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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 5321 | . . . . . . 7 ⊢ 〈(Base‘ndx), ℂ〉 ∈ V | |
2 | 1 | tpid1 4664 | . . . . . 6 ⊢ 〈(Base‘ndx), ℂ〉 ∈ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} |
3 | 2 | orci 862 | . . . . 5 ⊢ (〈(Base‘ndx), ℂ〉 ∈ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∨ 〈(Base‘ndx), ℂ〉 ∈ {〈(*𝑟‘ndx), ∗〉}) |
4 | elun 4076 | . . . . 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 234 | . . . 4 ⊢ 〈(Base‘ndx), ℂ〉 ∈ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) |
6 | 5 | orci 862 | . . 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 20092 | . . . . 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 2881 | . . . 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 4076 | . . . 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 278 | . . 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 234 | . 2 ⊢ 〈(Base‘ndx), ℂ〉 ∈ ℂfld |
12 | cffldtocusgr.p | . . . 4 ⊢ 𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} | |
13 | cnfldbas 20095 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
14 | 13 | pweqi 4515 | . . . . 5 ⊢ 𝒫 ℂ = 𝒫 (Base‘ℂfld) |
15 | 14 | rabeqi 3429 | . . . 4 ⊢ {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2} |
16 | 12, 15 | eqtri 2821 | . . 3 ⊢ 𝑃 = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2} |
17 | cnfldstr 20093 | . . . 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 6658 | . . . 4 ⊢ (Base‘ndx) ∈ V | |
21 | cnex 10607 | . . . 4 ⊢ ℂ ∈ V | |
22 | 20, 21 | opeldm 5740 | . . 3 ⊢ (〈(Base‘ndx), ℂ〉 ∈ ℂfld → (Base‘ndx) ∈ dom ℂfld) |
23 | 16, 18, 19, 22 | structtocusgr 27236 | . 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 844 = wceq 1538 ∈ wcel 2111 {crab 3110 ∪ cun 3879 𝒫 cpw 4497 {csn 4525 {ctp 4529 〈cop 4531 class class class wbr 5030 I cid 5424 ↾ cres 5521 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 1c1 10527 + caddc 10529 · cmul 10531 ≤ cle 10665 − cmin 10859 2c2 11680 3c3 11681 ;cdc 12086 ♯chash 13686 ∗ccj 14447 abscabs 14585 Struct cstr 16471 ndxcnx 16472 sSet csts 16473 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 *𝑟cstv 16559 TopSetcts 16563 lecple 16564 distcds 16566 UnifSetcunif 16567 MetOpencmopn 20081 metUnifcmetu 20082 ℂfldccnfld 20091 .efcedgf 26782 ComplUSGraphccusgr 27200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-cnfld 20092 df-edgf 26783 df-vtx 26791 df-iedg 26792 df-edg 26841 df-usgr 26944 df-nbgr 27123 df-uvtx 27176 df-cplgr 27201 df-cusgr 27202 |
This theorem is referenced by: (None) |
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