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Theorem u3lemob 632
Description: Lemma for Kalmbach implication study. (Contributed by NM, 15-Dec-1997.)
Assertion
Ref Expression
u3lemob ((a3 b) ∪ b) = (ab)

Proof of Theorem u3lemob
StepHypRef Expression
1 df-i3 46 . . 3 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
21ax-r5 38 . 2 ((a3 b) ∪ b) = ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∪ b)
3 or32 82 . . 3 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∪ b) = ((((ab) ∪ (ab )) ∪ b) ∪ (a ∩ (ab)))
4 or32 82 . . . . . 6 (((ab) ∪ (ab )) ∪ b) = (((ab) ∪ b) ∪ (ab ))
5 lear 161 . . . . . . . 8 (ab) ≤ b
65df-le2 131 . . . . . . 7 ((ab) ∪ b) = b
76ax-r5 38 . . . . . 6 (((ab) ∪ b) ∪ (ab )) = (b ∪ (ab ))
84, 7ax-r2 36 . . . . 5 (((ab) ∪ (ab )) ∪ b) = (b ∪ (ab ))
9 ancom 74 . . . . 5 (a ∩ (ab)) = ((ab) ∩ a)
108, 92or 72 . . . 4 ((((ab) ∪ (ab )) ∪ b) ∪ (a ∩ (ab))) = ((b ∪ (ab )) ∪ ((ab) ∩ a))
11 comor2 462 . . . . . . 7 (ab) C b
12 comor1 461 . . . . . . . 8 (ab) C a
1311comcom2 183 . . . . . . . 8 (ab) C b
1412, 13com2an 484 . . . . . . 7 (ab) C (ab )
1511, 14com2or 483 . . . . . 6 (ab) C (b ∪ (ab ))
1612comcom7 460 . . . . . 6 (ab) C a
1715, 16fh4 472 . . . . 5 ((b ∪ (ab )) ∪ ((ab) ∩ a)) = (((b ∪ (ab )) ∪ (ab)) ∩ ((b ∪ (ab )) ∪ a))
18 or32 82 . . . . . . . 8 ((b ∪ (ab )) ∪ (ab)) = ((b ∪ (ab)) ∪ (ab ))
19 or12 80 . . . . . . . . . . 11 (b ∪ (ab)) = (a ∪ (bb))
20 oridm 110 . . . . . . . . . . . 12 (bb) = b
2120lor 70 . . . . . . . . . . 11 (a ∪ (bb)) = (ab)
2219, 21ax-r2 36 . . . . . . . . . 10 (b ∪ (ab)) = (ab)
2322ax-r5 38 . . . . . . . . 9 ((b ∪ (ab)) ∪ (ab )) = ((ab) ∪ (ab ))
24 ax-a2 31 . . . . . . . . . 10 ((ab) ∪ (ab )) = ((ab ) ∪ (ab))
25 lea 160 . . . . . . . . . . . 12 (ab ) ≤ a
26 leo 158 . . . . . . . . . . . 12 a ≤ (ab)
2725, 26letr 137 . . . . . . . . . . 11 (ab ) ≤ (ab)
2827df-le2 131 . . . . . . . . . 10 ((ab ) ∪ (ab)) = (ab)
2924, 28ax-r2 36 . . . . . . . . 9 ((ab) ∪ (ab )) = (ab)
3023, 29ax-r2 36 . . . . . . . 8 ((b ∪ (ab)) ∪ (ab )) = (ab)
3118, 30ax-r2 36 . . . . . . 7 ((b ∪ (ab )) ∪ (ab)) = (ab)
32 or32 82 . . . . . . . 8 ((b ∪ (ab )) ∪ a) = ((ba) ∪ (ab ))
33 ancom 74 . . . . . . . . . . 11 (ab ) = (ba )
34 oran 87 . . . . . . . . . . . . 13 (ba) = (ba )
3534con2 67 . . . . . . . . . . . 12 (ba) = (ba )
3635ax-r1 35 . . . . . . . . . . 11 (ba ) = (ba)
3733, 36ax-r2 36 . . . . . . . . . 10 (ab ) = (ba)
3837lor 70 . . . . . . . . 9 ((ba) ∪ (ab )) = ((ba) ∪ (ba) )
39 df-t 41 . . . . . . . . . 10 1 = ((ba) ∪ (ba) )
4039ax-r1 35 . . . . . . . . 9 ((ba) ∪ (ba) ) = 1
4138, 40ax-r2 36 . . . . . . . 8 ((ba) ∪ (ab )) = 1
4232, 41ax-r2 36 . . . . . . 7 ((b ∪ (ab )) ∪ a) = 1
4331, 422an 79 . . . . . 6 (((b ∪ (ab )) ∪ (ab)) ∩ ((b ∪ (ab )) ∪ a)) = ((ab) ∩ 1)
44 an1 106 . . . . . 6 ((ab) ∩ 1) = (ab)
4543, 44ax-r2 36 . . . . 5 (((b ∪ (ab )) ∪ (ab)) ∩ ((b ∪ (ab )) ∪ a)) = (ab)
4617, 45ax-r2 36 . . . 4 ((b ∪ (ab )) ∪ ((ab) ∩ a)) = (ab)
4710, 46ax-r2 36 . . 3 ((((ab) ∪ (ab )) ∪ b) ∪ (a ∩ (ab))) = (ab)
483, 47ax-r2 36 . 2 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∪ b) = (ab)
492, 48ax-r2 36 1 ((a3 b) ∪ b) = (ab)
Colors of variables: term
Syntax hints:   = wb 1   wn 4  wo 6  wa 7  1wt 8  3 wi3 14
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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  u3lemnanb  657  neg3antlem2  865
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