Theorem cffldtocusgr 29427

Description: The field of complex numbers can be made a complete simple graph with the set of pairs of complex numbers regarded as edges. This theorem demonstrates the capabilities of the current definitions for graphs applied to extensible structures. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 17-Nov-2021.) Revise df-cnfld 21294. (Revised by GG, 31-Mar-2025.)

Hypotheses

Ref	Expression
cffldtocusgr.p	⊢ 𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2}
cffldtocusgr.g	⊢ 𝐺 = (ℂfld sSet ⟨(.ef‘ndx), ( I ↾ 𝑃)⟩)

Assertion

Ref	Expression
cffldtocusgr	⊢ 𝐺 ∈ ComplUSGraph

Distinct variable groups:   𝑥,𝐺   𝑥,𝑃

Proof of Theorem cffldtocusgr

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

Step	Hyp	Ref	Expression
1		opex 5407	. . . . . . 7 ⊢ ⟨(Base‘ndx), ℂ⟩ ∈ V
2	1	tpid1 4720	. . . . . 6 ⊢ ⟨(Base‘ndx), ℂ⟩ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩}
3	2	orci 865	. . . . 5 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∨ ⟨(Base‘ndx), ℂ⟩ ∈ {⟨(*𝑟‘ndx), ∗⟩})
4		elun 4102	. . . . 5 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ↔ (⟨(Base‘ndx), ℂ⟩ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∨ ⟨(Base‘ndx), ℂ⟩ ∈ {⟨(*𝑟‘ndx), ∗⟩}))
5	3, 4	mpbir 231	. . . 4 ⊢ ⟨(Base‘ndx), ℂ⟩ ∈ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩})
6	5	orci 865	. . 3 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∨ ⟨(Base‘ndx), ℂ⟩ ∈ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
7		df-cnfld 21294	. . . . 5 ⊢ 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 ∘ − ))⟩}))
8	7	eleq2i 2825	. . . 4 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ℂfld ↔ ⟨(Base‘ndx), ℂ⟩ ∈ (({⟨(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 ∘ − ))⟩})))
9		elun 4102	. . . 4 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ (({⟨(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⟩ ∈ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∨ ⟨(Base‘ndx), ℂ⟩ ∈ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})))
10	8, 9	bitri 275	. . 3 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ℂfld ↔ (⟨(Base‘ndx), ℂ⟩ ∈ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∨ ⟨(Base‘ndx), ℂ⟩ ∈ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})))
11	6, 10	mpbir 231	. 2 ⊢ ⟨(Base‘ndx), ℂ⟩ ∈ ℂfld
12		cffldtocusgr.p	. . . 4 ⊢ 𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2}
13		cnfldbas 21297	. . . . . 6 ⊢ ℂ = (Base‘ℂfld)
14	13	pweqi 4565	. . . . 5 ⊢ 𝒫 ℂ = 𝒫 (Base‘ℂfld)
15	14	rabeqi 3409	. . . 4 ⊢ {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2}
16	12, 15	eqtri 2756	. . 3 ⊢ 𝑃 = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2}
17		cnfldstr 21295	. . . 4 ⊢ ℂfld Struct ⟨1, ;13⟩
18	17	a1i 11	. . 3 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ℂfld → ℂfld Struct ⟨1, ;13⟩)
19		cffldtocusgr.g	. . 3 ⊢ 𝐺 = (ℂfld sSet ⟨(.ef‘ndx), ( I ↾ 𝑃)⟩)
20		fvex 6841	. . . 4 ⊢ (Base‘ndx) ∈ V
21		cnex 11094	. . . 4 ⊢ ℂ ∈ V
22	20, 21	opeldm 5851	. . 3 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ℂfld → (Base‘ndx) ∈ dom ℂfld)
23	16, 18, 19, 22	structtocusgr 29426	. 2 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ℂfld → 𝐺 ∈ ComplUSGraph)
24	11, 23	ax-mp 5	1 ⊢ 𝐺 ∈ ComplUSGraph

Syntax hints:   ∨ wo 847   = wceq 1541   ∈ wcel 2113  {crab 3396   ∪ cun 3896  𝒫 cpw 4549  {csn 4575  {ctp 4579  ⟨cop 4581   class class class wbr 5093   I cid 5513   ↾ cres 5621   ∘ ccom 5623  ‘cfv 6486  (class class class)co 7352   ∈ cmpo 7354  ℂcc 11011  1c1 11014   + caddc 11016   · cmul 11018   ≤ cle 11154   − cmin 11351  2c2 12187  3c3 12188  ;cdc 12594  ♯chash 14239  ∗ccj 15005  abscabs 15143   Struct cstr 17059   sSet csts 17076  ndxcnx 17106  Basecbs 17122  +_gcplusg 17163  ._rcmulr 17164  *_𝑟cstv 17165  TopSetcts 17169  lecple 17170  distcds 17172  UnifSetcunif 17173  MetOpencmopn 21283  metUnifcmetu 21284  ℂ_fldccnfld 21293  .efcedgf 28968  ComplUSGraphccusgr 29390

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-oadd 8395  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-dju 9801  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-5 12198  df-6 12199  df-7 12200  df-8 12201  df-9 12202  df-n0 12389  df-xnn0 12462  df-z 12476  df-dec 12595  df-uz 12739  df-fz 13410  df-hash 14240  df-struct 17060  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-plusg 17176  df-mulr 17177  df-starv 17178  df-tset 17182  df-ple 17183  df-ds 17185  df-unif 17186  df-cnfld 21294  df-edgf 28969  df-vtx 28978  df-iedg 28979  df-edg 29028  df-usgr 29131  df-nbgr 29313  df-uvtx 29366  df-cplgr 29391  df-cusgr 29392

This theorem is referenced by: (None)