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Theorem 2oalem1 825
 Description: Lemma for OA-like stuff with →2 instead of →0 . (Contributed by NM, 15-Nov-1998.)
Assertion
Ref Expression
2oalem1 ((a2 b) ∪ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = 1

Proof of Theorem 2oalem1
StepHypRef Expression
1 or12 80 . 2 ((a2 b) ∪ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = ((bc) ∪ ((a2 b) ∪ ((a2 b) ∩ (a2 c))))
2 ud2lem0c 278 . . . 4 (a2 b) = (b ∩ (ab))
3 df-i2 45 . . . . 5 (a2 b) = (b ∪ (ab ))
4 df-i2 45 . . . . 5 (a2 c) = (c ∪ (ac ))
53, 42an 79 . . . 4 ((a2 b) ∩ (a2 c)) = ((b ∪ (ab )) ∩ (c ∪ (ac )))
62, 52or 72 . . 3 ((a2 b) ∪ ((a2 b) ∩ (a2 c))) = ((b ∩ (ab)) ∪ ((b ∪ (ab )) ∩ (c ∪ (ac ))))
76lor 70 . 2 ((bc) ∪ ((a2 b) ∪ ((a2 b) ∩ (a2 c)))) = ((bc) ∪ ((b ∩ (ab)) ∪ ((b ∪ (ab )) ∩ (c ∪ (ac )))))
8 or32 82 . . . . 5 ((bc) ∪ (b ∩ (ab))) = ((b ∪ (b ∩ (ab))) ∪ c)
9 oml 445 . . . . . . 7 (b ∪ (b ∩ (ba))) = (ba)
10 ax-a2 31 . . . . . . . . 9 (ab) = (ba)
1110lan 77 . . . . . . . 8 (b ∩ (ab)) = (b ∩ (ba))
1211lor 70 . . . . . . 7 (b ∪ (b ∩ (ab))) = (b ∪ (b ∩ (ba)))
139, 12, 103tr1 63 . . . . . 6 (b ∪ (b ∩ (ab))) = (ab)
1413ax-r5 38 . . . . 5 ((b ∪ (b ∩ (ab))) ∪ c) = ((ab) ∪ c)
158, 14ax-r2 36 . . . 4 ((bc) ∪ (b ∩ (ab))) = ((ab) ∪ c)
1615ax-r5 38 . . 3 (((bc) ∪ (b ∩ (ab))) ∪ ((b ∪ (ab )) ∩ (c ∪ (ac )))) = (((ab) ∪ c) ∪ ((b ∪ (ab )) ∩ (c ∪ (ac ))))
17 ax-a3 32 . . 3 (((bc) ∪ (b ∩ (ab))) ∪ ((b ∪ (ab )) ∩ (c ∪ (ac )))) = ((bc) ∪ ((b ∩ (ab)) ∪ ((b ∪ (ab )) ∩ (c ∪ (ac )))))
18 oran 87 . . . . . . . . . . . . 13 (ab) = (ab )
1918lan 77 . . . . . . . . . . . 12 (b ∩ (ab)) = (b ∩ (ab ) )
20 anor3 90 . . . . . . . . . . . 12 (b ∩ (ab ) ) = (b ∪ (ab ))
2119, 20ax-r2 36 . . . . . . . . . . 11 (b ∩ (ab)) = (b ∪ (ab ))
2221ax-r1 35 . . . . . . . . . 10 (b ∪ (ab )) = (b ∩ (ab))
23 lear 161 . . . . . . . . . 10 (b ∩ (ab)) ≤ (ab)
2422, 23bltr 138 . . . . . . . . 9 (b ∪ (ab )) ≤ (ab)
25 leo 158 . . . . . . . . 9 (ab) ≤ ((ab) ∪ c)
2624, 25letr 137 . . . . . . . 8 (b ∪ (ab )) ≤ ((ab) ∪ c)
2726lecom 180 . . . . . . 7 (b ∪ (ab )) C ((ab) ∪ c)
2827comcom6 459 . . . . . 6 (b ∪ (ab )) C ((ab) ∪ c)
2928comcom 453 . . . . 5 ((ab) ∪ c) C (b ∪ (ab ))
30 lear 161 . . . . . . . . . 10 (c ∩ (ac)) ≤ (ac)
31 leo 158 . . . . . . . . . 10 (ac) ≤ ((ac) ∪ b)
3230, 31letr 137 . . . . . . . . 9 (c ∩ (ac)) ≤ ((ac) ∪ b)
33 oran 87 . . . . . . . . . . 11 (ac) = (ac )
3433lan 77 . . . . . . . . . 10 (c ∩ (ac)) = (c ∩ (ac ) )
35 anor3 90 . . . . . . . . . 10 (c ∩ (ac ) ) = (c ∪ (ac ))
3634, 35ax-r2 36 . . . . . . . . 9 (c ∩ (ac)) = (c ∪ (ac ))
37 or32 82 . . . . . . . . 9 ((ac) ∪ b) = ((ab) ∪ c)
3832, 36, 37le3tr2 141 . . . . . . . 8 (c ∪ (ac )) ≤ ((ab) ∪ c)
3938lecom 180 . . . . . . 7 (c ∪ (ac )) C ((ab) ∪ c)
4039comcom6 459 . . . . . 6 (c ∪ (ac )) C ((ab) ∪ c)
4140comcom 453 . . . . 5 ((ab) ∪ c) C (c ∪ (ac ))
4229, 41fh3 471 . . . 4 (((ab) ∪ c) ∪ ((b ∪ (ab )) ∩ (c ∪ (ac )))) = ((((ab) ∪ c) ∪ (b ∪ (ab ))) ∩ (((ab) ∪ c) ∪ (c ∪ (ac ))))
43 or12 80 . . . . . 6 (((ab) ∪ c) ∪ (b ∪ (ab ))) = (b ∪ (((ab) ∪ c) ∪ (ab )))
44 or32 82 . . . . . . . 8 (((ab) ∪ c) ∪ (ab )) = (((ab) ∪ (ab )) ∪ c)
45 ax-a2 31 . . . . . . . 8 (((ab) ∪ (ab )) ∪ c) = (c ∪ ((ab) ∪ (ab )))
46 anor3 90 . . . . . . . . . . . 12 (ab ) = (ab)
4746lor 70 . . . . . . . . . . 11 ((ab) ∪ (ab )) = ((ab) ∪ (ab) )
48 df-t 41 . . . . . . . . . . . 12 1 = ((ab) ∪ (ab) )
4948ax-r1 35 . . . . . . . . . . 11 ((ab) ∪ (ab) ) = 1
5047, 49ax-r2 36 . . . . . . . . . 10 ((ab) ∪ (ab )) = 1
5150lor 70 . . . . . . . . 9 (c ∪ ((ab) ∪ (ab ))) = (c ∪ 1)
52 or1 104 . . . . . . . . 9 (c ∪ 1) = 1
5351, 52ax-r2 36 . . . . . . . 8 (c ∪ ((ab) ∪ (ab ))) = 1
5444, 45, 533tr 65 . . . . . . 7 (((ab) ∪ c) ∪ (ab )) = 1
5554lor 70 . . . . . 6 (b ∪ (((ab) ∪ c) ∪ (ab ))) = (b ∪ 1)
56 or1 104 . . . . . 6 (b ∪ 1) = 1
5743, 55, 563tr 65 . . . . 5 (((ab) ∪ c) ∪ (b ∪ (ab ))) = 1
58 or12 80 . . . . . 6 (((ab) ∪ c) ∪ (c ∪ (ac ))) = (c ∪ (((ab) ∪ c) ∪ (ac )))
59 or32 82 . . . . . . . . . 10 ((ab) ∪ c) = ((ac) ∪ b)
60 ax-a2 31 . . . . . . . . . 10 ((ac) ∪ b) = (b ∪ (ac))
6159, 60ax-r2 36 . . . . . . . . 9 ((ab) ∪ c) = (b ∪ (ac))
6261ax-r5 38 . . . . . . . 8 (((ab) ∪ c) ∪ (ac )) = ((b ∪ (ac)) ∪ (ac ))
63 ax-a3 32 . . . . . . . 8 ((b ∪ (ac)) ∪ (ac )) = (b ∪ ((ac) ∪ (ac )))
64 anor3 90 . . . . . . . . . . . 12 (ac ) = (ac)
6564lor 70 . . . . . . . . . . 11 ((ac) ∪ (ac )) = ((ac) ∪ (ac) )
66 df-t 41 . . . . . . . . . . . 12 1 = ((ac) ∪ (ac) )
6766ax-r1 35 . . . . . . . . . . 11 ((ac) ∪ (ac) ) = 1
6865, 67ax-r2 36 . . . . . . . . . 10 ((ac) ∪ (ac )) = 1
6968lor 70 . . . . . . . . 9 (b ∪ ((ac) ∪ (ac ))) = (b ∪ 1)
7069, 56ax-r2 36 . . . . . . . 8 (b ∪ ((ac) ∪ (ac ))) = 1
7162, 63, 703tr 65 . . . . . . 7 (((ab) ∪ c) ∪ (ac )) = 1
7271lor 70 . . . . . 6 (c ∪ (((ab) ∪ c) ∪ (ac ))) = (c ∪ 1)
7358, 72, 523tr 65 . . . . 5 (((ab) ∪ c) ∪ (c ∪ (ac ))) = 1
7457, 732an 79 . . . 4 ((((ab) ∪ c) ∪ (b ∪ (ab ))) ∩ (((ab) ∪ c) ∪ (c ∪ (ac )))) = (1 ∩ 1)
75 anidm 111 . . . 4 (1 ∩ 1) = 1
7642, 74, 753tr 65 . . 3 (((ab) ∪ c) ∪ ((b ∪ (ab )) ∩ (c ∪ (ac )))) = 1
7716, 17, 763tr2 64 . 2 ((bc) ∪ ((b ∩ (ab)) ∪ ((b ∪ (ab )) ∩ (c ∪ (ac ))))) = 1
781, 7, 773tr 65 1 ((a2 b) ∪ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →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:  2oath1  826
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