Theorem oa3-3lem 981

Description: Lemma for 3-OA(3). Equivalence with substitution into 6-OA dual.

Assertion

Ref	Expression
oa3-3lem	(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))))))

Proof of Theorem oa3-3lem

Step	Hyp	Ref	Expression
1		dfb 94	. . . . . 6 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
2	1	ax-r1 35	. . . . 5 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) = (a ≡ b)
3		an1 106	. . . . . . . . 9 (a ∩ 1) = a
4		ax-a1 30	. . . . . . . . 9 a = a⊥ ⊥
5	3, 4	ax-r2 36	. . . . . . . 8 (a ∩ 1) = a⊥ ⊥
6	5	ax-r5 38	. . . . . . 7 ((a ∩ 1) ∪ (a⊥ ∩ c)) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
7		df-i1 44	. . . . . . . 8 (a⊥ →1 c) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
8	7	ax-r1 35	. . . . . . 7 (a⊥ ⊥ ∪ (a⊥ ∩ c)) = (a⊥ →1 c)
9	6, 8	ax-r2 36	. . . . . 6 ((a ∩ 1) ∪ (a⊥ ∩ c)) = (a⊥ →1 c)
10		an1 106	. . . . . . . . 9 (b ∩ 1) = b
11		ax-a1 30	. . . . . . . . 9 b = b⊥ ⊥
12	10, 11	ax-r2 36	. . . . . . . 8 (b ∩ 1) = b⊥ ⊥
13	12	ax-r5 38	. . . . . . 7 ((b ∩ 1) ∪ (b⊥ ∩ c)) = (b⊥ ⊥ ∪ (b⊥ ∩ c))
14		df-i1 44	. . . . . . . 8 (b⊥ →1 c) = (b⊥ ⊥ ∪ (b⊥ ∩ c))
15	14	ax-r1 35	. . . . . . 7 (b⊥ ⊥ ∪ (b⊥ ∩ c)) = (b⊥ →1 c)
16	13, 15	ax-r2 36	. . . . . 6 ((b ∩ 1) ∪ (b⊥ ∩ c)) = (b⊥ →1 c)
17	9, 16	2an 79	. . . . 5 (((a ∩ 1) ∪ (a⊥ ∩ c)) ∩ ((b ∩ 1) ∪ (b⊥ ∩ c))) = ((a⊥ →1 c) ∩ (b⊥ →1 c))
18	2, 17	2or 72	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (((a ∩ 1) ∪ (a⊥ ∩ c)) ∩ ((b ∩ 1) ∪ (b⊥ ∩ c)))) = ((a ≡ b) ∪ ((a⊥ →1 c) ∩ (b⊥ →1 c)))
19	18	lan 77	. . 3 (b ∩ (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (((a ∩ 1) ∪ (a⊥ ∩ c)) ∩ ((b ∩ 1) ∪ (b⊥ ∩ c))))) = (b ∩ ((a ≡ b) ∪ ((a⊥ →1 c) ∩ (b⊥ →1 c))))
20	19	lor 70	. 2 (a ∪ (b ∩ (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (((a ∩ 1) ∪ (a⊥ ∩ c)) ∩ ((b ∩ 1) ∪ (b⊥ ∩ c)))))) = (a ∪ (b ∩ ((a ≡ b) ∪ ((a⊥ →1 c) ∩ (b⊥ →1 c)))))
21	20	lan 77	1 (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))))))

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:  oa3-6to3  987