Theorem lem3.4.3 1076

Description: Equation 3.11 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton, 29-Jun-2005.) (Revised by Roy F. Longton, 3-Jul-2005.)

Hypothesis

Ref	Expression
lem3.4.3.1	(a →2 b) = 1

Assertion

Ref	Expression
lem3.4.3	(a →2 (a ≡5 b)) = 1

Proof of Theorem lem3.4.3

Step	Hyp	Ref	Expression
1		lem3.4.3.1	. . . . . 6 (a →2 b) = 1
2	1	2vwomr2a 364	. . . . 5 (a →1 b) = 1
3	2	ax-r1 35	. . . 4 1 = (a →1 b)
4		anidm 111	. . . . . . . . . 10 (a ∩ a) = a
5	4	ax-r1 35	. . . . . . . . 9 a = (a ∩ a)
6	5	ran 78	. . . . . . . 8 (a ∩ b) = ((a ∩ a) ∩ b)
7		lea 160	. . . . . . . . . 10 (a ∩ a) ≤ a
8	7	lel 151	. . . . . . . . 9 ((a ∩ a) ∩ b) ≤ a
9	7	leran 153	. . . . . . . . . 10 ((a ∩ a) ∩ b) ≤ (a ∩ b)
10	9	ler 149	. . . . . . . . 9 ((a ∩ a) ∩ b) ≤ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
11	8, 10	ler2an 173	. . . . . . . 8 ((a ∩ a) ∩ b) ≤ (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))
12	6, 11	bltr 138	. . . . . . 7 (a ∩ b) ≤ (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))
13		df-id5 1047	. . . . . . . . 9 (a ≡5 b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
14	13	ax-r1 35	. . . . . . . 8 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) = (a ≡5 b)
15	14	lan 77	. . . . . . 7 (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (a ∩ (a ≡5 b))
16	12, 15	lbtr 139	. . . . . 6 (a ∩ b) ≤ (a ∩ (a ≡5 b))
17	16	lelor 166	. . . . 5 (a⊥ ∪ (a ∩ b)) ≤ (a⊥ ∪ (a ∩ (a ≡5 b)))
18		df-i1 44	. . . . 5 (a →1 b) = (a⊥ ∪ (a ∩ b))
19		df-i1 44	. . . . 5 (a →1 (a ≡5 b)) = (a⊥ ∪ (a ∩ (a ≡5 b)))
20	17, 18, 19	le3tr1 140	. . . 4 (a →1 b) ≤ (a →1 (a ≡5 b))
21	3, 20	bltr 138	. . 3 1 ≤ (a →1 (a ≡5 b))
22	21	lem3.3.5lem 1054	. 2 (a →1 (a ≡5 b)) = 1
23	22	2vwomr1a 363	1 (a →2 (a ≡5 b)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12   →₂ wi2 13   ≡₅ wid5 22

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  ax-wom 361

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-id5 1047

This theorem is referenced by:  lem3.4.4  1077