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Theorem oalem1 1005
 Description: Lemma.
Assertion
Ref Expression
oalem1 ((bc) ∪ ((bc) ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))))) ≤ (a2 (bc))

Proof of Theorem oalem1
StepHypRef Expression
1 anidm 111 . . . . . . . . 9 (bb ) = b
21ran 78 . . . . . . . 8 ((bb ) ∩ c ) = (bc )
32ax-r1 35 . . . . . . 7 (bc ) = ((bb ) ∩ c )
4 anor3 90 . . . . . . 7 (bc ) = (bc)
5 an32 83 . . . . . . . 8 ((bb ) ∩ c ) = ((bc ) ∩ b )
64ran 78 . . . . . . . 8 ((bc ) ∩ b ) = ((bc)b )
75, 6ax-r2 36 . . . . . . 7 ((bb ) ∩ c ) = ((bc)b )
83, 4, 73tr2 64 . . . . . 6 (bc) = ((bc)b )
98ran 78 . . . . 5 ((bc) ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) = (((bc)b ) ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))))
10 anass 76 . . . . . 6 (((bc)b ) ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) = ((bc) ∩ (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))))
11 oalii 1002 . . . . . . 7 (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ a
1211lelan 167 . . . . . 6 ((bc) ∩ (b ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))))) ≤ ((bc)a )
1310, 12bltr 138 . . . . 5 (((bc)b ) ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ ((bc)a )
149, 13bltr 138 . . . 4 ((bc) ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ ((bc)a )
15 ancom 74 . . . 4 ((bc)a ) = (a ∩ (bc) )
1614, 15lbtr 139 . . 3 ((bc) ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))) ≤ (a ∩ (bc) )
1716lelor 166 . 2 ((bc) ∪ ((bc) ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))))) ≤ ((bc) ∪ (a ∩ (bc) ))
18 df-i2 45 . . 3 (a2 (bc)) = ((bc) ∪ (a ∩ (bc) ))
1918ax-r1 35 . 2 ((bc) ∪ (a ∩ (bc) )) = (a2 (bc))
2017, 19lbtr 139 1 ((bc) ∪ ((bc) ∩ ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))))) ≤ (a2 (bc))
 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-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 This theorem is referenced by: (None)
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