Theorem lem3.3.7i2e1 1063

Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 2, and this is the first part of the equation. (Contributed by Roy F. Longton, 28-Jun-2005.) (Revised by Roy F. Longton, 3-Jul-2005.)

Assertion

Ref	Expression
lem3.3.7i2e1	(a →2 (a ∩ b)) = (a ≡2 (a ∩ b))

Proof of Theorem lem3.3.7i2e1

Step	Hyp	Ref	Expression
1		or1r 105	. . . . . 6 (1 ∪ b⊥ ) = 1
2	1	ax-r1 35	. . . . 5 1 = (1 ∪ b⊥ )
3	2	ran 78	. . . 4 (1 ∩ ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ ))) = ((1 ∪ b⊥ ) ∩ ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )))
4		an1r 107	. . . 4 (1 ∩ ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ ))) = ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ ))
5		df-t 41	. . . . . 6 1 = (a ∪ a⊥ )
6	5	ax-r5 38	. . . . 5 (1 ∪ b⊥ ) = ((a ∪ a⊥ ) ∪ b⊥ )
7	6	ran 78	. . . 4 ((1 ∪ b⊥ ) ∩ ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ ))) = (((a ∪ a⊥ ) ∪ b⊥ ) ∩ ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )))
8	3, 4, 7	3tr2 64	. . 3 ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = (((a ∪ a⊥ ) ∪ b⊥ ) ∩ ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )))
9		ax-a3 32	. . . 4 ((a ∪ a⊥ ) ∪ b⊥ ) = (a ∪ (a⊥ ∪ b⊥ ))
10	9	ran 78	. . 3 (((a ∪ a⊥ ) ∪ b⊥ ) ∩ ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ ))) = ((a ∪ (a⊥ ∪ b⊥ )) ∩ ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )))
11		oran3 93	. . . . 5 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
12	11	lor 70	. . . 4 (a ∪ (a⊥ ∪ b⊥ )) = (a ∪ (a ∩ b)⊥ )
13	12	ran 78	. . 3 ((a ∪ (a⊥ ∪ b⊥ )) ∩ ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ ))) = ((a ∪ (a ∩ b)⊥ ) ∩ ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )))
14	8, 10, 13	3tr 65	. 2 ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = ((a ∪ (a ∩ b)⊥ ) ∩ ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )))
15		df-i2 45	. 2 (a →2 (a ∩ b)) = ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ ))
16		df-id2 51	. 2 (a ≡2 (a ∩ b)) = ((a ∪ (a ∩ b)⊥ ) ∩ ((a ∩ b) ∪ (a⊥ ∩ (a ∩ b)⊥ )))
17	14, 15, 16	3tr1 63	1 (a →2 (a ∩ b)) = (a ≡2 (a ∩ b))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₂ wi2 13   ≡₂ wid2 19

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

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i2 45  df-id2 51

This theorem is referenced by: (None)