Theorem cnfldeuf

Description: The Euclidean function component of the complex numbers. (Contributed by Thierry Arnoux, 9-Nov-2026)

	Ref	Expression
Assertion	cnfldeuf	⊢ abs = ( EuclF ‘ ℂfld )

Proof

Step	Hyp	Ref	Expression
1		absex	⊢ abs ∈ V
2		cnfldstr	⊢ ℂfld Struct ⟨ 1 , ; 2 1 ⟩
3		eufid	⊢ EuclF = Slot ( EuclF ‘ ndx )
4		snsspr2	⊢ { ⟨ ( EuclF ‘ ndx ) , abs ⟩ } ⊆ { ⟨ ( UnifSet ‘ ndx ) , ( metUnif ‘ ( abs ∘ − ) ) ⟩ , ⟨ ( EuclF ‘ ndx ) , abs ⟩ }
5		ssun2	⊢ { ⟨ ( UnifSet ‘ ndx ) , ( metUnif ‘ ( abs ∘ − ) ) ⟩ , ⟨ ( EuclF ‘ ndx ) , abs ⟩ } ⊆ ( { ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ ( abs ∘ − ) ) ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ , ⟨ ( dist ‘ ndx ) , ( abs ∘ − ) ⟩ } ∪ { ⟨ ( UnifSet ‘ ndx ) , ( metUnif ‘ ( abs ∘ − ) ) ⟩ , ⟨ ( EuclF ‘ ndx ) , abs ⟩ } )
6		ssun2	⊢ ( { ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ ( abs ∘ − ) ) ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ , ⟨ ( dist ‘ ndx ) , ( abs ∘ − ) ⟩ } ∪ { ⟨ ( UnifSet ‘ ndx ) , ( metUnif ‘ ( abs ∘ − ) ) ⟩ , ⟨ ( EuclF ‘ ndx ) , abs ⟩ } ) ⊆ ( ( { ⟨ ( Base ‘ ndx ) , ℂ ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑢 ∈ ℂ , 𝑣 ∈ ℂ ↦ ( 𝑢 + 𝑣 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑢 ∈ ℂ , 𝑣 ∈ ℂ ↦ ( 𝑢 · 𝑣 ) ) ⟩ } ∪ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } ) ∪ ( { ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ ( abs ∘ − ) ) ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ , ⟨ ( dist ‘ ndx ) , ( abs ∘ − ) ⟩ } ∪ { ⟨ ( UnifSet ‘ ndx ) , ( metUnif ‘ ( abs ∘ − ) ) ⟩ , ⟨ ( EuclF ‘ ndx ) , abs ⟩ } ) )
7		df-cnfld	⊢ ℂfld = ( ( { ⟨ ( Base ‘ ndx ) , ℂ ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑢 ∈ ℂ , 𝑣 ∈ ℂ ↦ ( 𝑢 + 𝑣 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑢 ∈ ℂ , 𝑣 ∈ ℂ ↦ ( 𝑢 · 𝑣 ) ) ⟩ } ∪ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } ) ∪ ( { ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ ( abs ∘ − ) ) ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ , ⟨ ( dist ‘ ndx ) , ( abs ∘ − ) ⟩ } ∪ { ⟨ ( UnifSet ‘ ndx ) , ( metUnif ‘ ( abs ∘ − ) ) ⟩ , ⟨ ( EuclF ‘ ndx ) , abs ⟩ } ) )
8	6 7	sseqtrri	⊢ ( { ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ ( abs ∘ − ) ) ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ , ⟨ ( dist ‘ ndx ) , ( abs ∘ − ) ⟩ } ∪ { ⟨ ( UnifSet ‘ ndx ) , ( metUnif ‘ ( abs ∘ − ) ) ⟩ , ⟨ ( EuclF ‘ ndx ) , abs ⟩ } ) ⊆ ℂfld
9	5 8	sstri	⊢ { ⟨ ( UnifSet ‘ ndx ) , ( metUnif ‘ ( abs ∘ − ) ) ⟩ , ⟨ ( EuclF ‘ ndx ) , abs ⟩ } ⊆ ℂfld
10	4 9	sstri	⊢ { ⟨ ( EuclF ‘ ndx ) , abs ⟩ } ⊆ ℂfld
11	2 3 10	strfv	⊢ ( abs ∈ V → abs = ( EuclF ‘ ℂfld ) )
12	1 11	ax-mp	⊢ abs = ( EuclF ‘ ℂfld )