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Theorem mlaconjolem 885
 Description: Lemma for OML proof of Mladen's conjecture, (Contributed by NM, 10-Mar-2002.)
Assertion
Ref Expression
mlaconjolem ((ac) ∪ (bc)) ≤ ((c ∩ (ab)) ∪ (c ∩ (ab )))

Proof of Theorem mlaconjolem
StepHypRef Expression
1 orbile 843 . 2 ((ac) ∪ (bc)) ≤ (((ab) →2 c) ∩ (c1 (ab)))
2 df-i2 45 . . . . 5 ((ab) →2 c) = (c ∪ ((ab)c ))
3 oran3 93 . . . . . . . 8 (ab ) = (ab)
43ran 78 . . . . . . 7 ((ab ) ∩ c ) = ((ab)c )
54lor 70 . . . . . 6 (c ∪ ((ab ) ∩ c )) = (c ∪ ((ab)c ))
65ax-r1 35 . . . . 5 (c ∪ ((ab)c )) = (c ∪ ((ab ) ∩ c ))
72, 6ax-r2 36 . . . 4 ((ab) →2 c) = (c ∪ ((ab ) ∩ c ))
8 df-i1 44 . . . 4 (c1 (ab)) = (c ∪ (c ∩ (ab)))
97, 82an 79 . . 3 (((ab) →2 c) ∩ (c1 (ab))) = ((c ∪ ((ab ) ∩ c )) ∩ (c ∪ (c ∩ (ab))))
10 comor1 461 . . . . 5 (c ∪ ((ab ) ∩ c )) C c
1110comcom2 183 . . . 4 (c ∪ ((ab ) ∩ c )) C c
12 leao1 162 . . . . . 6 (c ∩ (ab)) ≤ (c ∪ ((ab ) ∩ c ))
1312lecom 180 . . . . 5 (c ∩ (ab)) C (c ∪ ((ab ) ∩ c ))
1413comcom 453 . . . 4 (c ∪ ((ab ) ∩ c )) C (c ∩ (ab))
1511, 14fh1 469 . . 3 ((c ∪ ((ab ) ∩ c )) ∩ (c ∪ (c ∩ (ab)))) = (((c ∪ ((ab ) ∩ c )) ∩ c ) ∪ ((c ∪ ((ab ) ∩ c )) ∩ (c ∩ (ab))))
16 ancom 74 . . . . . . . 8 ((ab ) ∩ c ) = (c ∩ (ab ))
1716lor 70 . . . . . . 7 (c ∪ ((ab ) ∩ c )) = (c ∪ (c ∩ (ab )))
1817ran 78 . . . . . 6 ((c ∪ ((ab ) ∩ c )) ∩ c ) = ((c ∪ (c ∩ (ab ))) ∩ c )
19 ancom 74 . . . . . 6 ((c ∪ (c ∩ (ab ))) ∩ c ) = (c ∩ (c ∪ (c ∩ (ab ))))
20 omlan 448 . . . . . 6 (c ∩ (c ∪ (c ∩ (ab )))) = (c ∩ (ab ))
2118, 19, 203tr 65 . . . . 5 ((c ∪ ((ab ) ∩ c )) ∩ c ) = (c ∩ (ab ))
22 ancom 74 . . . . . 6 ((c ∪ ((ab ) ∩ c )) ∩ (c ∩ (ab))) = ((c ∩ (ab)) ∩ (c ∪ ((ab ) ∩ c )))
2312df2le2 136 . . . . . 6 ((c ∩ (ab)) ∩ (c ∪ ((ab ) ∩ c ))) = (c ∩ (ab))
2422, 23ax-r2 36 . . . . 5 ((c ∪ ((ab ) ∩ c )) ∩ (c ∩ (ab))) = (c ∩ (ab))
2521, 242or 72 . . . 4 (((c ∪ ((ab ) ∩ c )) ∩ c ) ∪ ((c ∪ ((ab ) ∩ c )) ∩ (c ∩ (ab)))) = ((c ∩ (ab )) ∪ (c ∩ (ab)))
26 ax-a2 31 . . . 4 ((c ∩ (ab )) ∪ (c ∩ (ab))) = ((c ∩ (ab)) ∪ (c ∩ (ab )))
2725, 26ax-r2 36 . . 3 (((c ∪ ((ab ) ∩ c )) ∩ c ) ∪ ((c ∪ ((ab ) ∩ c )) ∩ (c ∩ (ab)))) = ((c ∩ (ab)) ∪ (c ∩ (ab )))
289, 15, 273tr 65 . 2 (((ab) →2 c) ∩ (c1 (ab))) = ((c ∩ (ab)) ∪ (c ∩ (ab )))
291, 28lbtr 139 1 ((ac) ∪ (bc)) ≤ ((c ∩ (ab)) ∪ (c ∩ (ab )))
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7   →1 wi1 12   →2 wi2 13 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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  mlaconjo  886
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