Theorem oadistc0 1021

Description: Pre-distributive law. Note that the inference of the second hypothesis from the first may be an OM theorem. (Contributed by NM, 30-Nov-1998.)

Hypotheses

Ref	Expression
oadistc0.1	d ≤ ((a →2 b) ∩ (a →2 c))
oadistc0.2	((a →2 c) ∩ ((a →2 b) ∩ ((b ∪ c)⊥ ∪ d))) ≤ (((a →2 b) ∩ (b ∪ c)⊥ ) ∪ d)

Assertion

Ref	Expression
oadistc0	((a →2 b) ∩ ((b ∪ c)⊥ ∪ d)) = (((a →2 b) ∩ (b ∪ c)⊥ ) ∪ d)

Proof of Theorem oadistc0

Step	Hyp	Ref	Expression
1		ancom 74	. . . . 5 ((a →2 c) ∩ ((a →2 b) ∩ ((b ∪ c)⊥ ∪ d))) = (((a →2 b) ∩ ((b ∪ c)⊥ ∪ d)) ∩ (a →2 c))
2		oadistc0.1	. . . . . . . . 9 d ≤ ((a →2 b) ∩ (a →2 c))
3	2	lelor 166	. . . . . . . 8 ((b ∪ c)⊥ ∪ d) ≤ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
4	3	lelan 167	. . . . . . 7 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ d)) ≤ ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))))
5		oal2 999	. . . . . . 7 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) ≤ (a →2 c)
6	4, 5	letr 137	. . . . . 6 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ d)) ≤ (a →2 c)
7	6	df2le2 136	. . . . 5 (((a →2 b) ∩ ((b ∪ c)⊥ ∪ d)) ∩ (a →2 c)) = ((a →2 b) ∩ ((b ∪ c)⊥ ∪ d))
8	1, 7	ax-r2 36	. . . 4 ((a →2 c) ∩ ((a →2 b) ∩ ((b ∪ c)⊥ ∪ d))) = ((a →2 b) ∩ ((b ∪ c)⊥ ∪ d))
9	8	ax-r1 35	. . 3 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ d)) = ((a →2 c) ∩ ((a →2 b) ∩ ((b ∪ c)⊥ ∪ d)))
10		oadistc0.2	. . 3 ((a →2 c) ∩ ((a →2 b) ∩ ((b ∪ c)⊥ ∪ d))) ≤ (((a →2 b) ∩ (b ∪ c)⊥ ) ∪ d)
11	9, 10	bltr 138	. 2 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ d)) ≤ (((a →2 b) ∩ (b ∪ c)⊥ ) ∪ d)
12		ledior 177	. . 3 (((a →2 b) ∩ (b ∪ c)⊥ ) ∪ d) ≤ (((a →2 b) ∪ d) ∩ ((b ∪ c)⊥ ∪ d))
13		ax-a2 31	. . . . 5 ((a →2 b) ∪ d) = (d ∪ (a →2 b))
14		lea 160	. . . . . . 7 ((a →2 b) ∩ (a →2 c)) ≤ (a →2 b)
15	2, 14	letr 137	. . . . . 6 d ≤ (a →2 b)
16	15	df-le2 131	. . . . 5 (d ∪ (a →2 b)) = (a →2 b)
17	13, 16	ax-r2 36	. . . 4 ((a →2 b) ∪ d) = (a →2 b)
18	17	ran 78	. . 3 (((a →2 b) ∪ d) ∩ ((b ∪ c)⊥ ∪ d)) = ((a →2 b) ∩ ((b ∪ c)⊥ ∪ d))
19	12, 18	lbtr 139	. 2 (((a →2 b) ∩ (b ∪ c)⊥ ) ∪ d) ≤ ((a →2 b) ∩ ((b ∪ c)⊥ ∪ d))
20	11, 19	lebi 145	1 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ d)) = (((a →2 b) ∩ (b ∪ c)⊥ ) ∪ d)

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

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  ax-3oa 998

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

This theorem is referenced by: (None)