Theorem dyadval 25641

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 7398	. . . 4 ⊢ (𝑦 = 𝐵 → (2↑𝑦) = (2↑𝐵))
3	1, 2	oveqan12d 7409	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 / (2↑𝑦)) = (𝐴 / (2↑𝐵)))
4		oveq1 7397	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1))
5	4, 2	oveqan12d 7409	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 + 1) / (2↑𝑦)) = ((𝐴 + 1) / (2↑𝐵)))
6	3, 5	opeq12d 4836	. 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 5428	. 2 ⊢ ⟨(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))⟩ ∈ V
9	6, 7, 8	ovmpoa 7545	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0) → (𝐴𝐹𝐵) = ⟨(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))⟩)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ⟨cop 4585  (class class class)co 7390   ∈ cmpo 7392  1c1 11067   + caddc 11069   / cdiv 11837  2c2 12265  ℕ₀cn0 12474  ℤcz 12561  ↑cexp 14067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6471  df-fun 6517  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395

This theorem is referenced by:  dyadovol  25642  dyadss  25643  dyaddisjlem  25644  dyadmaxlem  25646  opnmbllem  25650  opnmbllem0  38115