Theorem elz12si 28469

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 2736	. . 3 ⊢ (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s𝑁))
2		oveq1 7365	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 /su (2s↑s𝑛)) = (𝐴 /su (2s↑s𝑛)))
3	2	eqeq2d 2747	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐴 /su (2s↑s𝑁)) = (𝑥 /su (2s↑s𝑛)) ↔ (𝐴 /su (2s↑s𝑁)) = (𝐴 /su (2s↑s𝑛))))
4		oveq2 7366	. . . . . 6 ⊢ (𝑛 = 𝑁 → (2s↑s𝑛) = (2s↑s𝑁))
5	4	oveq2d 7374	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝐴 /su (2s↑s𝑛)) = (𝐴 /su (2s↑s𝑁)))
6	5	eqeq2d 2747	. . . 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 1452	. 2 ⊢ ((𝐴 ∈ ℤs ∧ 𝑁 ∈ ℕ0s) → ∃𝑥 ∈ ℤs ∃𝑛 ∈ ℕ0s (𝐴 /su (2s↑s𝑁)) = (𝑥 /su (2s↑s𝑛)))
9		elz12s 28468	. 2 ⊢ ((𝐴 /su (2s↑s𝑁)) ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑛 ∈ ℕ0s (𝐴 /su (2s↑s𝑁)) = (𝑥 /su (2s↑s𝑛)))
10	8, 9	sylibr 234	1 ⊢ ((𝐴 ∈ ℤs ∧ 𝑁 ∈ ℕ0s) → (𝐴 /su (2s↑s𝑁)) ∈ ℤs[1/2])

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  (class class class)co 7358   /_su cdivs 28183  ℕ_0scn0s 28308  ℤ_sczs 28374  2_sc2s 28406  ↑_scexps 28408  ℤ_s[1/2]cz12s 28410

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-ext 2708  ax-nul 5251

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-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-z12s 28411

This theorem is referenced by:  bdayfinlem  28482