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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 5153 | . . . . . . 7 ⊢ 〈(Base‘ndx), ℂ〉 ∈ V | |
2 | 1 | tpid1 4521 | . . . . . 6 ⊢ 〈(Base‘ndx), ℂ〉 ∈ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} |
3 | 2 | orci 898 | . . . . 5 ⊢ (〈(Base‘ndx), ℂ〉 ∈ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∨ 〈(Base‘ndx), ℂ〉 ∈ {〈(*𝑟‘ndx), ∗〉}) |
4 | elun 3980 | . . . . 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 223 | . . . 4 ⊢ 〈(Base‘ndx), ℂ〉 ∈ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) |
6 | 5 | orci 898 | . . 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 20107 | . . . . 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 2898 | . . . 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 3980 | . . . 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 267 | . . 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 223 | . 2 ⊢ 〈(Base‘ndx), ℂ〉 ∈ ℂfld |
12 | cffldtocusgr.p | . . . 4 ⊢ 𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} | |
13 | cnfldbas 20110 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
14 | 13 | pweqi 4382 | . . . . 5 ⊢ 𝒫 ℂ = 𝒫 (Base‘ℂfld) |
15 | rabeq 3405 | . . . . 5 ⊢ (𝒫 ℂ = 𝒫 (Base‘ℂfld) → {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2}) | |
16 | 14, 15 | ax-mp 5 | . . . 4 ⊢ {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2} |
17 | 12, 16 | eqtri 2849 | . . 3 ⊢ 𝑃 = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2} |
18 | cnfldstr 20108 | . . . 4 ⊢ ℂfld Struct 〈1, ;13〉 | |
19 | 18 | a1i 11 | . . 3 ⊢ (〈(Base‘ndx), ℂ〉 ∈ ℂfld → ℂfld Struct 〈1, ;13〉) |
20 | cffldtocusgr.g | . . 3 ⊢ 𝐺 = (ℂfld sSet 〈(.ef‘ndx), ( I ↾ 𝑃)〉) | |
21 | fvex 6446 | . . . 4 ⊢ (Base‘ndx) ∈ V | |
22 | cnex 10333 | . . . 4 ⊢ ℂ ∈ V | |
23 | 21, 22 | opeldm 5560 | . . 3 ⊢ (〈(Base‘ndx), ℂ〉 ∈ ℂfld → (Base‘ndx) ∈ dom ℂfld) |
24 | 17, 19, 20, 23 | structtocusgr 26744 | . 2 ⊢ (〈(Base‘ndx), ℂ〉 ∈ ℂfld → 𝐺 ∈ ComplUSGraph) |
25 | 11, 24 | ax-mp 5 | 1 ⊢ 𝐺 ∈ ComplUSGraph |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 880 = wceq 1658 ∈ wcel 2166 {crab 3121 ∪ cun 3796 𝒫 cpw 4378 {csn 4397 {ctp 4401 〈cop 4403 class class class wbr 4873 I cid 5249 ↾ cres 5344 ∘ ccom 5346 ‘cfv 6123 (class class class)co 6905 ℂcc 10250 1c1 10253 + caddc 10255 · cmul 10257 ≤ cle 10392 − cmin 10585 2c2 11406 3c3 11407 ;cdc 11821 ♯chash 13410 ∗ccj 14213 abscabs 14351 Struct cstr 16218 ndxcnx 16219 sSet csts 16220 Basecbs 16222 +gcplusg 16305 .rcmulr 16306 *𝑟cstv 16307 TopSetcts 16311 lecple 16312 distcds 16314 UnifSetcunif 16315 MetOpencmopn 20096 metUnifcmetu 20097 ℂfldccnfld 20106 .efcedgf 26287 ComplUSGraphccusgr 26708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-card 9078 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-xnn0 11691 df-z 11705 df-dec 11822 df-uz 11969 df-fz 12620 df-hash 13411 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-plusg 16318 df-mulr 16319 df-starv 16320 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-cnfld 20107 df-edgf 26288 df-vtx 26296 df-iedg 26297 df-edg 26346 df-usgr 26450 df-nbgr 26630 df-uvtx 26684 df-cplgr 26709 df-cusgr 26710 |
This theorem is referenced by: (None) |
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