Theorem i2i1i1 800

Description: Equivalence to →₂ . (Contributed by NM, 5-Jul-2000.)

Assertion

Ref	Expression
i2i1i1	(a →2 b) = ((a →1 (a ∪ b)) ∩ ((a ∪ b) →1 b))

Proof of Theorem i2i1i1

Step	Hyp	Ref	Expression
1		an1r 107	. . 3 (1 ∩ (b ∪ (a⊥ ∩ b⊥ ))) = (b ∪ (a⊥ ∩ b⊥ ))
2	1	ax-r1 35	. 2 (b ∪ (a⊥ ∩ b⊥ )) = (1 ∩ (b ∪ (a⊥ ∩ b⊥ )))
3		df-i2 45	. 2 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
4		anabs 121	. . . . . 6 (a ∩ (a ∪ b)) = a
5	4	lor 70	. . . . 5 (a⊥ ∪ (a ∩ (a ∪ b))) = (a⊥ ∪ a)
6		ax-a2 31	. . . . 5 (a⊥ ∪ a) = (a ∪ a⊥ )
7	5, 6	ax-r2 36	. . . 4 (a⊥ ∪ (a ∩ (a ∪ b))) = (a ∪ a⊥ )
8		df-i1 44	. . . 4 (a →1 (a ∪ b)) = (a⊥ ∪ (a ∩ (a ∪ b)))
9		df-t 41	. . . 4 1 = (a ∪ a⊥ )
10	7, 8, 9	3tr1 63	. . 3 (a →1 (a ∪ b)) = 1
11		df-i1 44	. . . 4 ((a ∪ b) →1 b) = ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ b))
12		anor3 90	. . . . . 6 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
13		leor 159	. . . . . . . 8 b ≤ (a ∪ b)
14		leid 148	. . . . . . . 8 b ≤ b
15	13, 14	ler2an 173	. . . . . . 7 b ≤ ((a ∪ b) ∩ b)
16		lear 161	. . . . . . 7 ((a ∪ b) ∩ b) ≤ b
17	15, 16	lebi 145	. . . . . 6 b = ((a ∪ b) ∩ b)
18	12, 17	2or 72	. . . . 5 ((a⊥ ∩ b⊥ ) ∪ b) = ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ b))
19	18	ax-r1 35	. . . 4 ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ b)) = ((a⊥ ∩ b⊥ ) ∪ b)
20		ax-a2 31	. . . 4 ((a⊥ ∩ b⊥ ) ∪ b) = (b ∪ (a⊥ ∩ b⊥ ))
21	11, 19, 20	3tr 65	. . 3 ((a ∪ b) →1 b) = (b ∪ (a⊥ ∩ b⊥ ))
22	10, 21	2an 79	. 2 ((a →1 (a ∪ b)) ∩ ((a ∪ b) →1 b)) = (1 ∩ (b ∪ (a⊥ ∩ b⊥ )))
23	2, 3, 22	3tr1 63	1 (a →2 b) = ((a →1 (a ∪ b)) ∩ ((a ∪ b) →1 b))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by:  mlaconj  845