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Theorem dpval 32854
Description: Define the value of the decimal point operator. See df-dp 32853. (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 32836 . . 3 𝑥𝑦 = (𝑥 + (𝑦 / 10))
2 oveq1 7455 . . 3 (𝑥 = 𝐴 → (𝑥 + (𝑦 / 10)) = (𝐴 + (𝑦 / 10)))
31, 2eqtrid 2792 . 2 (𝑥 = 𝐴𝑥𝑦 = (𝐴 + (𝑦 / 10)))
4 oveq1 7455 . . . 4 (𝑦 = 𝐵 → (𝑦 / 10) = (𝐵 / 10))
54oveq2d 7464 . . 3 (𝑦 = 𝐵 → (𝐴 + (𝑦 / 10)) = (𝐴 + (𝐵 / 10)))
6 df-dp2 32836 . . 3 𝐴𝐵 = (𝐴 + (𝐵 / 10))
75, 6eqtr4di 2798 . 2 (𝑦 = 𝐵 → (𝐴 + (𝑦 / 10)) = 𝐴𝐵)
8 df-dp 32853 . 2 . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ 𝑥𝑦)
96ovexi 7482 . 2 𝐴𝐵 ∈ V
103, 7, 8, 9ovmpo 7610 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  (class class class)co 7448  cr 11183  0cc0 11184  1c1 11185   + caddc 11187   / cdiv 11947  0cn0 12553  cdc 12758  cdp2 32835  .cdp 32852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-dp2 32836  df-dp 32853
This theorem is referenced by:  dpcl  32855  dpfrac1  32856  dpval2  32857  dpmul1000  32863  dpadd2  32874
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