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Theorem d6oa 997
 Description: Derivation of 6-variable orthoarguesian law from OA distributive law.
Hypotheses
Ref Expression
d6oa.1 ab
d6oa.2 cd
d6oa.3 ef
Assertion
Ref Expression
d6oa (((ab) ∩ (cd)) ∩ (ef)) ≤ (b ∪ (a ∩ (c ∪ (((ac) ∩ (bd)) ∩ (((ae) ∩ (bf)) ∪ ((ce) ∩ (df)))))))

Proof of Theorem d6oa
StepHypRef Expression
1 d6oa.1 . 2 ab
2 d6oa.2 . 2 cd
3 d6oa.3 . 2 ef
4 id 59 . 2 (((ab ) ∪ (cd )) ∪ (ef )) = (((ab ) ∪ (cd )) ∪ (ef ))
5 id 59 . 2 a = a
6 id 59 . 2 c = c
7 id 59 . 2 e = e
8 id 59 . . . . 5 (((c1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (a1 (((ab ) ∪ (cd )) ∪ (ef )))) ∪ (((c1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ ((a1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef ))))) = (((c1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (a1 (((ab ) ∪ (cd )) ∪ (ef )))) ∪ (((c1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ ((a1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef )))))
9 id 59 . . . . 5 ((((c1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (e1 (((ab ) ∪ (cd )) ∪ (ef )))) ∪ (((c1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ ((e1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef ))))) ∩ (((a1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (e1 (((ab ) ∪ (cd )) ∪ (ef )))) ∪ (((a1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ ((e1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef )))))) = ((((c1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (e1 (((ab ) ∪ (cd )) ∪ (ef )))) ∪ (((c1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ ((e1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef ))))) ∩ (((a1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (e1 (((ab ) ∪ (cd )) ∪ (ef )))) ∪ (((a1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ ((e1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef ))))))
108, 9d4oa 996 . . . 4 (((c1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ ((((c1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (a1 (((ab ) ∪ (cd )) ∪ (ef )))) ∪ (((c1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ ((a1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef ))))) ∪ ((((c1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (e1 (((ab ) ∪ (cd )) ∪ (ef )))) ∪ (((c1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ ((e1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef ))))) ∩ (((a1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (e1 (((ab ) ∪ (cd )) ∪ (ef )))) ∪ (((a1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ ((e1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef )))))))) ≤ ((a1 (((ab ) ∪ (cd )) ∪ (ef ))) →1 (((ab ) ∪ (cd )) ∪ (ef )))
11 id 59 . . . 4 (a1 (((ab ) ∪ (cd )) ∪ (ef ))) = (a1 (((ab ) ∪ (cd )) ∪ (ef )))
12 id 59 . . . 4 (c1 (((ab ) ∪ (cd )) ∪ (ef ))) = (c1 (((ab ) ∪ (cd )) ∪ (ef )))
13 id 59 . . . 4 (e1 (((ab ) ∪ (cd )) ∪ (ef ))) = (e1 (((ab ) ∪ (cd )) ∪ (ef )))
1410, 11, 12, 13oa4gto4u 976 . . 3 ((a1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ ((a 1 (((ab ) ∪ (cd )) ∪ (ef ))) ∪ ((c 1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ ((((a1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (c1 (((ab ) ∪ (cd )) ∪ (ef )))) ∪ ((a 1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (c 1 (((ab ) ∪ (cd )) ∪ (ef ))))) ∪ ((((a1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (e1 (((ab ) ∪ (cd )) ∪ (ef )))) ∪ ((a 1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (e 1 (((ab ) ∪ (cd )) ∪ (ef ))))) ∩ (((c1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (e1 (((ab ) ∪ (cd )) ∪ (ef )))) ∪ ((c 1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (e 1 (((ab ) ∪ (cd )) ∪ (ef )))))))))) ≤ (((ab ) ∪ (cd )) ∪ (ef ))
1514oa4uto4 977 . 2 ((a1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (a ∪ (c ∩ (((ac ) ∪ ((a1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (c1 (((ab ) ∪ (cd )) ∪ (ef ))))) ∪ (((ae ) ∪ ((a1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (e1 (((ab ) ∪ (cd )) ∪ (ef ))))) ∩ ((ce ) ∪ ((c1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (e1 (((ab ) ∪ (cd )) ∪ (ef )))))))))) ≤ (((ab ) ∪ (cd )) ∪ (ef ))
161, 2, 3, 4, 5, 6, 7, 15oa4to6 965 1 (((ab) ∩ (cd)) ∩ (ef)) ≤ (b ∪ (a ∩ (c ∪ (((ac) ∩ (bd)) ∩ (((ae) ∩ (bf)) ∪ ((ce) ∩ (df)))))))
 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-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439  ax-oadist 994 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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