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Theorem dpval 30114
Description: Define the value of the decimal point operator. See df-dp 30113. (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 30096 . . 3 𝑥𝑦 = (𝑥 + (𝑦 / 10))
2 oveq1 6885 . . 3 (𝑥 = 𝐴 → (𝑥 + (𝑦 / 10)) = (𝐴 + (𝑦 / 10)))
31, 2syl5eq 2845 . 2 (𝑥 = 𝐴𝑥𝑦 = (𝐴 + (𝑦 / 10)))
4 oveq1 6885 . . . 4 (𝑦 = 𝐵 → (𝑦 / 10) = (𝐵 / 10))
54oveq2d 6894 . . 3 (𝑦 = 𝐵 → (𝐴 + (𝑦 / 10)) = (𝐴 + (𝐵 / 10)))
6 df-dp2 30096 . . 3 𝐴𝐵 = (𝐴 + (𝐵 / 10))
75, 6syl6eqr 2851 . 2 (𝑦 = 𝐵 → (𝐴 + (𝑦 / 10)) = 𝐴𝐵)
8 df-dp 30113 . 2 . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ 𝑥𝑦)
96ovexi 6911 . 2 𝐴𝐵 ∈ V
103, 7, 8, 9ovmpt2 7030 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  (class class class)co 6878  cr 10223  0cc0 10224  1c1 10225   + caddc 10227   / cdiv 10976  0cn0 11580  cdc 11783  cdp2 30095  .cdp 30112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-dp2 30096  df-dp 30113
This theorem is referenced by:  dpcl  30115  dpfrac1  30116  dpval2  30117  dpmul1000  30123  dpadd2  30134
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