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Theorem oa3to4lem3 947
 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))
Assertion
Ref Expression
oa3to4lem3 (a ∩ (b ∪ (d ∩ ((ac) ∪ (bd))))) ≤ (a ∩ ((a1 g) ∪ ((c1 g) ∩ ((ac) ∪ ((a1 g) ∩ (c1 g))))))

Proof of Theorem oa3to4lem3
StepHypRef Expression
1 oa3to4lem.1 . . . 4 ab
2 oa3to4lem.2 . . . 4 cd
3 oa3to4lem.3 . . . 4 g = ((ab) ∪ (cd))
41, 2, 3oa3to4lem1 945 . . 3 b ≤ (a1 g)
51, 2, 3oa3to4lem2 946 . . . 4 d ≤ (c1 g)
64, 5le2an 169 . . . . 5 (bd) ≤ ((a1 g) ∩ (c1 g))
76lelor 166 . . . 4 ((ac) ∪ (bd)) ≤ ((ac) ∪ ((a1 g) ∩ (c1 g)))
85, 7le2an 169 . . 3 (d ∩ ((ac) ∪ (bd))) ≤ ((c1 g) ∩ ((ac) ∪ ((a1 g) ∩ (c1 g))))
94, 8le2or 168 . 2 (b ∪ (d ∩ ((ac) ∪ (bd)))) ≤ ((a1 g) ∪ ((c1 g) ∩ ((ac) ∪ ((a1 g) ∩ (c1 g)))))
109lelan 167 1 (a ∩ (b ∪ (d ∩ ((ac) ∪ (bd))))) ≤ (a ∩ ((a1 g) ∪ ((c1 g) ∩ ((ac) ∪ ((a1 g) ∩ (c1 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:  oa3to4lem4  948
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