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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 = ((a ^ b) v (a ^ b'))
31comcom3 454 . . . . 5 a' C b
43df-c2 133 . . . 4 a' = ((a' ^ b) v (a' ^ b'))
52, 42or 72 . . 3 (a v a') = (((a ^ b) v (a ^ b')) v ((a' ^ b) v (a' ^ b')))
65ax-r1 35 . 2 (((a ^ b) v (a ^ b')) v ((a' ^ b) v (a' ^ b'))) = (a v a')
7 df-cmtr 134 . 2 C (a, b) = (((a ^ b) v (a ^ b')) v ((a' ^ b) v (a' ^ b')))
8 df-t 41 . 2 1 = (a v a')
96, 7, 83tr1 63 1 C (a, b) = 1
Colors of variables: term
Syntax hints:   = wb 1   C wc 3  'wn 4   v 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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