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Theorem dyadval 23580
Description: Value of the dyadic rational function 𝐹. (Contributed by Mario Carneiro, 26-Mar-2015.)
Hypothesis
Ref Expression
dyadmbl.1 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
Assertion
Ref Expression
dyadval ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0) → (𝐴𝐹𝐵) = ⟨(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))⟩)
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐴,𝑦   𝑥,𝐹,𝑦

Proof of Theorem dyadval
StepHypRef Expression
1 id 22 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
2 oveq2 6801 . . . 4 (𝑦 = 𝐵 → (2↑𝑦) = (2↑𝐵))
31, 2oveqan12d 6812 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 / (2↑𝑦)) = (𝐴 / (2↑𝐵)))
4 oveq1 6800 . . . 4 (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1))
54, 2oveqan12d 6812 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 + 1) / (2↑𝑦)) = ((𝐴 + 1) / (2↑𝐵)))
63, 5opeq12d 4547 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ = ⟨(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))⟩)
7 dyadmbl.1 . 2 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
8 opex 5060 . 2 ⟨(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))⟩ ∈ V
96, 7, 8ovmpt2a 6938 1 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0) → (𝐴𝐹𝐵) = ⟨(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cop 4322  (class class class)co 6793  cmpt2 6795  1c1 10139   + caddc 10141   / cdiv 10886  2c2 11272  0cn0 11494  cz 11579  cexp 13067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798
This theorem is referenced by:  dyadovol  23581  dyadss  23582  dyaddisjlem  23583  dyadmaxlem  23585  opnmbllem  23589  opnmbllem0  33778
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