Theorem cffldtocusgr 29594

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 21405. (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 5430	. . . . . . 7 ⊢ ⟨(Base‘ndx), ℂ⟩ ∈ V
2	1	tpid1 4726	. . . . . 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 876	. . . . 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 4106	. . . . 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 233	. . . 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 876	. . 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 21405	. . . . 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 2853	. . . 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 4106	. . . 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 277	. . 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 233	. 2 ⊢ ⟨(Base‘ndx), ℂ⟩ ∈ ℂfld
12		cffldtocusgr.p	. . . 4 ⊢ 𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2}
13		cnfldbas 21408	. . . . . 6 ⊢ ℂ = (Base‘ℂfld)
14	13	pweqi 4570	. . . . 5 ⊢ 𝒫 ℂ = 𝒫 (Base‘ℂfld)
15	14	rabeqi 3426	. . . 4 ⊢ {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2}
16	12, 15	eqtri 2784	. . 3 ⊢ 𝑃 = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2}
17		cnfldstr 21406	. . . 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 6876	. . . 4 ⊢ (Base‘ndx) ∈ V
21		cnex 11151	. . . 4 ⊢ ℂ ∈ V
22	20, 21	opeldm 5881	. . 3 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ℂfld → (Base‘ndx) ∈ dom ℂfld)
23	16, 18, 19, 22	structtocusgr 29593	. 2 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ℂfld → 𝐺 ∈ ComplUSGraph)
24	11, 23	ax-mp 5	1 ⊢ 𝐺 ∈ ComplUSGraph

Syntax hints:   ∨ wo 858   = wceq 1559   ∈ wcel 2141  {crab 3413   ∪ cun 3902  𝒫 cpw 4554  {csn 4581  {ctp 4585  ⟨cop 4587   class class class wbr 5099   I cid 5539   ↾ cres 5647   ∘ ccom 5649  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  ℂcc 11068  1c1 11071   + caddc 11073   · cmul 11075   ≤ cle 11214   − cmin 11411  2c2 12269  3c3 12270  ;cdc 12685  ♯chash 14340  ∗ccj 15106  abscabs 15244   Struct cstr 17165   sSet csts 17182  ndxcnx 17212  Basecbs 17228  +_gcplusg 17269  ._rcmulr 17270  *_𝑟cstv 17271  TopSetcts 17275  lecple 17276  distcds 17278  UnifSetcunif 17279  MetOpencmopn 21394  metUnifcmetu 21395  ℂ_fldccnfld 21404  .efcedgf 29135  ComplUSGraphccusgr 29557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-oadd 8436  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-dju 9856  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12479  df-xnn0 12552  df-z 12566  df-dec 12686  df-uz 12837  df-fz 13510  df-hash 14341  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-plusg 17282  df-mulr 17283  df-starv 17284  df-tset 17288  df-ple 17289  df-ds 17291  df-unif 17292  df-cnfld 21405  df-edgf 29136  df-vtx 29145  df-iedg 29146  df-edg 29195  df-usgr 29298  df-nbgr 29480  df-uvtx 29533  df-cplgr 29558  df-cusgr 29559

This theorem is referenced by: (None)