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Theorem dpval 32857
Description: Define the value of the decimal point operator. See df-dp 32856. (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 32839 . . 3 𝑥𝑦 = (𝑥 + (𝑦 / 10))
2 oveq1 7438 . . 3 (𝑥 = 𝐴 → (𝑥 + (𝑦 / 10)) = (𝐴 + (𝑦 / 10)))
31, 2eqtrid 2787 . 2 (𝑥 = 𝐴𝑥𝑦 = (𝐴 + (𝑦 / 10)))
4 oveq1 7438 . . . 4 (𝑦 = 𝐵 → (𝑦 / 10) = (𝐵 / 10))
54oveq2d 7447 . . 3 (𝑦 = 𝐵 → (𝐴 + (𝑦 / 10)) = (𝐴 + (𝐵 / 10)))
6 df-dp2 32839 . . 3 𝐴𝐵 = (𝐴 + (𝐵 / 10))
75, 6eqtr4di 2793 . 2 (𝑦 = 𝐵 → (𝐴 + (𝑦 / 10)) = 𝐴𝐵)
8 df-dp 32856 . 2 . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ 𝑥𝑦)
96ovexi 7465 . 2 𝐴𝐵 ∈ V
103, 7, 8, 9ovmpo 7593 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  (class class class)co 7431  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   / cdiv 11918  0cn0 12524  cdc 12731  cdp2 32838  .cdp 32855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-dp2 32839  df-dp 32856
This theorem is referenced by:  dpcl  32858  dpfrac1  32859  dpval2  32860  dpmul1000  32866  dpadd2  32877
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