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Theorem oat 927
 Description: Transformation lemma for studying the orthoarguesian law.
Hypothesis
Ref Expression
oat.1 (a ∩ (ab)) ≤ c
Assertion
Ref Expression
oat b ≤ (a1 c)

Proof of Theorem oat
StepHypRef Expression
1 leor 159 . . 3 b ≤ (ab)
2 oml 445 . . . . 5 (a ∪ (a ∩ (ab))) = (ab)
32ax-r1 35 . . . 4 (ab) = (a ∪ (a ∩ (ab)))
4 lea 160 . . . . . 6 (a ∩ (ab)) ≤ a
5 oat.1 . . . . . 6 (a ∩ (ab)) ≤ c
64, 5ler2an 173 . . . . 5 (a ∩ (ab)) ≤ (ac)
76lelor 166 . . . 4 (a ∪ (a ∩ (ab))) ≤ (a ∪ (ac))
83, 7bltr 138 . . 3 (ab) ≤ (a ∪ (ac))
91, 8letr 137 . 2 b ≤ (a ∪ (ac))
10 ax-a1 30 . . . 4 a = a
1110ax-r5 38 . . 3 (a ∪ (ac)) = (a ∪ (ac))
12 df-i1 44 . . . 4 (a1 c) = (a ∪ (ac))
1312ax-r1 35 . . 3 (a ∪ (ac)) = (a1 c)
1411, 13ax-r2 36 . 2 (a ∪ (ac)) = (a1 c)
159, 14lbtr 139 1 b ≤ (a1 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:  oa4ctod  968
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