Mathbox for David A. Wheeler < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dpval Structured version   Visualization version   GIF version

Theorem dpval 42310
 Description: Define the value of the decimal point operator. See df-dp 42308. (Contributed by David A. Wheeler, 15-May-2015.)
Assertion
Ref Expression
dpval ((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = 𝐴𝐵)

Proof of Theorem dpval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dp2 42306 . . 3 𝑥𝑦 = (𝑥 + (𝑦 / 10))
2 oveq1 6556 . . 3 (𝑥 = 𝐴 → (𝑥 + (𝑦 / 10)) = (𝐴 + (𝑦 / 10)))
31, 2syl5eq 2656 . 2 (𝑥 = 𝐴𝑥𝑦 = (𝐴 + (𝑦 / 10)))
4 oveq1 6556 . . . 4 (𝑦 = 𝐵 → (𝑦 / 10) = (𝐵 / 10))
54oveq2d 6565 . . 3 (𝑦 = 𝐵 → (𝐴 + (𝑦 / 10)) = (𝐴 + (𝐵 / 10)))
6 df-dp2 42306 . . 3 𝐴𝐵 = (𝐴 + (𝐵 / 10))
75, 6syl6eqr 2662 . 2 (𝑦 = 𝐵 → (𝐴 + (𝑦 / 10)) = 𝐴𝐵)
8 df-dp 42308 . 2 . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ 𝑥𝑦)
96ovexi 6578 . 2 𝐴𝐵 ∈ V
103, 7, 8, 9ovmpt2 6694 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   / cdiv 10563  ℕ0cn0 11169  ;cdc 11369  _cdp2 42304  .cdp 42305 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-dp2 42306  df-dp 42308 This theorem is referenced by:  dpcl  42311  dpfrac1  42312  dpfrac1OLD  42313
 Copyright terms: Public domain W3C validator