Theorem elz12si 28543

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 2761	. . 3 ⊢ (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s𝑁))
2		oveq1 7399	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 /su (2s↑s𝑛)) = (𝐴 /su (2s↑s𝑛)))
3	2	eqeq2d 2772	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐴 /su (2s↑s𝑁)) = (𝑥 /su (2s↑s𝑛)) ↔ (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s𝑛))))
4		oveq2 7400	. . . . . 6 ⊢ (𝑛 = 𝑁 → (2s↑s𝑛) = (2s↑s𝑁))
5	4	oveq2d 7408	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝐴 /su (2s↑s𝑛)) = (𝐴 /su (2s↑s𝑁)))
6	5	eqeq2d 2772	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s𝑛)) ↔ (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s𝑁))))
7	3, 6	rspc2ev 3594	. . 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 1470	. 2 ⊢ ((𝐴 ∈ ℤs ∧ 𝑁 ∈ ℕ0s) → ∃𝑥 ∈ ℤs ∃𝑛 ∈ ℕ0s (𝐴 /su (2s↑s𝑁)) = (𝑥 /su (2s↑s𝑛)))
9		elz12s 28542	. 2 ⊢ ((𝐴 /su (2s↑s𝑁)) ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑛 ∈ ℕ0s (𝐴 /su (2s↑s𝑁)) = (𝑥 /su (2s↑s𝑛)))
10	8, 9	sylibr 236	1 ⊢ ((𝐴 ∈ ℤs ∧ 𝑁 ∈ ℕ0s) → (𝐴 /su (2s↑s𝑁)) ∈ ℤs[1/2])

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  (class class class)co 7392   /_su cdivs 28257  ℕ_0scn0s 28382  ℤ_sczs 28448  2_sc2s 28480  ↑_scexps 28482  ℤ_s[1/2]cz12s 28484

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-ext 2733  ax-nul 5255

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-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6473  df-fv 6525  df-ov 7395  df-z12s 28485

This theorem is referenced by:  bdayfinlem  28556