Theorem dfcnfldOLD 21287

Description: Obsolete version of df-cnfld 21272 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 21272	. 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 2731	. . . . . 6 ⊢ (⊤ → ⟨(Base‘ndx), ℂ⟩ = ⟨(Base‘ndx), ℂ⟩)
3		ax-addf 11154	. . . . . . . . . . 11 ⊢ + :(ℂ × ℂ)⟶ℂ
4		ffn 6691	. . . . . . . . . . 11 ⊢ ( + :(ℂ × ℂ)⟶ℂ → + Fn (ℂ × ℂ))
5	3, 4	ax-mp 5	. . . . . . . . . 10 ⊢ + Fn (ℂ × ℂ)
6		fnov 7523	. . . . . . . . . 10 ⊢ ( + Fn (ℂ × ℂ) ↔ + = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣)))
7	5, 6	mpbi 230	. . . . . . . . 9 ⊢ + = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))
8		eqcom 2737	. . . . . . . . 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 4844	. . . . . . 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 11155	. . . . . . . . . . 11 ⊢ · :(ℂ × ℂ)⟶ℂ
13		ffn 6691	. . . . . . . . . . 11 ⊢ ( · :(ℂ × ℂ)⟶ℂ → · Fn (ℂ × ℂ))
14	12, 13	ax-mp 5	. . . . . . . . . 10 ⊢ · Fn (ℂ × ℂ)
15		fnov 7523	. . . . . . . . . 10 ⊢ ( · Fn (ℂ × ℂ) ↔ · = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)))
16	14, 15	mpbi 230	. . . . . . . . 9 ⊢ · = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))
17		eqcom 2737	. . . . . . . . 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 4844	. . . . . . 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 4715	. . . . 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 1547	. . . 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 4130	. . 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 4130	. 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 2753	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 1540  ⊤wtru 1541   ∪ cun 3915  {csn 4592  {ctp 4596  ⟨cop 4598   × cxp 5639   ∘ ccom 5645   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  ℂcc 11073   + caddc 11078   · cmul 11080   ≤ cle 11216   − cmin 11412  ∗ccj 15069  abscabs 15207  ndxcnx 17170  Basecbs 17186  +_gcplusg 17227  ._rcmulr 17228  *_𝑟cstv 17229  TopSetcts 17233  lecple 17234  distcds 17236  UnifSetcunif 17237  MetOpencmopn 21261  metUnifcmetu 21262  ℂ_fldccnfld 21271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-addf 11154  ax-mulf 11155

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-cnfld 21272

This theorem is referenced by:  cnfldstrOLD  21288  cnfldbasOLD  21290  cnfldaddOLD  21291  cnfldmulOLD  21292  cnfldcjOLD  21293  cnfldtsetOLD  21294  cnfldleOLD  21295  cnflddsOLD  21296  cnfldunifOLD  21297  cnfldfunOLD  21298  cnfldfunALTOLD  21299  cffldtocusgrOLD  29382