Theorem lei3 246

Description: L.e. to Kalmbach implication. (Contributed by NM, 3-Nov-1997.)

Hypothesis

Ref	Expression
lei3.1	a ≤ b

Assertion

Ref	Expression
lei3	(a →3 b) = 1

Proof of Theorem lei3

Step	Hyp	Ref	Expression
1		ax-a3 32	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = ((a⊥ ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b))))
2		ax-a2 31	. . . . 5 (b⊥ ∪ a) = (a ∪ b⊥ )
3		ancom 74	. . . . . . 7 (a⊥ ∩ b⊥ ) = (b⊥ ∩ a⊥ )
4		lei3.1	. . . . . . . . 9 a ≤ b
5	4	lecon 154	. . . . . . . 8 b⊥ ≤ a⊥
6	5	df2le2 136	. . . . . . 7 (b⊥ ∩ a⊥ ) = b⊥
7	3, 6	ax-r2 36	. . . . . 6 (a⊥ ∩ b⊥ ) = b⊥
8	4	sklem 230	. . . . . . . 8 (a⊥ ∪ b) = 1
9	8	lan 77	. . . . . . 7 (a ∩ (a⊥ ∪ b)) = (a ∩ 1)
10		an1 106	. . . . . . 7 (a ∩ 1) = a
11	9, 10	ax-r2 36	. . . . . 6 (a ∩ (a⊥ ∪ b)) = a
12	7, 11	2or 72	. . . . 5 ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b))) = (b⊥ ∪ a)
13		anor2 89	. . . . . 6 (a⊥ ∩ b) = (a ∪ b⊥ )⊥
14	13	con2 67	. . . . 5 (a⊥ ∩ b)⊥ = (a ∪ b⊥ )
15	2, 12, 14	3tr1 63	. . . 4 ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∩ b)⊥
16	15	lor 70	. . 3 ((a⊥ ∩ b) ∪ ((a⊥ ∩ b⊥ ) ∪ (a ∩ (a⊥ ∪ b)))) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b)⊥ )
17	1, 16	ax-r2 36	. 2 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b)⊥ )
18		df-i3 46	. 2 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
19		df-t 41	. 2 1 = ((a⊥ ∩ b) ∪ (a⊥ ∩ b)⊥ )
20	17, 18, 19	3tr1 63	1 (a →3 b) = 1

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₃ wi3 14

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-i3 46  df-le1 130  df-le2 131

This theorem is referenced by:  bina3  284  bina4  285  bina5  286  bii3  516  binr1  517  binr2  518  binr3  519  i3ri3  538  i3li3  539  i32i3  540  i3th5  547  i3th7  549  i3th8  550