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Theorem lem3.3.5 1055
 Description: Equation 3.5 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton, 3-Jul-05.)
Hypothesis
Ref Expression
lem3.3.5.1 (a5 b) = 1
Assertion
Ref Expression
lem3.3.5 (a1 (bc)) = 1

Proof of Theorem lem3.3.5
StepHypRef Expression
1 df-b1 1048 . . . . . 6 (a1 b) = ((a1 b) ∩ (b1 a))
2 lea 160 . . . . . 6 ((a1 b) ∩ (b1 a)) ≤ (a1 b)
31, 2bltr 138 . . . . 5 (a1 b) ≤ (a1 b)
4 df-i1 44 . . . . 5 (a1 b) = (a ∪ (ab))
53, 4lbtr 139 . . . 4 (a1 b) ≤ (a ∪ (ab))
6 leo 158 . . . . . 6 b ≤ (bc)
76lelan 167 . . . . 5 (ab) ≤ (a ∩ (bc))
87lelor 166 . . . 4 (a ∪ (ab)) ≤ (a ∪ (a ∩ (bc)))
95, 8letr 137 . . 3 (a1 b) ≤ (a ∪ (a ∩ (bc)))
10 lem3.3.5.1 . . . . 5 (a5 b) = 1
11 lem3.3.3 1052 . . . . 5 ((a5 b) →0 (a1 b)) = 1
1210, 11lem3.3.2 1046 . . . 4 (a1 b) = 1
1312ax-r1 35 . . 3 1 = (a1 b)
14 df-i1 44 . . 3 (a1 (bc)) = (a ∪ (a ∩ (bc)))
159, 13, 14le3tr1 140 . 2 1 ≤ (a1 (bc))
1615lem3.3.5lem 1054 1 (a1 (bc)) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →1 wi1 12   ≡5 wid5 22   ↔1 wb1 24 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-i0 43  df-i1 44  df-le1 130  df-le2 131  df-id5 1047  df-b1 1048 This theorem is referenced by:  lem3.4.5  1078
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