Theorem lem3.3.7i4e1 1069

Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 4, 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
lem3.3.7i4e1	(a →4 (a ∩ b)) = (a ≡4 (a ∩ b))

Proof of Theorem lem3.3.7i4e1

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	5	ax-r5 38	. . . 4 ((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) = ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)))
7	6	ax-r5 38	. . 3 (((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )) = (((a ∩ b) ∪ (a⊥ ∩ (a ∩ b))) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ))
8	2	lecon 154	. . . . . . . 8 a⊥ ≤ (a ∩ b)⊥
9	8	ortha 438	. . . . . . 7 (a⊥ ∩ (a ∩ b)) = 0
10	9	lor 70	. . . . . 6 ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b))) = ((a ∩ b) ∪ 0)
11	10	ax-r5 38	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ (a ∩ b))) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )) = (((a ∩ b) ∪ 0) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ))
12		or0 102	. . . . . 6 ((a ∩ b) ∪ 0) = (a ∩ b)
13	12	ax-r5 38	. . . . 5 (((a ∩ b) ∪ 0) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )) = ((a ∩ b) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ))
14		leor 159	. . . . . . 7 (a ∩ b) ≤ (a⊥ ∪ (a ∩ b))
15		lea 160	. . . . . . 7 ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ) ≤ (a⊥ ∪ (a ∩ b))
16	14, 15	lel2or 170	. . . . . 6 ((a ∩ b) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )) ≤ (a⊥ ∪ (a ∩ b))
17		leo 158	. . . . . . . . 9 a⊥ ≤ (a⊥ ∪ (a ∩ b))
18	17, 8	ler2an 173	. . . . . . . 8 a⊥ ≤ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )
19	18	lerr 150	. . . . . . 7 a⊥ ≤ ((a ∩ b) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ))
20		leo 158	. . . . . . 7 (a ∩ b) ≤ ((a ∩ b) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ))
21	19, 20	lel2or 170	. . . . . 6 (a⊥ ∪ (a ∩ b)) ≤ ((a ∩ b) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ))
22	16, 21	lebi 145	. . . . 5 ((a ∩ b) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )) = (a⊥ ∪ (a ∩ b))
23	11, 13, 22	3tr 65	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ (a ∩ b))) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )) = (a⊥ ∪ (a ∩ b))
24		an1 106	. . . . 5 ((a⊥ ∪ (a ∩ b)) ∩ 1) = (a⊥ ∪ (a ∩ b))
25	24	ax-r1 35	. . . 4 (a⊥ ∪ (a ∩ b)) = ((a⊥ ∪ (a ∩ b)) ∩ 1)
26	3	sklem 230	. . . . . 6 ((a ∩ b)⊥ ∪ (a ∩ b)) = 1
27	26	ax-r1 35	. . . . 5 1 = ((a ∩ b)⊥ ∪ (a ∩ b))
28	27	lan 77	. . . 4 ((a⊥ ∪ (a ∩ b)) ∩ 1) = ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ (a ∩ b)))
29	23, 25, 28	3tr 65	. . 3 (((a ∩ b) ∪ (a⊥ ∩ (a ∩ b))) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )) = ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ (a ∩ b)))
30	4, 1	lebi 145	. . . . 5 (a ∩ b) = (a ∩ (a ∩ b))
31	30	lor 70	. . . 4 ((a ∩ b)⊥ ∪ (a ∩ b)) = ((a ∩ b)⊥ ∪ (a ∩ (a ∩ b)))
32	31	lan 77	. . 3 ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ (a ∩ b))) = ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ (a ∩ (a ∩ b))))
33	7, 29, 32	3tr 65	. 2 (((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ )) = ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ (a ∩ (a ∩ b))))
34		df-i4 47	. 2 (a →4 (a ∩ b)) = (((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b))) ∪ ((a⊥ ∪ (a ∩ b)) ∩ (a ∩ b)⊥ ))
35		df-id4 53	. 2 (a ≡4 (a ∩ b)) = ((a⊥ ∪ (a ∩ b)) ∩ ((a ∩ b)⊥ ∪ (a ∩ (a ∩ b))))
36	33, 34, 35	3tr1 63	1 (a →4 (a ∩ b)) = (a ≡4 (a ∩ b))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →₄ wi4 15   ≡₄ wid4 21

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-i4 47  df-id4 53  df-le1 130  df-le2 131

This theorem is referenced by: (None)