Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dp2eq1 Structured version   Visualization version   GIF version

Theorem dp2eq1 30619
 Description: Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
Assertion
Ref Expression
dp2eq1 (𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)

Proof of Theorem dp2eq1
StepHypRef Expression
1 oveq1 7152 . 2 (𝐴 = 𝐵 → (𝐴 + (𝐶 / 10)) = (𝐵 + (𝐶 / 10)))
2 df-dp2 30618 . 2 𝐴𝐶 = (𝐴 + (𝐶 / 10))
3 df-dp2 30618 . 2 𝐵𝐶 = (𝐵 + (𝐶 / 10))
41, 2, 33eqtr4g 2858 1 (𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  (class class class)co 7145  0cc0 10544  1c1 10545   + caddc 10547   / cdiv 11304  ;cdc 12106  _cdp2 30617 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3444  df-un 3888  df-in 3890  df-ss 3900  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-iota 6291  df-fv 6340  df-ov 7148  df-dp2 30618 This theorem is referenced by:  dp2eq1i  30621
 Copyright terms: Public domain W3C validator