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Theorem comcmtr1 494
 Description: Commutation implies commutator equal to 1. Theorem 2.11 of Beran, p. 86.
Hypothesis
Ref Expression
comcmtr1.1 a C b
Assertion
Ref Expression
comcmtr1 C (a, b) = 1

Proof of Theorem comcmtr1
StepHypRef Expression
1 comcmtr1.1 . . . . 5 a C b
21df-c2 133 . . . 4 a = ((ab) ∪ (ab ))
31comcom3 454 . . . . 5 a C b
43df-c2 133 . . . 4 a = ((ab) ∪ (ab ))
52, 42or 72 . . 3 (aa ) = (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab )))
65ax-r1 35 . 2 (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab ))) = (aa )
7 df-cmtr 134 . 2 C (a, b) = (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab )))
8 df-t 41 . 2 1 = (aa )
96, 7, 83tr1 63 1 C (a, b) = 1
 Colors of variables: term Syntax hints:   = wb 1   C wc 3  ⊥ wn 4   ∪ 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-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-le1 130  df-le2 131  df-c1 132  df-c2 133  df-cmtr 134 This theorem is referenced by: (None)
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