Theorem d4oa 996

Description: Variant of proper 4-OA proved from OA distributive law. (Contributed by NM, 30-Dec-1998.)

Hypotheses

Ref	Expression
d4oa.2	e = ((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d)))
d4oa.1	f = (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))

Assertion

Ref	Expression
d4oa	((a →1 d) ∩ (e ∪ f)) ≤ (b →1 d)

Proof of Theorem d4oa

Step	Hyp	Ref	Expression
1		ax-a2 31	. . . 4 (e ∪ f) = (f ∪ e)
2	1	lan 77	. . 3 ((a →1 d) ∩ (e ∪ f)) = ((a →1 d) ∩ (f ∪ e))
3		id 59	. . . 4 (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))) = (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))
4		d4oa.2	. . . . 5 e = ((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d)))
5		d4oa.1	. . . . 5 f = (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))
6	4, 5	2or 72	. . . 4 (e ∪ f) = (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))))
7		leid 148	. . . 4 (a →1 d) ≤ (a →1 d)
8		leor 159	. . . 4 f ≤ (e ∪ f)
9		leo 158	. . . 4 e ≤ (e ∪ f)
10		leor 159	. . . . 5 ((a →1 d) ∩ (b →1 d)) ≤ ((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d)))
11	4	ax-r1 35	. . . . 5 ((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) = e
12	10, 11	lbtr 139	. . . 4 ((a →1 d) ∩ (b →1 d)) ≤ e
13	3, 6, 7, 8, 9, 12	ax-oadist 994	. . 3 ((a →1 d) ∩ (f ∪ e)) = (((a →1 d) ∩ f) ∪ ((a →1 d) ∩ e))
14	2, 13	ax-r2 36	. 2 ((a →1 d) ∩ (e ∪ f)) = (((a →1 d) ∩ f) ∪ ((a →1 d) ∩ e))
15	5	lan 77	. . . . . 6 ((a →1 d) ∩ f) = ((a →1 d) ∩ (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))))
16		anass 76	. . . . . . 7 (((a →1 d) ∩ ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d)))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))) = ((a →1 d) ∩ (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))))
17	16	ax-r1 35	. . . . . 6 ((a →1 d) ∩ (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))) = (((a →1 d) ∩ ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d)))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))
18	15, 17	ax-r2 36	. . . . 5 ((a →1 d) ∩ f) = (((a →1 d) ∩ ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d)))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))
19		id 59	. . . . . . 7 ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) = ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d)))
20	19	d3oa 995	. . . . . 6 ((a →1 d) ∩ ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d)))) ≤ (c →1 d)
21	20	leran 153	. . . . 5 (((a →1 d) ∩ ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d)))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))) ≤ ((c →1 d) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))
22	18, 21	bltr 138	. . . 4 ((a →1 d) ∩ f) ≤ ((c →1 d) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))
23		ancom 74	. . . . . 6 (b ∩ c) = (c ∩ b)
24		ancom 74	. . . . . 6 ((b →1 d) ∩ (c →1 d)) = ((c →1 d) ∩ (b →1 d))
25	23, 24	2or 72	. . . . 5 ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))) = ((c ∩ b) ∪ ((c →1 d) ∩ (b →1 d)))
26	25	d3oa 995	. . . 4 ((c →1 d) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))) ≤ (b →1 d)
27	22, 26	letr 137	. . 3 ((a →1 d) ∩ f) ≤ (b →1 d)
28	4	d3oa 995	. . 3 ((a →1 d) ∩ e) ≤ (b →1 d)
29	27, 28	lel2or 170	. 2 (((a →1 d) ∩ f) ∪ ((a →1 d) ∩ e)) ≤ (b →1 d)
30	14, 29	bltr 138	1 ((a →1 d) ∩ (e ∪ f)) ≤ (b →1 d)

This proof depends on syntax axioms:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7   →₁ wi1 12

This proof depends on 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-r3 439  ax-oadist 994

This proof depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is used by:  d6oa  997