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Theorem i3le 515
 Description: L.e. to Kalmbach implication.
Hypothesis
Ref Expression
i3le.1 (a3 b) = 1
Assertion
Ref Expression
i3le ab

Proof of Theorem i3le
StepHypRef Expression
1 ancom 74 . . . 4 (1 ∩ b ) = (b ∩ 1)
2 i3le.1 . . . . . 6 (a3 b) = 1
32i3lem3 506 . . . . 5 ((ab) ∩ b ) = (ab )
42i3lem4 507 . . . . . 6 (ab) = 1
54ran 78 . . . . 5 ((ab) ∩ b ) = (1 ∩ b )
6 ancom 74 . . . . 5 (ab ) = (ba )
73, 5, 63tr2 64 . . . 4 (1 ∩ b ) = (ba )
8 an1 106 . . . 4 (b ∩ 1) = b
91, 7, 83tr2 64 . . 3 (ba ) = b
109df2le1 135 . 2 ba
1110lecon1 155 1 ab
 Colors of variables: term 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
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