Theorem elz12si 28767

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 2760	. . 3 ⊢ (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s𝑁))
2		oveq1 7422	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 /su (2s↑s𝑛)) = (𝐴 /su (2s↑s𝑛)))
3	2	eqeq2d 2771	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐴 /su (2s↑s𝑁)) = (𝑥 /su (2s↑s𝑛)) ↔ (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s𝑛))))
4		oveq2 7423	. . . . . 6 ⊢ (𝑛 = 𝑁 → (2s↑s𝑛) = (2s↑s𝑁))
5	4	oveq2d 7431	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝐴 /su (2s↑s𝑛)) = (𝐴 /su (2s↑s𝑁)))
6	5	eqeq2d 2771	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s𝑛)) ↔ (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s𝑁))))
7	3, 6	rspc2ev 3589	. . 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 28766	. 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 2145  ∃wrex 3086  (class class class)co 7415   /_su cdivs 28481  ℕ_0scn0s 28606  ℤ_sczs 28672  2_sc2s 28704  ↑_scexps 28706  ℤ_s[1/2]cz12s 28708

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 2147  ax-9 2155  ax-ext 2732  ax-nul 5263

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 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6490  df-fv 6542  df-ov 7418  df-z12s 28709

This theorem is used by:  bdayfinlem  28780