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Theorem oa3to4lem4 948
 Description: Lemma for orthoarguesian law (Godowski/Greechie 3-variable to 4-variable proof).
Hypotheses
Ref Expression
oa3to4lem.1 ab
oa3to4lem.2 cd
oa3to4lem.3 g = ((ab) ∪ (cd))
oa3to4lem.oa3 (a ∩ ((a1 g) ∪ ((c1 g) ∩ ((ac) ∪ ((a1 g) ∩ (c1 g)))))) ≤ ((ag) ∪ (cg))
Assertion
Ref Expression
oa3to4lem4 (a ∩ (b ∪ (d ∩ ((ac) ∪ (bd))))) ≤ g

Proof of Theorem oa3to4lem4
StepHypRef Expression
1 oa3to4lem.1 . . 3 ab
2 oa3to4lem.2 . . 3 cd
3 oa3to4lem.3 . . 3 g = ((ab) ∪ (cd))
41, 2, 3oa3to4lem3 947 . 2 (a ∩ (b ∪ (d ∩ ((ac) ∪ (bd))))) ≤ (a ∩ ((a1 g) ∪ ((c1 g) ∩ ((ac) ∪ ((a1 g) ∩ (c1 g))))))
5 oa3to4lem.oa3 . . 3 (a ∩ ((a1 g) ∪ ((c1 g) ∩ ((ac) ∪ ((a1 g) ∩ (c1 g)))))) ≤ ((ag) ∪ (cg))
6 lear 161 . . . 4 (ag) ≤ g
7 lear 161 . . . 4 (cg) ≤ g
86, 7lel2or 170 . . 3 ((ag) ∪ (cg)) ≤ g
95, 8letr 137 . 2 (a ∩ ((a1 g) ∪ ((c1 g) ∩ ((ac) ∪ ((a1 g) ∩ (c1 g)))))) ≤ g
104, 9letr 137 1 (a ∩ (b ∪ (d ∩ ((ac) ∪ (bd))))) ≤ g
 Colors of variables: term Syntax hints:   = wb 1   ≤ 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-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:  oa3to4lem6  950
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