Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  i0cmtrcom GIF version

Theorem i0cmtrcom 495
 Description: Commutator element →0 commutator implies commutation.
Hypothesis
Ref Expression
i0cmtrcom.1 (a0 C (a, b)) = 1
Assertion
Ref Expression
i0cmtrcom a C b

Proof of Theorem i0cmtrcom
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 df-cmtr 134 . . . . . . . 8 C (a, b) = (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab )))
65lor 70 . . . . . . 7 (a C (a, b)) = (a ∪ (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab ))))
76ax-r1 35 . . . . . 6 (a ∪ (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab )))) = (a C (a, b))
8 ax-a2 31 . . . . . . 7 (a ∪ ((ab) ∪ (ab ))) = (((ab) ∪ (ab )) ∪ a )
9 ax-a2 31 . . . . . . . . . 10 (a ∪ ((ab) ∪ (ab ))) = (((ab) ∪ (ab )) ∪ a )
10 lea 160 . . . . . . . . . . . 12 (ab) ≤ a
11 lea 160 . . . . . . . . . . . 12 (ab ) ≤ a
1210, 11lel2or 170 . . . . . . . . . . 11 ((ab) ∪ (ab )) ≤ a
1312df-le2 131 . . . . . . . . . 10 (((ab) ∪ (ab )) ∪ a ) = a
149, 13ax-r2 36 . . . . . . . . 9 (a ∪ ((ab) ∪ (ab ))) = a
1514lor 70 . . . . . . . 8 (((ab) ∪ (ab )) ∪ (a ∪ ((ab) ∪ (ab )))) = (((ab) ∪ (ab )) ∪ a )
1615ax-r1 35 . . . . . . 7 (((ab) ∪ (ab )) ∪ a ) = (((ab) ∪ (ab )) ∪ (a ∪ ((ab) ∪ (ab ))))
17 or12 80 . . . . . . 7 (((ab) ∪ (ab )) ∪ (a ∪ ((ab) ∪ (ab )))) = (a ∪ (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab ))))
188, 16, 173tr 65 . . . . . 6 (a ∪ ((ab) ∪ (ab ))) = (a ∪ (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab ))))
19 df-i0 43 . . . . . 6 (a0 C (a, b)) = (a C (a, b))
207, 18, 193tr1 63 . . . . 5 (a ∪ ((ab) ∪ (ab ))) = (a0 C (a, b))
21 i0cmtrcom.1 . . . . 5 (a0 C (a, b)) = 1
2220, 21ax-r2 36 . . . 4 (a ∪ ((ab) ∪ (ab ))) = 1
234, 22lem3.1 443 . . 3 ((ab) ∪ (ab )) = a
2423ax-r1 35 . 2 a = ((ab) ∪ (ab ))
2524df-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   →0 wi0 11   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-i0 43  df-le1 130  df-le2 131  df-c1 132  df-cmtr 134 This theorem is referenced by:  3vded3  819
 Copyright terms: Public domain W3C validator