Theorem lazval 49719

Description: An adjunction of a complex number to the ring of integers. (Contributed by AV, 24-Oct-2026.)

Assertion

Ref	Expression
lazval	⊢ (𝑆 ∈ ℂ → (Z[]l‘𝑆) = {𝑥 ∈ ℂ ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = (𝑎 + (𝑏 · 𝑆))})

Distinct variable group:   𝑆,𝑎,𝑏,𝑥

Proof of Theorem lazval

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7457	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑏 · 𝑠) = (𝑏 · 𝑆))
2	1	oveq2d 7465	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑎 + (𝑏 · 𝑠)) = (𝑎 + (𝑏 · 𝑆)))
3	2	eqeq2d 2774	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑥 = (𝑎 + (𝑏 · 𝑠)) ↔ 𝑥 = (𝑎 + (𝑏 · 𝑆))))
4	3	2rexbidv 3231	. . 3 ⊢ (𝑠 = 𝑆 → (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = (𝑎 + (𝑏 · 𝑠)) ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = (𝑎 + (𝑏 · 𝑆))))
5	4	rabbidv 3423	. 2 ⊢ (𝑠 = 𝑆 → {𝑥 ∈ ℂ ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = (𝑎 + (𝑏 · 𝑠))} = {𝑥 ∈ ℂ ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = (𝑎 + (𝑏 · 𝑆))})
6		df-laz 49718	. 2 ⊢ Z[]l = (𝑠 ∈ ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = (𝑎 + (𝑏 · 𝑠))})
7		cnex 11318	. . 3 ⊢ ℂ ∈ V
8	7	rabex 5305	. 2 ⊢ {𝑥 ∈ ℂ ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = (𝑎 + (𝑏 · 𝑆))} ∈ V
9	5, 6, 8	fvmpt 7022	1 ⊢ (𝑆 ∈ ℂ → (Z[]l‘𝑆) = {𝑥 ∈ ℂ ∣ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = (𝑎 + (𝑏 · 𝑆))})

This proof depends on syntax axioms:   → wi 4   = 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:  ellaz  49720