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Theorem dyadval 25472
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 7412 . . . 4 (𝑦 = 𝐵 → (2↑𝑦) = (2↑𝐵))
31, 2oveqan12d 7423 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 / (2↑𝑦)) = (𝐴 / (2↑𝐵)))
4 oveq1 7411 . . . 4 (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1))
54, 2oveqan12d 7423 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 + 1) / (2↑𝑦)) = ((𝐴 + 1) / (2↑𝐵)))
63, 5opeq12d 4876 . 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 5457 . 2 ⟨(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))⟩ ∈ V
96, 7, 8ovmpoa 7558 1 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0) → (𝐴𝐹𝐵) = ⟨(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cop 4629  (class class class)co 7404  cmpo 7406  1c1 11110   + caddc 11112   / cdiv 11872  2c2 12268  0cn0 12473  cz 12559  cexp 14030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409
This theorem is referenced by:  dyadovol  25473  dyadss  25474  dyaddisjlem  25475  dyadmaxlem  25477  opnmbllem  25481  opnmbllem0  37035
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