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Theorem id5leid0 351
Description: Quantum identity is less than classical identity.
Assertion
Ref Expression
id5leid0 (a == b) =< (a ==0 b)
Proof of Theorem id5leid0
StepHypRef Expression
1 ax-a2 31 . . 3 ((a ^ b) v (a' ^ b')) = ((a' ^ b') v (a ^ b))
2 lea 160 . . . . 5 (a' ^ b') =< a'
3 lear 161 . . . . 5 (a ^ b) =< b
42, 3le2or 168 . . . 4 ((a' ^ b') v (a ^ b)) =< (a' v b)
5 lear 161 . . . . 5 (a' ^ b') =< b'
6 lea 160 . . . . 5 (a ^ b) =< a
75, 6le2or 168 . . . 4 ((a' ^ b') v (a ^ b)) =< (b' v a)
84, 7ler2an 173 . . 3 ((a' ^ b') v (a ^ b)) =< ((a' v b) ^ (b' v a))
91, 8bltr 138 . 2 ((a ^ b) v (a' ^ b')) =< ((a' v b) ^ (b' v a))
10 dfb 94 . 2 (a == b) = ((a ^ b) v (a' ^ b'))
11 df-id0 49 . 2 (a ==0 b) = ((a' v b) ^ (b' v a))
129, 10, 11le3tr1 140 1 (a == b) =< (a ==0 b)
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   == tb 5   v wo 6   ^ wa 7   ==0 wid0 17
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
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-id0 49  df-le1 130  df-le2 131
This theorem is referenced by:  id5id0  352
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