Theorem ellaz 49720

Description: Elementhood in an adjunction of a complex number to the ring of integers. Such an element is also called "extended integer" in the following. (Contributed by AV, 24-Oct-2026.)

Assertion

Ref	Expression
ellaz	⊢ (𝑆 ∈ ℂ → (𝐴 ∈ (Z[]l‘𝑆) ↔ (𝐴 ∈ ℂ ∧ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝐴 = (𝑎 + (𝑏 · 𝑆)))))

Distinct variable groups:   𝐴,𝑎,𝑏   𝑆,𝑎,𝑏

Proof of Theorem ellaz

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lazval 49719	. . 3 ⊢ (𝑆 ∈ ℂ → (Z[]l‘𝑆) = {𝑥 ∈ ℂ ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = (𝑎 + (𝑏 · 𝑆))})
2	1	eleq2d 2850	. 2 ⊢ (𝑆 ∈ ℂ → (𝐴 ∈ (Z[]l‘𝑆) ↔ 𝐴 ∈ {𝑥 ∈ ℂ ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = (𝑎 + (𝑏 · 𝑆))}))
3		eqeq1 2767	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = (𝑎 + (𝑏 · 𝑆)) ↔ 𝐴 = (𝑎 + (𝑏 · 𝑆))))
4	3	2rexbidv 3231	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = (𝑎 + (𝑏 · 𝑆)) ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝐴 = (𝑎 + (𝑏 · 𝑆))))
5		eqid 2763	. . 3 ⊢ {𝑥 ∈ ℂ ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = (𝑎 + (𝑏 · 𝑆))} = {𝑥 ∈ ℂ ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = (𝑎 + (𝑏 · 𝑆))}
6	4, 5	elrab2 3652	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ ℂ ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = (𝑎 + (𝑏 · 𝑆))} ↔ (𝐴 ∈ ℂ ∧ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝐴 = (𝑎 + (𝑏 · 𝑆))))
7	2, 6	bitrdi 290	1 ⊢ (𝑆 ∈ ℂ → (𝐴 ∈ (Z[]l‘𝑆) ↔ (𝐴 ∈ ℂ ∧ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝐴 = (𝑎 + (𝑏 · 𝑆)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  {crab 3416  ‘cfv 6564  (class class class)co 7449  ℂcc 11235   + caddc 11240   · cmul 11242  ℤcz 12731  Z[]lclaz 49717

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396  ax-cnex 11293

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-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-iota 6518  df-fun 6566  df-fv 6572  df-ov 7452  df-laz 49718

This theorem is used by:  ellazim  49721  lazcn  49722  zellaz  49723  adcnmulzlaz  49725  lazlinz  49727  adcnlaz  49728  gzilaz  49729