Theorem azval 49710

Description: An adjunction of a complex number to the ring of integers. (Contributed by AV, 30-Oct-2026.)

Assertion

Ref	Expression
azval	⊢ (𝑆 ∈ ℂ → (Z[]‘𝑆) = ((RingSpan‘ℂfld)‘(ℤ ∪ {𝑆})))

Proof of Theorem azval

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4594	. . . 4 ⊢ (𝑠 = 𝑆 → {𝑠} = {𝑆})
2	1	uneq2d 4170	. . 3 ⊢ (𝑠 = 𝑆 → (ℤ ∪ {𝑠}) = (ℤ ∪ {𝑆}))
3	2	fveq2d 6918	. 2 ⊢ (𝑠 = 𝑆 → ((RingSpan‘ℂfld)‘(ℤ ∪ {𝑠})) = ((RingSpan‘ℂfld)‘(ℤ ∪ {𝑆})))
4		df-az 49709	. 2 ⊢ Z[] = (𝑠 ∈ ℂ ↦ ((RingSpan‘ℂfld)‘(ℤ ∪ {𝑠})))
5		fvex 6927	. 2 ⊢ ((RingSpan‘ℂfld)‘(ℤ ∪ {𝑆})) ∈ V
6	3, 4, 5	fvmpt 7022	1 ⊢ (𝑆 ∈ ℂ → (Z[]‘𝑆) = ((RingSpan‘ℂfld)‘(ℤ ∪ {𝑆})))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   ∪ cun 4152  {csn 4588  ‘cfv 6564  ℂcc 11235  ℤcz 12731  RingSpancrgspn 20904  ℂ_fldccnfld 21720  Z[]caz 49708

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-nul 5264  ax-pr 5396

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-ne 2960  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-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-az 49709

This theorem is used by:  elaz  49711  azssid  49712  azcn  49714