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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 (ab) ≤ a
2 lea 160 . . . . . 6 (ab ) ≤ a
31, 2lel2or 170 . . . . 5 ((ab) ∪ (ab )) ≤ a
43df-le2 131 . . . 4 (((ab) ∪ (ab )) ∪ a) = a
5 le1 146 . . . . 5 (a ∪ ((ab) ∪ (ab ))) ≤ 1
6 df-cmtr 134 . . . . . . 7 C (a, b) = (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab )))
7 cmtr1com.1 . . . . . . 7 C (a, b) = 1
8 ax-a2 31 . . . . . . 7 (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab ))) = (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab )))
96, 7, 83tr2 64 . . . . . 6 1 = (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab )))
10 lea 160 . . . . . . . 8 (ab) ≤ a
11 lea 160 . . . . . . . 8 (ab ) ≤ a
1210, 11lel2or 170 . . . . . . 7 ((ab) ∪ (ab )) ≤ a
1312leror 152 . . . . . 6 (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab ))) ≤ (a ∪ ((ab) ∪ (ab )))
149, 13bltr 138 . . . . 5 1 ≤ (a ∪ ((ab) ∪ (ab )))
155, 14lebi 145 . . . 4 (a ∪ ((ab) ∪ (ab ))) = 1
164, 15lem3.1 443 . . 3 ((ab) ∪ (ab )) = a
1716ax-r1 35 . 2 a = ((ab) ∪ (ab ))
1817df-c1 132 1 a C b
 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-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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