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Theorem dp2eq2 30579
Description: Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)
Assertion
Ref Expression
dp2eq2 (𝐴 = 𝐵𝐶𝐴 = 𝐶𝐵)

Proof of Theorem dp2eq2
StepHypRef Expression
1 oveq1 7146 . . 3 (𝐴 = 𝐵 → (𝐴 / 10) = (𝐵 / 10))
21oveq2d 7155 . 2 (𝐴 = 𝐵 → (𝐶 + (𝐴 / 10)) = (𝐶 + (𝐵 / 10)))
3 df-dp2 30577 . 2 𝐶𝐴 = (𝐶 + (𝐴 / 10))
4 df-dp2 30577 . 2 𝐶𝐵 = (𝐶 + (𝐵 / 10))
52, 3, 43eqtr4g 2861 1 (𝐴 = 𝐵𝐶𝐴 = 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  (class class class)co 7139  0cc0 10530  1c1 10531   + caddc 10533   / cdiv 11290  cdc 12090  cdp2 30576
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 2114  ax-9 2122  ax-ext 2773
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 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142  df-dp2 30577
This theorem is referenced by:  dp2eq2i  30581
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