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| Mirrors > Home > MPE Home > Th. List > cffldtocusgrOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of cffldtocusgr 29465 as of 27-Apr-2025. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 17-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| cffldtocusgrOLD.p | ⊢ 𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} | 
| cffldtocusgrOLD.g | ⊢ 𝐺 = (ℂfld sSet 〈(.ef‘ndx), ( I ↾ 𝑃)〉) | 
| Ref | Expression | 
|---|---|
| cffldtocusgrOLD | ⊢ 𝐺 ∈ ComplUSGraph | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | opex 5468 | . . . . . . 7 ⊢ 〈(Base‘ndx), ℂ〉 ∈ V | |
| 2 | 1 | tpid1 4767 | . . . . . 6 ⊢ 〈(Base‘ndx), ℂ〉 ∈ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} | 
| 3 | 2 | orci 865 | . . . . 5 ⊢ (〈(Base‘ndx), ℂ〉 ∈ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∨ 〈(Base‘ndx), ℂ〉 ∈ {〈(*𝑟‘ndx), ∗〉}) | 
| 4 | elun 4152 | . . . . 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 231 | . . . 4 ⊢ 〈(Base‘ndx), ℂ〉 ∈ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) | 
| 6 | 5 | orci 865 | . . 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 | dfcnfldOLD 21381 | . . . . 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 2832 | . . . 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 4152 | . . . 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 275 | . . 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 231 | . 2 ⊢ 〈(Base‘ndx), ℂ〉 ∈ ℂfld | 
| 12 | cffldtocusgrOLD.p | . . . 4 ⊢ 𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} | |
| 13 | cnfldbas 21369 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
| 14 | 13 | pweqi 4615 | . . . . 5 ⊢ 𝒫 ℂ = 𝒫 (Base‘ℂfld) | 
| 15 | 14 | rabeqi 3449 | . . . 4 ⊢ {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2} | 
| 16 | 12, 15 | eqtri 2764 | . . 3 ⊢ 𝑃 = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2} | 
| 17 | cnfldstr 21367 | . . . 4 ⊢ ℂfld Struct 〈1, ;13〉 | |
| 18 | 17 | a1i 11 | . . 3 ⊢ (〈(Base‘ndx), ℂ〉 ∈ ℂfld → ℂfld Struct 〈1, ;13〉) | 
| 19 | cffldtocusgrOLD.g | . . 3 ⊢ 𝐺 = (ℂfld sSet 〈(.ef‘ndx), ( I ↾ 𝑃)〉) | |
| 20 | fvex 6918 | . . . 4 ⊢ (Base‘ndx) ∈ V | |
| 21 | cnex 11237 | . . . 4 ⊢ ℂ ∈ V | |
| 22 | 20, 21 | opeldm 5917 | . . 3 ⊢ (〈(Base‘ndx), ℂ〉 ∈ ℂfld → (Base‘ndx) ∈ dom ℂfld) | 
| 23 | 16, 18, 19, 22 | structtocusgr 29464 | . 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 847 = wceq 1539 ∈ wcel 2107 {crab 3435 ∪ cun 3948 𝒫 cpw 4599 {csn 4625 {ctp 4629 〈cop 4631 class class class wbr 5142 I cid 5576 ↾ cres 5686 ∘ ccom 5688 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 1c1 11157 + caddc 11159 · cmul 11161 ≤ cle 11297 − cmin 11493 2c2 12322 3c3 12323 ;cdc 12735 ♯chash 14370 ∗ccj 15136 abscabs 15274 Struct cstr 17184 sSet csts 17201 ndxcnx 17231 Basecbs 17248 +gcplusg 17298 .rcmulr 17299 *𝑟cstv 17300 TopSetcts 17304 lecple 17305 distcds 17307 UnifSetcunif 17308 MetOpencmopn 21355 metUnifcmetu 21356 ℂfldccnfld 21365 .efcedgf 29004 ComplUSGraphccusgr 29428 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-addf 11235 ax-mulf 11236 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-oadd 8511 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-dju 9942 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-xnn0 12602 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-hash 14371 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-cnfld 21366 df-edgf 29005 df-vtx 29016 df-iedg 29017 df-edg 29066 df-usgr 29169 df-nbgr 29351 df-uvtx 29404 df-cplgr 29429 df-cusgr 29430 | 
| This theorem is referenced by: (None) | 
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