Theorem elz12si 28722

Description: Inference form of membership in the dyadic fractions. (Contributed by Scott Fenton, 21-Feb-2026.)

Assertion

Ref	Expression
elz12si	⊢ ((𝐴 ∈ ℤs ∧ 𝑁 ∈ ℕ0s) → (𝐴 /su (2s↑s𝑁)) ∈ ℤs[1/2])

Proof of Theorem elz12si

Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2765	. . 3 ⊢ (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s𝑁))
2		oveq1 7427	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 /su (2s↑s𝑛)) = (𝐴 /su (2s↑s𝑛)))
3	2	eqeq2d 2776	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐴 /su (2s↑s𝑁)) = (𝑥 /su (2s↑s𝑛)) ↔ (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s𝑛))))
4		oveq2 7428	. . . . . 6 ⊢ (𝑛 = 𝑁 → (2s↑s𝑛) = (2s↑s𝑁))
5	4	oveq2d 7436	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝐴 /su (2s↑s𝑛)) = (𝐴 /su (2s↑s𝑁)))
6	5	eqeq2d 2776	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s𝑛)) ↔ (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s𝑁))))
7	3, 6	rspc2ev 3596	. . 3 ⊢ ((𝐴 ∈ ℤs ∧ 𝑁 ∈ ℕ0s ∧ (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s𝑁))) → ∃𝑥 ∈ ℤs ∃𝑛 ∈ ℕ0s (𝐴 /su (2s↑s𝑁)) = (𝑥 /su (2s↑s𝑛)))
8	1, 7	mp3an3 1479	. 2 ⊢ ((𝐴 ∈ ℤs ∧ 𝑁 ∈ ℕ0s) → ∃𝑥 ∈ ℤs ∃𝑛 ∈ ℕ0s (𝐴 /su (2s↑s𝑁)) = (𝑥 /su (2s↑s𝑛)))
9		elz12s 28721	. 2 ⊢ ((𝐴 /su (2s↑s𝑁)) ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑛 ∈ ℕ0s (𝐴 /su (2s↑s𝑁)) = (𝑥 /su (2s↑s𝑛)))
10	8, 9	sylibr 237	1 ⊢ ((𝐴 ∈ ℤs ∧ 𝑁 ∈ ℕ0s) → (𝐴 /su (2s↑s𝑁)) ∈ ℤs[1/2])

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3091  (class class class)co 7420   /_su cdivs 28436  ℕ_0scn0s 28561  ℤ_sczs 28627  2_sc2s 28659  ↑_scexps 28661  ℤ_s[1/2]cz12s 28663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-nul 5271

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6497  df-fv 6549  df-ov 7423  df-z12s 28664

This theorem is used by:  bdayfinlem  28735