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Theorem dpval 30595
Description: Define the value of the decimal point operator. See df-dp 30594. (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 30577 . . 3 𝑥𝑦 = (𝑥 + (𝑦 / 10))
2 oveq1 7146 . . 3 (𝑥 = 𝐴 → (𝑥 + (𝑦 / 10)) = (𝐴 + (𝑦 / 10)))
31, 2syl5eq 2848 . 2 (𝑥 = 𝐴𝑥𝑦 = (𝐴 + (𝑦 / 10)))
4 oveq1 7146 . . . 4 (𝑦 = 𝐵 → (𝑦 / 10) = (𝐵 / 10))
54oveq2d 7155 . . 3 (𝑦 = 𝐵 → (𝐴 + (𝑦 / 10)) = (𝐴 + (𝐵 / 10)))
6 df-dp2 30577 . . 3 𝐴𝐵 = (𝐴 + (𝐵 / 10))
75, 6eqtr4di 2854 . 2 (𝑦 = 𝐵 → (𝐴 + (𝑦 / 10)) = 𝐴𝐵)
8 df-dp 30594 . 2 . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ 𝑥𝑦)
96ovexi 7173 . 2 𝐴𝐵 ∈ V
103, 7, 8, 9ovmpo 7293 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℝ) → (𝐴.𝐵) = 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  (class class class)co 7139  cr 10529  0cc0 10530  1c1 10531   + caddc 10533   / cdiv 11290  0cn0 11889  cdc 12090  cdp2 30576  .cdp 30593
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-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-dp2 30577  df-dp 30594
This theorem is referenced by:  dpcl  30596  dpfrac1  30597  dpval2  30598  dpmul1000  30604  dpadd2  30615
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