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Mirrors > Home > MPE Home > Th. List > cffldtocusgrOLD | Structured version Visualization version GIF version |
Description: Obsolete version of cffldtocusgr 29479 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 5475 | . . . . . . 7 ⊢ 〈(Base‘ndx), ℂ〉 ∈ V | |
2 | 1 | tpid1 4773 | . . . . . 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 4163 | . . . . 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 21398 | . . . . 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 2831 | . . . 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 4163 | . . . 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 21386 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
14 | 13 | pweqi 4621 | . . . . 5 ⊢ 𝒫 ℂ = 𝒫 (Base‘ℂfld) |
15 | 14 | rabeqi 3447 | . . . 4 ⊢ {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2} |
16 | 12, 15 | eqtri 2763 | . . 3 ⊢ 𝑃 = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2} |
17 | cnfldstr 21384 | . . . 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 6920 | . . . 4 ⊢ (Base‘ndx) ∈ V | |
21 | cnex 11234 | . . . 4 ⊢ ℂ ∈ V | |
22 | 20, 21 | opeldm 5921 | . . 3 ⊢ (〈(Base‘ndx), ℂ〉 ∈ ℂfld → (Base‘ndx) ∈ dom ℂfld) |
23 | 16, 18, 19, 22 | structtocusgr 29478 | . 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 1537 ∈ wcel 2106 {crab 3433 ∪ cun 3961 𝒫 cpw 4605 {csn 4631 {ctp 4635 〈cop 4637 class class class wbr 5148 I cid 5582 ↾ cres 5691 ∘ ccom 5693 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 1c1 11154 + caddc 11156 · cmul 11158 ≤ cle 11294 − cmin 11490 2c2 12319 3c3 12320 ;cdc 12731 ♯chash 14366 ∗ccj 15132 abscabs 15270 Struct cstr 17180 sSet csts 17197 ndxcnx 17227 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 *𝑟cstv 17300 TopSetcts 17304 lecple 17305 distcds 17307 UnifSetcunif 17308 MetOpencmopn 21372 metUnifcmetu 21373 ℂfldccnfld 21382 .efcedgf 29018 ComplUSGraphccusgr 29442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-cnfld 21383 df-edgf 29019 df-vtx 29030 df-iedg 29031 df-edg 29080 df-usgr 29183 df-nbgr 29365 df-uvtx 29418 df-cplgr 29443 df-cusgr 29444 |
This theorem is referenced by: (None) |
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