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Theorem oatr 928
 Description: Reverse transformation lemma for studying the orthoarguesian law.
Hypothesis
Ref Expression
oatr.1 b ≤ (a1 c)
Assertion
Ref Expression
oatr (a ∩ (ab)) ≤ c

Proof of Theorem oatr
StepHypRef Expression
1 leo 158 . . . . 5 a ≤ (a ∪ (ac))
2 oatr.1 . . . . . 6 b ≤ (a1 c)
3 df-i1 44 . . . . . . 7 (a1 c) = (a ∪ (ac))
4 ax-a1 30 . . . . . . . . 9 a = a
54ax-r5 38 . . . . . . . 8 (a ∪ (ac)) = (a ∪ (ac))
65ax-r1 35 . . . . . . 7 (a ∪ (ac)) = (a ∪ (ac))
73, 6ax-r2 36 . . . . . 6 (a1 c) = (a ∪ (ac))
82, 7lbtr 139 . . . . 5 b ≤ (a ∪ (ac))
91, 8lel2or 170 . . . 4 (ab) ≤ (a ∪ (ac))
109lelan 167 . . 3 (a ∩ (ab)) ≤ (a ∩ (a ∪ (ac)))
11 omlan 448 . . 3 (a ∩ (a ∪ (ac))) = (ac)
1210, 11lbtr 139 . 2 (a ∩ (ab)) ≤ (ac)
13 lear 161 . 2 (ac) ≤ c
1412, 13letr 137 1 (a ∩ (ab)) ≤ c
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  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-i1 44  df-le1 130  df-le2 131 This theorem is referenced by:  oa4dtoc  969
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