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Theorem wdf-c2 384
Description: Show that commutator is a 'commutes' analogue for analogue of =. (Contributed by NM, 27-Jan-2002.)
Hypothesis
Ref Expression
wdf-c2.1 C (a, b) = 1
Assertion
Ref Expression
wdf-c2 (a ≡ ((ab) ∪ (ab ))) = 1

Proof of Theorem wdf-c2
StepHypRef Expression
1 le1 146 . 2 (a ≡ ((ab) ∪ (ab ))) ≤ 1
2 lea 160 . . . . 5 (ab) ≤ a
3 lea 160 . . . . 5 (ab ) ≤ a
42, 3lel2or 170 . . . 4 ((ab) ∪ (ab )) ≤ a
54lelor 166 . . 3 (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab ))) ≤ (((ab) ∪ (ab )) ∪ a )
6 wdf-c2.1 . . . . 5 C (a, b) = 1
76ax-r1 35 . . . 4 1 = C (a, b)
8 df-cmtr 134 . . . 4 C (a, b) = (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab )))
97, 8ax-r2 36 . . 3 1 = (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab )))
10 dfb 94 . . . 4 (a ≡ ((ab) ∪ (ab ))) = ((a ∩ ((ab) ∪ (ab ))) ∪ (a ∩ ((ab) ∪ (ab )) ))
11 ancom 74 . . . . . 6 (a ∩ ((ab) ∪ (ab ))) = (((ab) ∪ (ab )) ∩ a)
12 lea 160 . . . . . . . 8 (ab) ≤ a
13 lea 160 . . . . . . . 8 (ab ) ≤ a
1412, 13lel2or 170 . . . . . . 7 ((ab) ∪ (ab )) ≤ a
1514df2le2 136 . . . . . 6 (((ab) ∪ (ab )) ∩ a) = ((ab) ∪ (ab ))
1611, 15ax-r2 36 . . . . 5 (a ∩ ((ab) ∪ (ab ))) = ((ab) ∪ (ab ))
17 anandi 114 . . . . . 6 (a ∩ ((ab ) ∩ (ab))) = ((a ∩ (ab )) ∩ (a ∩ (ab)))
18 oran3 93 . . . . . . . . 9 (ab ) = (ab)
19 oran2 92 . . . . . . . . 9 (ab) = (ab )
2018, 192an 79 . . . . . . . 8 ((ab ) ∩ (ab)) = ((ab) ∩ (ab ) )
21 anor3 90 . . . . . . . 8 ((ab) ∩ (ab ) ) = ((ab) ∪ (ab ))
2220, 21ax-r2 36 . . . . . . 7 ((ab ) ∩ (ab)) = ((ab) ∪ (ab ))
2322lan 77 . . . . . 6 (a ∩ ((ab ) ∩ (ab))) = (a ∩ ((ab) ∪ (ab )) )
24 anabs 121 . . . . . . . 8 (a ∩ (ab )) = a
25 anabs 121 . . . . . . . 8 (a ∩ (ab)) = a
2624, 252an 79 . . . . . . 7 ((a ∩ (ab )) ∩ (a ∩ (ab))) = (aa )
27 anidm 111 . . . . . . 7 (aa ) = a
2826, 27ax-r2 36 . . . . . 6 ((a ∩ (ab )) ∩ (a ∩ (ab))) = a
2917, 23, 283tr2 64 . . . . 5 (a ∩ ((ab) ∪ (ab )) ) = a
3016, 292or 72 . . . 4 ((a ∩ ((ab) ∪ (ab ))) ∪ (a ∩ ((ab) ∪ (ab )) )) = (((ab) ∪ (ab )) ∪ a )
3110, 30ax-r2 36 . . 3 (a ≡ ((ab) ∪ (ab ))) = (((ab) ∪ (ab )) ∪ a )
325, 9, 31le3tr1 140 . 2 1 ≤ (a ≡ ((ab) ∪ (ab )))
331, 32lebi 145 1 (a ≡ ((ab) ∪ (ab ))) = 1
Colors of variables: term
Syntax hints:   = wb 1   wn 4  tb 5  wo 6  wa 7  1wt 8   C wcmtr 29
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
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-cmtr 134
This theorem is referenced by:  wbctr  410  wcbtr  411  wcomcom2  415  wcomd  418  wcomcom5  420  wcom2or  427
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