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Theorem oaur 930
 Description: Transformation lemma for studying the orthoarguesian law.
Hypothesis
Ref Expression
oaur.1 b ≤ (a1 c)
Assertion
Ref Expression
oaur (a ∩ ((a1 c) ∪ b)) ≤ c

Proof of Theorem oaur
StepHypRef Expression
1 leid 148 . . . . 5 (a1 c) ≤ (a1 c)
2 oaur.1 . . . . 5 b ≤ (a1 c)
31, 2lel2or 170 . . . 4 ((a1 c) ∪ b) ≤ (a1 c)
43lelan 167 . . 3 (a ∩ ((a1 c) ∪ b)) ≤ (a ∩ (a1 c))
5 ancom 74 . . . 4 (a ∩ (a1 c)) = ((a1 c) ∩ a)
6 u1lemaa 600 . . . 4 ((a1 c) ∩ a) = (ac)
75, 6ax-r2 36 . . 3 (a ∩ (a1 c)) = (ac)
84, 7lbtr 139 . 2 (a ∩ ((a1 c) ∪ b)) ≤ (ac)
9 lear 161 . 2 (ac) ≤ c
108, 9letr 137 1 (a ∩ ((a1 c) ∪ b)) ≤ c
 Colors of variables: term Syntax hints:   ≤ wle 2   ∪ wo 6   ∩ wa 7   →1 wi1 12 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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  oa4gto4u  976
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