Theorem dfcnfldOLD 21403

Description: Obsolete version of df-cnfld 21388 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 21388	. 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 2741	. . . . . 6 ⊢ (⊤ → ⟨(Base‘ndx), ℂ⟩ = ⟨(Base‘ndx), ℂ⟩)
3		ax-addf 11263	. . . . . . . . . . 11 ⊢ + :(ℂ × ℂ)⟶ℂ
4		ffn 6747	. . . . . . . . . . 11 ⊢ ( + :(ℂ × ℂ)⟶ℂ → + Fn (ℂ × ℂ))
5	3, 4	ax-mp 5	. . . . . . . . . 10 ⊢ + Fn (ℂ × ℂ)
6		fnov 7581	. . . . . . . . . 10 ⊢ ( + Fn (ℂ × ℂ) ↔ + = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣)))
7	5, 6	mpbi 230	. . . . . . . . 9 ⊢ + = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))
8		eqcom 2747	. . . . . . . . 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 4901	. . . . . . 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 11264	. . . . . . . . . . 11 ⊢ · :(ℂ × ℂ)⟶ℂ
13		ffn 6747	. . . . . . . . . . 11 ⊢ ( · :(ℂ × ℂ)⟶ℂ → · Fn (ℂ × ℂ))
14	12, 13	ax-mp 5	. . . . . . . . . 10 ⊢ · Fn (ℂ × ℂ)
15		fnov 7581	. . . . . . . . . 10 ⊢ ( · Fn (ℂ × ℂ) ↔ · = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)))
16	14, 15	mpbi 230	. . . . . . . . 9 ⊢ · = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))
17		eqcom 2747	. . . . . . . . 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 4901	. . . . . . 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 4773	. . . . 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 1544	. . . 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 4187	. . 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 4187	. 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 2768	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 1537  ⊤wtru 1538   ∪ cun 3974  {csn 4648  {ctp 4652  ⟨cop 4654   × cxp 5698   ∘ ccom 5704   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  ℂcc 11182   + caddc 11187   · cmul 11189   ≤ cle 11325   − cmin 11520  ∗ccj 15145  abscabs 15283  ndxcnx 17240  Basecbs 17258  +_gcplusg 17311  ._rcmulr 17312  *_𝑟cstv 17313  TopSetcts 17317  lecple 17318  distcds 17320  UnifSetcunif 17321  MetOpencmopn 21377  metUnifcmetu 21378  ℂ_fldccnfld 21387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-addf 11263  ax-mulf 11264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-cnfld 21388

This theorem is referenced by:  cnfldstrOLD  21404  cnfldbasOLD  21406  cnfldaddOLD  21407  cnfldmulOLD  21408  cnfldcjOLD  21409  cnfldtsetOLD  21410  cnfldleOLD  21411  cnflddsOLD  21412  cnfldunifOLD  21413  cnfldfunOLD  21414  cnfldfunALTOLD  21415  cnfldfunALTOLDOLD  21416  cffldtocusgrOLD  29483