Theorem oa3-2to4 988

Description: Derivation of 3-OA variant (4) from (2). (Contributed by NM, 24-Dec-1998.)

Hypothesis

Ref	Expression
oa3-2to4.1	((a →1 c) ∩ (a ∪ (b ∩ ((a ∩ b) ∪ ((a →1 c) ∩ (b →1 c)))))) ≤ c

Assertion

Ref	Expression
oa3-2to4	(a⊥ ∩ (a ∪ (b ∩ ((a ≡ b) ∪ ((a →1 c) ∩ (b →1 c)))))) ≤ c

Proof of Theorem oa3-2to4

Step	Hyp	Ref	Expression
1		oa3-4lem 983	. . 3 (a⊥ ∩ (a ∪ (b ∩ (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (((a ∩ c) ∪ (a⊥ ∩ 1)) ∩ ((b ∩ c) ∪ (b⊥ ∩ 1))))))) = (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 ∩ c) ∪ (a⊥ ∩ 1)) ∩ ((b ∩ c) ∪ (b⊥ ∩ 1)))))))
3		leid 148	. . 3 a⊥ ≤ a⊥
4		leid 148	. . 3 b⊥ ≤ b⊥
5		le1 146	. . 3 c⊥ ≤ 1
6		an1 106	. . . . . . 7 (c ∩ 1) = c
7		dff 101	. . . . . . . . . 10 0 = (a ∩ a⊥ )
8		dff 101	. . . . . . . . . 10 0 = (b ∩ b⊥ )
9	7, 8	2or 72	. . . . . . . . 9 (0 ∪ 0) = ((a ∩ a⊥ ) ∪ (b ∩ b⊥ ))
10	9	ax-r1 35	. . . . . . . 8 ((a ∩ a⊥ ) ∪ (b ∩ b⊥ )) = (0 ∪ 0)
11		or0 102	. . . . . . . 8 (0 ∪ 0) = 0
12	10, 11	ax-r2 36	. . . . . . 7 ((a ∩ a⊥ ) ∪ (b ∩ b⊥ )) = 0
13	6, 12	2or 72	. . . . . 6 ((c ∩ 1) ∪ ((a ∩ a⊥ ) ∪ (b ∩ b⊥ ))) = (c ∪ 0)
14		or0 102	. . . . . 6 (c ∪ 0) = c
15	13, 14	ax-r2 36	. . . . 5 ((c ∩ 1) ∪ ((a ∩ a⊥ ) ∪ (b ∩ b⊥ ))) = c
16	15	ax-r1 35	. . . 4 c = ((c ∩ 1) ∪ ((a ∩ a⊥ ) ∪ (b ∩ b⊥ )))
17		ax-a2 31	. . . 4 ((c ∩ 1) ∪ ((a ∩ a⊥ ) ∪ (b ∩ b⊥ ))) = (((a ∩ a⊥ ) ∪ (b ∩ b⊥ )) ∪ (c ∩ 1))
18	16, 17	ax-r2 36	. . 3 c = (((a ∩ a⊥ ) ∪ (b ∩ b⊥ )) ∪ (c ∩ 1))
19		oa3-2lemb 979	. . . 4 ((a →1 c) ∩ (a ∪ (b ∩ (((a ∩ b) ∪ ((a →1 c) ∩ (b →1 c))) ∪ (((a ∩ c) ∪ ((a →1 c) ∩ (c →1 c))) ∩ ((b ∩ c) ∪ ((b →1 c) ∩ (c →1 c)))))))) = ((a →1 c) ∩ (a ∪ (b ∩ ((a ∩ b) ∪ ((a →1 c) ∩ (b →1 c))))))
20		oa3-2to4.1	. . . 4 ((a →1 c) ∩ (a ∪ (b ∩ ((a ∩ b) ∪ ((a →1 c) ∩ (b →1 c)))))) ≤ c
21	19, 20	bltr 138	. . 3 ((a →1 c) ∩ (a ∪ (b ∩ (((a ∩ b) ∪ ((a →1 c) ∩ (b →1 c))) ∪ (((a ∩ c) ∪ ((a →1 c) ∩ (c →1 c))) ∩ ((b ∩ c) ∪ ((b →1 c) ∩ (c →1 c)))))))) ≤ c
22	3, 4, 5, 18, 21	oa4to6dual 964	. 2 (a⊥ ∩ (a ∪ (b ∩ (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (((a ∩ c) ∪ (a⊥ ∩ 1)) ∩ ((b ∩ c) ∪ (b⊥ ∩ 1))))))) ≤ c
23	2, 22	bltr 138	1 (a⊥ ∩ (a ∪ (b ∩ ((a ≡ b) ∪ ((a →1 c) ∩ (b →1 c)))))) ≤ c

This proof depends on syntax axioms:   ≤ wle 2  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →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

This proof 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 used by: (None)