Theorem d4oa 976

Description: Variant of proper 4-OA proved from OA distributive law.

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 30	. . . 4 (e ∪ f) = (f ∪ e)
2	1	lan 70	. . 3 ((a →1 d) ∩ (e ∪ f)) = ((a →1 d) ∩ (f ∪ e))
3		id 58	. . . 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 67	. . . 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 140	. . . 4 (a →1 d) ≤ (a →1 d)
8		leor 151	. . . 4 f ≤ (e ∪ f)
9		leo 150	. . . 4 e ≤ (e ∪ f)
10		leor 151	. . . . 5 ((a →1 d) ∩ (b →1 d)) ≤ ((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d)))
11	4	ax-r1 34	. . . . 5 ((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) = e
12	10, 11	lbtr 131	. . . 4 ((a →1 d) ∩ (b →1 d)) ≤ e
13	3, 6, 7, 8, 9, 12	ax-oadist 974	. . 3 ((a →1 d) ∩ (f ∪ e)) = (((a →1 d) ∩ f) ∪ ((a →1 d) ∩ e))
14	2, 13	ax-r2 35	. 2 ((a →1 d) ∩ (e ∪ f)) = (((a →1 d) ∩ f) ∪ ((a →1 d) ∩ e))
15	5	lan 70	. . . . . 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 69	. . . . . . 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 34	. . . . . 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 35	. . . . 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 58	. . . . . . 7 ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) = ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d)))
20	19	d3oa 975	. . . . . 6 ((a →1 d) ∩ ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d)))) ≤ (c →1 d)
21	20	leran 145	. . . . 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 130	. . . 4 ((a →1 d) ∩ f) ≤ ((c →1 d) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))
23		ancom 68	. . . . . 6 (b ∩ c) = (c ∩ b)
24		ancom 68	. . . . . 6 ((b →1 d) ∩ (c →1 d)) = ((c →1 d) ∩ (b →1 d))
25	23, 24	2or 67	. . . . 5 ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))) = ((c ∩ b) ∪ ((c →1 d) ∩ (b →1 d)))
26	25	d3oa 975	. . . 4 ((c →1 d) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))) ≤ (b →1 d)
27	22, 26	letr 129	. . 3 ((a →1 d) ∩ f) ≤ (b →1 d)
28	4	d3oa 975	. . 3 ((a →1 d) ∩ e) ≤ (b →1 d)
29	27, 28	lel2or 162	. 2 (((a →1 d) ∩ f) ∪ ((a →1 d) ∩ e)) ≤ (b →1 d)
30	14, 29	bltr 130	1 ((a →1 d) ∩ (e ∪ f)) ≤ (b →1 d)

Syntax hints:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7   →₁ wi1 13

This theorem is referenced by:  d6oa 977

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421  ax-oadist 974

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i0 42  df-i1 43  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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