Theorem oa3-4lem 983

Description: Lemma for 3-OA(4). Equivalence with substitution into 6-OA dual. (Contributed by NM, 25-Dec-1998.)

Assertion

Ref	Expression
oa3-4lem	(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))))))

Proof of Theorem oa3-4lem

Step	Hyp	Ref	Expression
1		dfb 94	. . . . . 6 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
2		ax-a2 31	. . . . . . . 8 (a⊥ ∪ (a ∩ c)) = ((a ∩ c) ∪ a⊥ )
3		df-i1 44	. . . . . . . 8 (a →1 c) = (a⊥ ∪ (a ∩ c))
4		an1 106	. . . . . . . . 9 (a⊥ ∩ 1) = a⊥
5	4	lor 70	. . . . . . . 8 ((a ∩ c) ∪ (a⊥ ∩ 1)) = ((a ∩ c) ∪ a⊥ )
6	2, 3, 5	3tr1 63	. . . . . . 7 (a →1 c) = ((a ∩ c) ∪ (a⊥ ∩ 1))
7		ax-a2 31	. . . . . . . 8 (b⊥ ∪ (b ∩ c)) = ((b ∩ c) ∪ b⊥ )
8		df-i1 44	. . . . . . . 8 (b →1 c) = (b⊥ ∪ (b ∩ c))
9		an1 106	. . . . . . . . 9 (b⊥ ∩ 1) = b⊥
10	9	lor 70	. . . . . . . 8 ((b ∩ c) ∪ (b⊥ ∩ 1)) = ((b ∩ c) ∪ b⊥ )
11	7, 8, 10	3tr1 63	. . . . . . 7 (b →1 c) = ((b ∩ c) ∪ (b⊥ ∩ 1))
12	6, 11	2an 79	. . . . . 6 ((a →1 c) ∩ (b →1 c)) = (((a ∩ c) ∪ (a⊥ ∩ 1)) ∩ ((b ∩ c) ∪ (b⊥ ∩ 1)))
13	1, 12	2or 72	. . . . 5 ((a ≡ b) ∪ ((a →1 c) ∩ (b →1 c))) = (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (((a ∩ c) ∪ (a⊥ ∩ 1)) ∩ ((b ∩ c) ∪ (b⊥ ∩ 1))))
14	13	ax-r1 35	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (((a ∩ c) ∪ (a⊥ ∩ 1)) ∩ ((b ∩ c) ∪ (b⊥ ∩ 1)))) = ((a ≡ b) ∪ ((a →1 c) ∩ (b →1 c)))
15	14	lan 77	. . 3 (b ∩ (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (((a ∩ c) ∪ (a⊥ ∩ 1)) ∩ ((b ∩ c) ∪ (b⊥ ∩ 1))))) = (b ∩ ((a ≡ b) ∪ ((a →1 c) ∩ (b →1 c))))
16	15	lor 70	. 2 (a ∪ (b ∩ (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (((a ∩ c) ∪ (a⊥ ∩ 1)) ∩ ((b ∩ c) ∪ (b⊥ ∩ 1)))))) = (a ∪ (b ∩ ((a ≡ b) ∪ ((a →1 c) ∩ (b →1 c)))))
17	16	lan 77	1 (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))))))

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44

This theorem is referenced by:  oa3-2to4  988