Theorem oa3-6to3 987

Description: Derivation of 3-OA variant (3) from (6). (Contributed by NM, 24-Dec-1998.)

Hypothesis

Ref	Expression
oa3-6to3.1	((a →1 c) ∩ (a ∪ (b ∩ (((a⊥ →1 c) ∩ (b⊥ →1 c)) ∪ ((a →1 c) ∩ (b →1 c)))))) ≤ c

Assertion

Ref	Expression
oa3-6to3	(a⊥ ∩ (a ∪ (b ∩ ((a ≡ b) ∪ ((a⊥ →1 c) ∩ (b⊥ →1 c)))))) ≤ c

Proof of Theorem oa3-6to3

Step	Hyp	Ref	Expression
1		oa3-3lem 981	. . 3 (a⊥ ∩ (a ∪ (b ∩ (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (((a ∩ 1) ∪ (a⊥ ∩ c)) ∩ ((b ∩ 1) ∪ (b⊥ ∩ c))))))) = (a⊥ ∩ (a ∪ (b ∩ ((a ≡ b) ∪ ((a⊥ →1 c) ∩ (b⊥ →1 c))))))
2	1	ax-r1 35	. 2 (a⊥ ∩ (a ∪ (b ∩ ((a ≡ b) ∪ ((a⊥ →1 c) ∩ (b⊥ →1 c)))))) = (a⊥ ∩ (a ∪ (b ∩ (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (((a ∩ 1) ∪ (a⊥ ∩ c)) ∩ ((b ∩ 1) ∪ (b⊥ ∩ c)))))))
3		leid 148	. . 3 a⊥ ≤ a⊥
4		leid 148	. . 3 b⊥ ≤ b⊥
5		df-f 42	. . . . 5 0 = 1⊥
6	5	ax-r1 35	. . . 4 1⊥ = 0
7		le0 147	. . . 4 0 ≤ c
8	6, 7	bltr 138	. . 3 1⊥ ≤ c
9		ancom 74	. . . . . . . 8 (1 ∩ c) = (c ∩ 1)
10		an1 106	. . . . . . . 8 (c ∩ 1) = c
11	9, 10	ax-r2 36	. . . . . . 7 (1 ∩ c) = c
12		dff 101	. . . . . . . . . 10 0 = (a ∩ a⊥ )
13		dff 101	. . . . . . . . . 10 0 = (b ∩ b⊥ )
14	12, 13	2or 72	. . . . . . . . 9 (0 ∪ 0) = ((a ∩ a⊥ ) ∪ (b ∩ b⊥ ))
15	14	ax-r1 35	. . . . . . . 8 ((a ∩ a⊥ ) ∪ (b ∩ b⊥ )) = (0 ∪ 0)
16		or0 102	. . . . . . . 8 (0 ∪ 0) = 0
17	15, 16	ax-r2 36	. . . . . . 7 ((a ∩ a⊥ ) ∪ (b ∩ b⊥ )) = 0
18	11, 17	2or 72	. . . . . 6 ((1 ∩ c) ∪ ((a ∩ a⊥ ) ∪ (b ∩ b⊥ ))) = (c ∪ 0)
19		or0 102	. . . . . 6 (c ∪ 0) = c
20	18, 19	ax-r2 36	. . . . 5 ((1 ∩ c) ∪ ((a ∩ a⊥ ) ∪ (b ∩ b⊥ ))) = c
21	20	ax-r1 35	. . . 4 c = ((1 ∩ c) ∪ ((a ∩ a⊥ ) ∪ (b ∩ b⊥ )))
22		ax-a2 31	. . . 4 ((1 ∩ c) ∪ ((a ∩ a⊥ ) ∪ (b ∩ b⊥ ))) = (((a ∩ a⊥ ) ∪ (b ∩ b⊥ )) ∪ (1 ∩ c))
23	21, 22	ax-r2 36	. . 3 c = (((a ∩ a⊥ ) ∪ (b ∩ b⊥ )) ∪ (1 ∩ c))
24		oa3-6lem 980	. . . 4 ((a →1 c) ∩ (a ∪ (b ∩ (((a ∩ b) ∪ ((a →1 c) ∩ (b →1 c))) ∪ (((a ∩ 1) ∪ ((a →1 c) ∩ (1 →1 c))) ∩ ((b ∩ 1) ∪ ((b →1 c) ∩ (1 →1 c)))))))) = ((a →1 c) ∩ (a ∪ (b ∩ (((a⊥ →1 c) ∩ (b⊥ →1 c)) ∪ ((a →1 c) ∩ (b →1 c))))))
25		oa3-6to3.1	. . . 4 ((a →1 c) ∩ (a ∪ (b ∩ (((a⊥ →1 c) ∩ (b⊥ →1 c)) ∪ ((a →1 c) ∩ (b →1 c)))))) ≤ c
26	24, 25	bltr 138	. . 3 ((a →1 c) ∩ (a ∪ (b ∩ (((a ∩ b) ∪ ((a →1 c) ∩ (b →1 c))) ∪ (((a ∩ 1) ∪ ((a →1 c) ∩ (1 →1 c))) ∩ ((b ∩ 1) ∪ ((b →1 c) ∩ (1 →1 c)))))))) ≤ c
27	3, 4, 8, 23, 26	oa4to6dual 964	. 2 (a⊥ ∩ (a ∪ (b ∩ (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (((a ∩ 1) ∪ (a⊥ ∩ c)) ∩ ((b ∩ 1) ∪ (b⊥ ∩ c))))))) ≤ c
28	2, 27	bltr 138	1 (a⊥ ∩ (a ∪ (b ∩ ((a ≡ b) ∪ ((a⊥ →1 c) ∩ (b⊥ →1 c)))))) ≤ c

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →₁ wi1 12

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-r3 439

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

This theorem is referenced by: (None)