Theorem cffldtocusgr 29706

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 21483. (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 5436	. . . . . . 7 ⊢ ⟨(Base‘ndx), ℂ⟩ ∈ V
2	1	tpid1 4730	. . . . . 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 878	. . . . 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 4109	. . . . 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 234	. . . 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 878	. . 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 21483	. . . . 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 2857	. . . 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 4109	. . . 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 278	. . 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 234	. 2 ⊢ ⟨(Base‘ndx), ℂ⟩ ∈ ℂfld
12		cffldtocusgr.p	. . . 4 ⊢ 𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2}
13		cnfldbas 21486	. . . . . 6 ⊢ ℂ = (Base‘ℂfld)
14	13	pweqi 4574	. . . . 5 ⊢ 𝒫 ℂ = 𝒫 (Base‘ℂfld)
15	14	rabeqi 3430	. . . 4 ⊢ {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2}
16	12, 15	eqtri 2788	. . 3 ⊢ 𝑃 = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2}
17		cnfldstr 21484	. . . 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 6884	. . . 4 ⊢ (Base‘ndx) ∈ V
21		cnex 11169	. . . 4 ⊢ ℂ ∈ V
22	20, 21	opeldm 5888	. . 3 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ℂfld → (Base‘ndx) ∈ dom ℂfld)
23	16, 18, 19, 22	structtocusgr 29705	. 2 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ℂfld → 𝐺 ∈ ComplUSGraph)
24	11, 23	ax-mp 5	1 ⊢ 𝐺 ∈ ComplUSGraph

Syntax hints:   ∨ wo 860   = wceq 1563   ∈ wcel 2145  {crab 3417   ∪ cun 3905  𝒫 cpw 4558  {csn 4585  {ctp 4589  ⟨cop 4591   class class class wbr 5105   I cid 5546   ↾ cres 5654   ∘ ccom 5656  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  ℂcc 11086  1c1 11089   + caddc 11091   · cmul 11093   ≤ cle 11232   − cmin 11429  2c2 12286  3c3 12287  ;cdc 12702  ♯chash 14357  ∗ccj 15137  abscabs 15275   Struct cstr 17196   sSet csts 17213  ndxcnx 17243  Basecbs 17259  +_gcplusg 17300  ._rcmulr 17301  *_𝑟cstv 17302  TopSetcts 17306  lecple 17307  distcds 17309  UnifSetcunif 17310  MetOpencmopn 21472  metUnifcmetu 21473  ℂ_fldccnfld 21482  .efcedgf 29247  ComplUSGraphccusgr 29669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-oadd 8445  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-dju 9875  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-9 12301  df-n0 12496  df-xnn0 12569  df-z 12583  df-dec 12703  df-uz 12854  df-fz 13527  df-hash 14358  df-struct 17197  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-plusg 17313  df-mulr 17314  df-starv 17315  df-tset 17319  df-ple 17320  df-ds 17322  df-unif 17323  df-cnfld 21483  df-edgf 29248  df-vtx 29257  df-iedg 29258  df-edg 29307  df-usgr 29410  df-nbgr 29592  df-uvtx 29645  df-cplgr 29670  df-cusgr 29671

This theorem is referenced by: (None)