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Theorem oale 829
 Description: Relation for studying OA.
Assertion
Ref Expression
oale ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))) ) ≤ (a2 c)

Proof of Theorem oale
StepHypRef Expression
1 df-i2 45 . . . . . . 7 ((bc) →2 ((a2 b) ∩ (a2 c))) = (((a2 b) ∩ (a2 c)) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) ))
21lan 77 . . . . . 6 ((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (((a2 b) ∩ (a2 c)) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) )))
3 coman1 185 . . . . . . 7 ((a2 b) ∩ (a2 c)) C (a2 b)
4 comanr2 465 . . . . . . . 8 ((a2 b) ∩ (a2 c)) C ((bc) ∩ ((a2 b) ∩ (a2 c)) )
54comcom6 459 . . . . . . 7 ((a2 b) ∩ (a2 c)) C ((bc) ∩ ((a2 b) ∩ (a2 c)) )
63, 5fh2 470 . . . . . 6 ((a2 b) ∩ (((a2 b) ∩ (a2 c)) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) ))) = (((a2 b) ∩ ((a2 b) ∩ (a2 c))) ∪ ((a2 b) ∩ ((bc) ∩ ((a2 b) ∩ (a2 c)) )))
7 anass 76 . . . . . . . . . 10 (((a2 b) ∩ (a2 b)) ∩ (a2 c)) = ((a2 b) ∩ ((a2 b) ∩ (a2 c)))
87ax-r1 35 . . . . . . . . 9 ((a2 b) ∩ ((a2 b) ∩ (a2 c))) = (((a2 b) ∩ (a2 b)) ∩ (a2 c))
9 anidm 111 . . . . . . . . . 10 ((a2 b) ∩ (a2 b)) = (a2 b)
109ran 78 . . . . . . . . 9 (((a2 b) ∩ (a2 b)) ∩ (a2 c)) = ((a2 b) ∩ (a2 c))
118, 10ax-r2 36 . . . . . . . 8 ((a2 b) ∩ ((a2 b) ∩ (a2 c))) = ((a2 b) ∩ (a2 c))
12 anor3 90 . . . . . . . . 9 ((bc) ∩ ((a2 b) ∩ (a2 c)) ) = ((bc) ∪ ((a2 b) ∩ (a2 c)))
1312lan 77 . . . . . . . 8 ((a2 b) ∩ ((bc) ∩ ((a2 b) ∩ (a2 c)) )) = ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))) )
1411, 132or 72 . . . . . . 7 (((a2 b) ∩ ((a2 b) ∩ (a2 c))) ∪ ((a2 b) ∩ ((bc) ∩ ((a2 b) ∩ (a2 c)) ))) = (((a2 b) ∩ (a2 c)) ∪ ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))) ))
15 ax-a2 31 . . . . . . 7 (((a2 b) ∩ (a2 c)) ∪ ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))) )) = (((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))) ) ∪ ((a2 b) ∩ (a2 c)))
1614, 15ax-r2 36 . . . . . 6 (((a2 b) ∩ ((a2 b) ∩ (a2 c))) ∪ ((a2 b) ∩ ((bc) ∩ ((a2 b) ∩ (a2 c)) ))) = (((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))) ) ∪ ((a2 b) ∩ (a2 c)))
172, 6, 163tr 65 . . . . 5 ((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c)))) = (((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))) ) ∪ ((a2 b) ∩ (a2 c)))
1817ax-r1 35 . . . 4 (((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))) ) ∪ ((a2 b) ∩ (a2 c))) = ((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c))))
19 2oath1 826 . . . 4 ((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (a2 c))
2018, 19ax-r2 36 . . 3 (((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))) ) ∪ ((a2 b) ∩ (a2 c))) = ((a2 b) ∩ (a2 c))
2120df-le1 130 . 2 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))) ) ≤ ((a2 b) ∩ (a2 c))
22 lear 161 . 2 ((a2 b) ∩ (a2 c)) ≤ (a2 c)
2321, 22letr 137 1 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))) ) ≤ (a2 c)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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