Theorem dfcnfldOLD 21305

Description: Obsolete version of df-cnfld 21290 as of 27-Apr-2025. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Thierry Arnoux, 15-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dfcnfldOLD	⊢ ℂfld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))

Proof of Theorem dfcnfldOLD

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cnfld 21290	. 2 ⊢ 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 ∘ − ))⟩}))
2		eqidd 2732	. . . . . 6 ⊢ (⊤ → ⟨(Base‘ndx), ℂ⟩ = ⟨(Base‘ndx), ℂ⟩)
3		ax-addf 11082	. . . . . . . . . . 11 ⊢ + :(ℂ × ℂ)⟶ℂ
4		ffn 6651	. . . . . . . . . . 11 ⊢ ( + :(ℂ × ℂ)⟶ℂ → + Fn (ℂ × ℂ))
5	3, 4	ax-mp 5	. . . . . . . . . 10 ⊢ + Fn (ℂ × ℂ)
6		fnov 7477	. . . . . . . . . 10 ⊢ ( + Fn (ℂ × ℂ) ↔ + = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣)))
7	5, 6	mpbi 230	. . . . . . . . 9 ⊢ + = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))
8		eqcom 2738	. . . . . . . . 9 ⊢ ( + = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣)) ↔ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣)) = + )
9	7, 8	mpbi 230	. . . . . . . 8 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣)) = +
10	9	opeq2i 4829	. . . . . . 7 ⊢ ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩ = ⟨(+g‘ndx), + ⟩
11	10	a1i 11	. . . . . 6 ⊢ (⊤ → ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩ = ⟨(+g‘ndx), + ⟩)
12		ax-mulf 11083	. . . . . . . . . . 11 ⊢ · :(ℂ × ℂ)⟶ℂ
13		ffn 6651	. . . . . . . . . . 11 ⊢ ( · :(ℂ × ℂ)⟶ℂ → · Fn (ℂ × ℂ))
14	12, 13	ax-mp 5	. . . . . . . . . 10 ⊢ · Fn (ℂ × ℂ)
15		fnov 7477	. . . . . . . . . 10 ⊢ ( · Fn (ℂ × ℂ) ↔ · = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)))
16	14, 15	mpbi 230	. . . . . . . . 9 ⊢ · = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))
17		eqcom 2738	. . . . . . . . 9 ⊢ ( · = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ↔ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) = · )
18	16, 17	mpbi 230	. . . . . . . 8 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) = ·
19	18	opeq2i 4829	. . . . . . 7 ⊢ ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩ = ⟨(.r‘ndx), · ⟩
20	19	a1i 11	. . . . . 6 ⊢ (⊤ → ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩ = ⟨(.r‘ndx), · ⟩)
21	2, 11, 20	tpeq123d 4701	. . . . 5 ⊢ (⊤ → {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} = {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩})
22	21	mptru 1548	. . . 4 ⊢ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} = {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩}
23	22	uneq1i 4114	. . 3 ⊢ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩})
24	23	uneq1i 4114	. 2 ⊢ (({⟨(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 ∘ − ))⟩})) = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
25	1, 24	eqtri 2754	1 ⊢ ℂfld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))

Syntax hints:   = wceq 1541  ⊤wtru 1542   ∪ cun 3900  {csn 4576  {ctp 4580  ⟨cop 4582   × cxp 5614   ∘ ccom 5620   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  ℂcc 11001   + caddc 11006   · cmul 11008   ≤ cle 11144   − cmin 11341  ∗ccj 15000  abscabs 15138  ndxcnx 17101  Basecbs 17117  +_gcplusg 17158  ._rcmulr 17159  *_𝑟cstv 17160  TopSetcts 17164  lecple 17165  distcds 17167  UnifSetcunif 17168  MetOpencmopn 21279  metUnifcmetu 21280  ℂ_fldccnfld 21289

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-addf 11082  ax-mulf 11083

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-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-cnfld 21290

This theorem is referenced by:  cnfldstrOLD  21306  cnfldbasOLD  21308  cnfldaddOLD  21309  cnfldmulOLD  21310  cnfldcjOLD  21311  cnfldtsetOLD  21312  cnfldleOLD  21313  cnflddsOLD  21314  cnfldunifOLD  21315  cnfldfunOLD  21316  cnfldfunALTOLD  21317  cffldtocusgrOLD  29424