Theorem i3le 515

Description: L.e. to Kalmbach implication. (Contributed by NM, 7-Nov-1997.)

Hypothesis

Ref	Expression
i3le.1	(a →3 b) = 1

Assertion

Ref	Expression
i3le	a ≤ b

Proof of Theorem i3le

Step	Hyp	Ref	Expression
1		ancom 74	. . . 4 (1 ∩ b⊥ ) = (b⊥ ∩ 1)
2		i3le.1	. . . . . 6 (a →3 b) = 1
3	2	i3lem3 506	. . . . 5 ((a⊥ ∪ b) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
4	2	i3lem4 507	. . . . . 6 (a⊥ ∪ b) = 1
5	4	ran 78	. . . . 5 ((a⊥ ∪ b) ∩ b⊥ ) = (1 ∩ b⊥ )
6		ancom 74	. . . . 5 (a⊥ ∩ b⊥ ) = (b⊥ ∩ a⊥ )
7	3, 5, 6	3tr2 64	. . . 4 (1 ∩ b⊥ ) = (b⊥ ∩ a⊥ )
8		an1 106	. . . 4 (b⊥ ∩ 1) = b⊥
9	1, 7, 8	3tr2 64	. . 3 (b⊥ ∩ a⊥ ) = b⊥
10	9	df2le1 135	. 2 b⊥ ≤ a⊥
11	10	lecon1 155	1 a ≤ b

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  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  binr1  517  binr2  518  binr3  519  i3ri3  538  i3li3  539  i32i3  540  u3lemle2  717