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Theorem comm0 178
Description: Commutation with 0. Kalmbach 83 p. 20.
Assertion
Ref Expression
comm0 a C 0
Proof of Theorem comm0
StepHypRef Expression
1 ax-a2 31 . . . . 5 (0 v a) = (a v 0)
2 or0 102 . . . . 5 (a v 0) = a
31, 2ax-r2 36 . . . 4 (0 v a) = a
43ax-r1 35 . . 3 a = (0 v a)
5 an0 108 . . . . 5 (a ^ 0) = 0
6 df-f 42 . . . . . . . 8 0 = 1'
76con2 67 . . . . . . 7 0' = 1
87lan 77 . . . . . 6 (a ^ 0') = (a ^ 1)
9 an1 106 . . . . . 6 (a ^ 1) = a
108, 9ax-r2 36 . . . . 5 (a ^ 0') = a
115, 102or 72 . . . 4 ((a ^ 0) v (a ^ 0')) = (0 v a)
1211ax-r1 35 . . 3 (0 v a) = ((a ^ 0) v (a ^ 0'))
134, 12ax-r2 36 . 2 a = ((a ^ 0) v (a ^ 0'))
1413df-c1 132 1 a C 0
Colors of variables: term
Syntax hints:   C wc 3  'wn 4   v wo 6   ^ wa 7  1wt 8  0wf 9
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-c1 132
This theorem is referenced by:  wcom0  407
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