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Theorem cmtr1com 493
Description: Commutator equal to 1 commutes. Theorem 2.11 of Beran, p. 86.
Hypothesis
Ref Expression
cmtr1com.1 C (a, b) = 1
Assertion
Ref Expression
cmtr1com a C b
Proof of Theorem cmtr1com
StepHypRef Expression
1 lea 160 . . . . . 6 (a ^ b) =< a
2 lea 160 . . . . . 6 (a ^ b') =< a
31, 2lel2or 170 . . . . 5 ((a ^ b) v (a ^ b')) =< a
43df-le2 131 . . . 4 (((a ^ b) v (a ^ b')) v a) = a
5 le1 146 . . . . 5 (a' v ((a ^ b) v (a ^ b'))) =< 1
6 df-cmtr 134 . . . . . . 7 C (a, b) = (((a ^ b) v (a ^ b')) v ((a' ^ b) v (a' ^ b')))
7 cmtr1com.1 . . . . . . 7 C (a, b) = 1
8 ax-a2 31 . . . . . . 7 (((a ^ b) v (a ^ b')) v ((a' ^ b) v (a' ^ b'))) = (((a' ^ b) v (a' ^ b')) v ((a ^ b) v (a ^ b')))
96, 7, 83tr2 64 . . . . . 6 1 = (((a' ^ b) v (a' ^ b')) v ((a ^ b) v (a ^ b')))
10 lea 160 . . . . . . . 8 (a' ^ b) =< a'
11 lea 160 . . . . . . . 8 (a' ^ b') =< a'
1210, 11lel2or 170 . . . . . . 7 ((a' ^ b) v (a' ^ b')) =< a'
1312leror 152 . . . . . 6 (((a' ^ b) v (a' ^ b')) v ((a ^ b) v (a ^ b'))) =< (a' v ((a ^ b) v (a ^ b')))
149, 13bltr 138 . . . . 5 1 =< (a' v ((a ^ b) v (a ^ b')))
155, 14lebi 145 . . . 4 (a' v ((a ^ b) v (a ^ b'))) = 1
164, 15lem3.1 443 . . 3 ((a ^ b) v (a ^ b')) = a
1716ax-r1 35 . 2 a = ((a ^ b) v (a ^ b'))
1817df-c1 132 1 a C b
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-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-le1 130  df-le2 131  df-c1 132  df-cmtr 134
This theorem is referenced by: (None)
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