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Mirrors > Home > MPE Home > Th. List > cffldtocusgr | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
cffldtocusgr.p | ⊢ 𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} |
cffldtocusgr.g | ⊢ 𝐺 = (ℂfld sSet ⟨(.ef‘ndx), ( I ↾ 𝑃)⟩) |
Ref | Expression |
---|---|
cffldtocusgr | ⊢ 𝐺 ∈ ComplUSGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5463 | . . . . . . 7 ⊢ ⟨(Base‘ndx), ℂ⟩ ∈ V | |
2 | 1 | tpid1 4771 | . . . . . 6 ⊢ ⟨(Base‘ndx), ℂ⟩ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} |
3 | 2 | orci 861 | . . . . 5 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∨ ⟨(Base‘ndx), ℂ⟩ ∈ {⟨(*𝑟‘ndx), ∗⟩}) |
4 | elun 4147 | . . . . 5 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ↔ (⟨(Base‘ndx), ℂ⟩ ∈ {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∨ ⟨(Base‘ndx), ℂ⟩ ∈ {⟨(*𝑟‘ndx), ∗⟩})) | |
5 | 3, 4 | mpbir 230 | . . . 4 ⊢ ⟨(Base‘ndx), ℂ⟩ ∈ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) |
6 | 5 | orci 861 | . . 3 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∨ ⟨(Base‘ndx), ℂ⟩ ∈ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) |
7 | df-cnfld 21145 | . . . . 5 ⊢ ℂfld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) | |
8 | 7 | eleq2i 2823 | . . . 4 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ℂfld ↔ ⟨(Base‘ndx), ℂ⟩ ∈ (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))) |
9 | elun 4147 | . . . 4 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) ↔ (⟨(Base‘ndx), ℂ⟩ ∈ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∨ ⟨(Base‘ndx), ℂ⟩ ∈ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))) | |
10 | 8, 9 | bitri 274 | . . 3 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ℂfld ↔ (⟨(Base‘ndx), ℂ⟩ ∈ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∨ ⟨(Base‘ndx), ℂ⟩ ∈ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))) |
11 | 6, 10 | mpbir 230 | . 2 ⊢ ⟨(Base‘ndx), ℂ⟩ ∈ ℂfld |
12 | cffldtocusgr.p | . . . 4 ⊢ 𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} | |
13 | cnfldbas 21148 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
14 | 13 | pweqi 4617 | . . . . 5 ⊢ 𝒫 ℂ = 𝒫 (Base‘ℂfld) |
15 | 14 | rabeqi 3443 | . . . 4 ⊢ {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2} |
16 | 12, 15 | eqtri 2758 | . . 3 ⊢ 𝑃 = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2} |
17 | cnfldstr 21146 | . . . 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 6903 | . . . 4 ⊢ (Base‘ndx) ∈ V | |
21 | cnex 11193 | . . . 4 ⊢ ℂ ∈ V | |
22 | 20, 21 | opeldm 5906 | . . 3 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ℂfld → (Base‘ndx) ∈ dom ℂfld) |
23 | 16, 18, 19, 22 | structtocusgr 28970 | . 2 ⊢ (⟨(Base‘ndx), ℂ⟩ ∈ ℂfld → 𝐺 ∈ ComplUSGraph) |
24 | 11, 23 | ax-mp 5 | 1 ⊢ 𝐺 ∈ ComplUSGraph |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 843 = wceq 1539 ∈ wcel 2104 {crab 3430 ∪ cun 3945 𝒫 cpw 4601 {csn 4627 {ctp 4631 ⟨cop 4633 class class class wbr 5147 I cid 5572 ↾ cres 5677 ∘ ccom 5679 ‘cfv 6542 (class class class)co 7411 ℂcc 11110 1c1 11113 + caddc 11115 · cmul 11117 ≤ cle 11253 − cmin 11448 2c2 12271 3c3 12272 ;cdc 12681 ♯chash 14294 ∗ccj 15047 abscabs 15185 Struct cstr 17083 sSet csts 17100 ndxcnx 17130 Basecbs 17148 +gcplusg 17201 .rcmulr 17202 *𝑟cstv 17203 TopSetcts 17207 lecple 17208 distcds 17210 UnifSetcunif 17211 MetOpencmopn 21134 metUnifcmetu 21135 ℂfldccnfld 21144 .efcedgf 28513 ComplUSGraphccusgr 28934 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-hash 14295 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-plusg 17214 df-mulr 17215 df-starv 17216 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-cnfld 21145 df-edgf 28514 df-vtx 28525 df-iedg 28526 df-edg 28575 df-usgr 28678 df-nbgr 28857 df-uvtx 28910 df-cplgr 28935 df-cusgr 28936 |
This theorem is referenced by: (None) |
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