Theorem oa6v4v 913

Description: 6-variable OA to 4-variable OA.

Hypotheses

Ref	Expression
oa6v4v.1	(((a ∪ b) ∩ (c ∪ d)) ∩ (e ∪ f)) ≤ (b ∪ (a ∩ (c ∪ (((a ∪ c) ∩ (b ∪ d)) ∩ (((a ∪ e) ∩ (b ∪ f)) ∪ ((c ∪ e) ∩ (d ∪ f)))))))
oa6v4v.2	e = 0
oa6v4v.3	f = 1

Assertion

Ref	Expression
oa6v4v	((a ∪ b) ∩ (c ∪ d)) ≤ (b ∪ (a ∩ (c ∪ ((a ∪ c) ∩ (b ∪ d)))))

Proof of Theorem oa6v4v

Step	Hyp	Ref	Expression
1		oa6v4v.1	. 2 (((a ∪ b) ∩ (c ∪ d)) ∩ (e ∪ f)) ≤ (b ∪ (a ∩ (c ∪ (((a ∪ c) ∩ (b ∪ d)) ∩ (((a ∪ e) ∩ (b ∪ f)) ∪ ((c ∪ e) ∩ (d ∪ f)))))))
2		oa6v4v.2	. . . . . 6 e = 0
3		oa6v4v.3	. . . . . 6 f = 1
4	2, 3	2or 67	. . . . 5 (e ∪ f) = (0 ∪ 1)
5		or0r 95	. . . . 5 (0 ∪ 1) = 1
6	4, 5	ax-r2 35	. . . 4 (e ∪ f) = 1
7	6	lan 70	. . 3 (((a ∪ b) ∩ (c ∪ d)) ∩ (e ∪ f)) = (((a ∪ b) ∩ (c ∪ d)) ∩ 1)
8		an1 98	. . 3 (((a ∪ b) ∩ (c ∪ d)) ∩ 1) = ((a ∪ b) ∩ (c ∪ d))
9	7, 8	ax-r2 35	. 2 (((a ∪ b) ∩ (c ∪ d)) ∩ (e ∪ f)) = ((a ∪ b) ∩ (c ∪ d))
10	2	lor 66	. . . . . . . . . . 11 (a ∪ e) = (a ∪ 0)
11		or0 94	. . . . . . . . . . 11 (a ∪ 0) = a
12	10, 11	ax-r2 35	. . . . . . . . . 10 (a ∪ e) = a
13	3	lor 66	. . . . . . . . . . 11 (b ∪ f) = (b ∪ 1)
14		or1 96	. . . . . . . . . . 11 (b ∪ 1) = 1
15	13, 14	ax-r2 35	. . . . . . . . . 10 (b ∪ f) = 1
16	12, 15	2an 72	. . . . . . . . 9 ((a ∪ e) ∩ (b ∪ f)) = (a ∩ 1)
17		an1 98	. . . . . . . . 9 (a ∩ 1) = a
18	16, 17	ax-r2 35	. . . . . . . 8 ((a ∪ e) ∩ (b ∪ f)) = a
19	2	lor 66	. . . . . . . . . . 11 (c ∪ e) = (c ∪ 0)
20		or0 94	. . . . . . . . . . 11 (c ∪ 0) = c
21	19, 20	ax-r2 35	. . . . . . . . . 10 (c ∪ e) = c
22	3	lor 66	. . . . . . . . . . 11 (d ∪ f) = (d ∪ 1)
23		or1 96	. . . . . . . . . . 11 (d ∪ 1) = 1
24	22, 23	ax-r2 35	. . . . . . . . . 10 (d ∪ f) = 1
25	21, 24	2an 72	. . . . . . . . 9 ((c ∪ e) ∩ (d ∪ f)) = (c ∩ 1)
26		an1 98	. . . . . . . . 9 (c ∩ 1) = c
27	25, 26	ax-r2 35	. . . . . . . 8 ((c ∪ e) ∩ (d ∪ f)) = c
28	18, 27	2or 67	. . . . . . 7 (((a ∪ e) ∩ (b ∪ f)) ∪ ((c ∪ e) ∩ (d ∪ f))) = (a ∪ c)
29	28	lan 70	. . . . . 6 (((a ∪ c) ∩ (b ∪ d)) ∩ (((a ∪ e) ∩ (b ∪ f)) ∪ ((c ∪ e) ∩ (d ∪ f)))) = (((a ∪ c) ∩ (b ∪ d)) ∩ (a ∪ c))
30		an32 76	. . . . . . 7 (((a ∪ c) ∩ (b ∪ d)) ∩ (a ∪ c)) = (((a ∪ c) ∩ (a ∪ c)) ∩ (b ∪ d))
31		anidm 103	. . . . . . . 8 ((a ∪ c) ∩ (a ∪ c)) = (a ∪ c)
32	31	ran 71	. . . . . . 7 (((a ∪ c) ∩ (a ∪ c)) ∩ (b ∪ d)) = ((a ∪ c) ∩ (b ∪ d))
33	30, 32	ax-r2 35	. . . . . 6 (((a ∪ c) ∩ (b ∪ d)) ∩ (a ∪ c)) = ((a ∪ c) ∩ (b ∪ d))
34	29, 33	ax-r2 35	. . . . 5 (((a ∪ c) ∩ (b ∪ d)) ∩ (((a ∪ e) ∩ (b ∪ f)) ∪ ((c ∪ e) ∩ (d ∪ f)))) = ((a ∪ c) ∩ (b ∪ d))
35	34	lor 66	. . . 4 (c ∪ (((a ∪ c) ∩ (b ∪ d)) ∩ (((a ∪ e) ∩ (b ∪ f)) ∪ ((c ∪ e) ∩ (d ∪ f))))) = (c ∪ ((a ∪ c) ∩ (b ∪ d)))
36	35	lan 70	. . 3 (a ∩ (c ∪ (((a ∪ c) ∩ (b ∪ d)) ∩ (((a ∪ e) ∩ (b ∪ f)) ∪ ((c ∪ e) ∩ (d ∪ f)))))) = (a ∩ (c ∪ ((a ∪ c) ∩ (b ∪ d))))
37	36	lor 66	. 2 (b ∪ (a ∩ (c ∪ (((a ∪ c) ∩ (b ∪ d)) ∩ (((a ∪ e) ∩ (b ∪ f)) ∪ ((c ∪ e) ∩ (d ∪ f))))))) = (b ∪ (a ∩ (c ∪ ((a ∪ c) ∩ (b ∪ d)))))
38	1, 9, 37	le3tr2 133	1 ((a ∪ b) ∩ (c ∪ d)) ≤ (b ∪ (a ∩ (c ∪ ((a ∪ c) ∩ (b ∪ d)))))

Syntax hints:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7  1wt 9  0wf 10

This theorem is referenced by:  oa64v 1010

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

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123

metamath.org