48  df-le1 130  df-le2 131  df-id5 1047

This theorem is referenced by: (None)