Theorem dyadval 25523

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

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
2		oveq2 7362	. . . 4 ⊢ (𝑦 = 𝐵 → (2↑𝑦) = (2↑𝐵))
3	1, 2	oveqan12d 7373	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 / (2↑𝑦)) = (𝐴 / (2↑𝐵)))
4		oveq1 7361	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1))
5	4, 2	oveqan12d 7373	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 + 1) / (2↑𝑦)) = ((𝐴 + 1) / (2↑𝐵)))
6	3, 5	opeq12d 4834	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 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 5409	. 2 ⊢ ⟨(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))⟩ ∈ V
9	6, 7, 8	ovmpoa 7509	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0) → (𝐴𝐹𝐵) = ⟨(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ⟨cop 4583  (class class class)co 7354   ∈ cmpo 7356  1c1 11016   + caddc 11018   / cdiv 11783  2c2 12189  ℕ₀cn0 12390  ℤcz 12477  ↑cexp 13972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6444  df-fun 6490  df-fv 6496  df-ov 7357  df-oprab 7358  df-mpo 7359

This theorem is referenced by:  dyadovol  25524  dyadss  25525  dyaddisjlem  25526  dyadmaxlem  25528  opnmbllem  25532  opnmbllem0  37719