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Theorem mloa 1018
Assertion
Ref Expression
mloa ((ab) ∩ ((bc) ∪ ((b ∪ (ab)) ∩ (c ∪ (ac))))) ≤ (c ∪ (ac))

Proof of Theorem mloa
StepHypRef Expression
1 lea 160 . . . 4 ((a2 b) ∩ (b2 a)) ≤ (a2 b)
2 ax-a3 32 . . . . . 6 (((bc) ∪ (bc )) ∪ ((a2 b) ∩ (a2 c))) = ((bc) ∪ ((bc ) ∪ ((a2 b) ∩ (a2 c))))
3 or12 80 . . . . . . 7 ((bc) ∪ ((bc ) ∪ ((a2 b) ∩ (a2 c)))) = ((bc ) ∪ ((bc) ∪ ((a2 b) ∩ (a2 c))))
4 anor3 90 . . . . . . . 8 (bc ) = (bc)
54ax-r5 38 . . . . . . 7 ((bc ) ∪ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = ((bc) ∪ ((bc) ∪ ((a2 b) ∩ (a2 c))))
63, 5ax-r2 36 . . . . . 6 ((bc) ∪ ((bc ) ∪ ((a2 b) ∩ (a2 c)))) = ((bc) ∪ ((bc) ∪ ((a2 b) ∩ (a2 c))))
72, 6ax-r2 36 . . . . 5 (((bc) ∪ (bc )) ∪ ((a2 b) ∩ (a2 c))) = ((bc) ∪ ((bc) ∪ ((a2 b) ∩ (a2 c))))
8 leo 158 . . . . . . . . 9 b ≤ (b ∪ (ab ))
9 df-i2 45 . . . . . . . . . 10 (a2 b) = (b ∪ (ab ))
109ax-r1 35 . . . . . . . . 9 (b ∪ (ab )) = (a2 b)
118, 10lbtr 139 . . . . . . . 8 b ≤ (a2 b)
12 leo 158 . . . . . . . . 9 c ≤ (c ∪ (ac ))
13 df-i2 45 . . . . . . . . . 10 (a2 c) = (c ∪ (ac ))
1413ax-r1 35 . . . . . . . . 9 (c ∪ (ac )) = (a2 c)
1512, 14lbtr 139 . . . . . . . 8 c ≤ (a2 c)
1611, 15le2an 169 . . . . . . 7 (bc) ≤ ((a2 b) ∩ (a2 c))
17 id 59 . . . . . . . 8 ((a2 b) ∩ (a2 c)) = ((a2 b) ∩ (a2 c))
1817bile 142 . . . . . . 7 ((a2 b) ∩ (a2 c)) ≤ ((a2 b) ∩ (a2 c))
1916, 18lel2or 170 . . . . . 6 ((bc) ∪ ((a2 b) ∩ (a2 c))) ≤ ((a2 b) ∩ (a2 c))
2019lelor 166 . . . . 5 ((bc) ∪ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ≤ ((bc) ∪ ((a2 b) ∩ (a2 c)))
217, 20bltr 138 . . . 4 (((bc) ∪ (bc )) ∪ ((a2 b) ∩ (a2 c))) ≤ ((bc) ∪ ((a2 b) ∩ (a2 c)))
221, 21le2an 169 . . 3 (((a2 b) ∩ (b2 a)) ∩ (((bc) ∪ (bc )) ∪ ((a2 b) ∩ (a2 c)))) ≤ ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))
23 oal2 999 . . 3 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ≤ (a2 c)
2422, 23letr 137 . 2 (((a2 b) ∩ (b2 a)) ∩ (((bc) ∪ (bc )) ∪ ((a2 b) ∩ (a2 c)))) ≤ (a2 c)
25 u2lembi 721 . . 3 ((a2 b) ∩ (b2 a)) = (ab)
26 dfb 94 . . . . 5 (bc) = ((bc) ∪ (bc ))
2726ax-r1 35 . . . 4 ((bc) ∪ (bc )) = (bc)
28 i2bi 722 . . . . 5 (a2 b) = (b ∪ (ab))
29 i2bi 722 . . . . 5 (a2 c) = (c ∪ (ac))
3028, 292an 79 . . . 4 ((a2 b) ∩ (a2 c)) = ((b ∪ (ab)) ∩ (c ∪ (ac)))
3127, 302or 72 . . 3 (((bc) ∪ (bc )) ∪ ((a2 b) ∩ (a2 c))) = ((bc) ∪ ((b ∪ (ab)) ∩ (c ∪ (ac))))
3225, 312an 79 . 2 (((a2 b) ∩ (b2 a)) ∩ (((bc) ∪ (bc )) ∪ ((a2 b) ∩ (a2 c)))) = ((ab) ∩ ((bc) ∪ ((b ∪ (ab)) ∩ (c ∪ (ac)))))
3324, 32, 29le3tr2 141 1 ((ab) ∩ ((bc) ∪ ((b ∪ (ab)) ∩ (c ∪ (ac))))) ≤ (c ∪ (ac))
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ≡ tb 5   ∪ 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  ax-3oa 998 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: (None)
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