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Mirrors > Home > MPE Home > Th. List > cnflduss | Structured version Visualization version GIF version |
Description: The uniform structure of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
Ref | Expression |
---|---|
cnflduss.1 | ⊢ 𝑈 = (UnifSt‘ℂfld) |
Ref | Expression |
---|---|
cnflduss | ⊢ 𝑈 = (metUnif‘(abs ∘ − )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnflduss.1 | . 2 ⊢ 𝑈 = (UnifSt‘ℂfld) | |
2 | 0cn 11245 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
3 | 2 | ne0ii 4338 | . . . . . 6 ⊢ ℂ ≠ ∅ |
4 | cnxmet 24775 | . . . . . . 7 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
5 | xmetpsmet 24340 | . . . . . . 7 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (abs ∘ − ) ∈ (PsMet‘ℂ)) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ (abs ∘ − ) ∈ (PsMet‘ℂ) |
7 | metuust 24555 | . . . . . 6 ⊢ ((ℂ ≠ ∅ ∧ (abs ∘ − ) ∈ (PsMet‘ℂ)) → (metUnif‘(abs ∘ − )) ∈ (UnifOn‘ℂ)) | |
8 | 3, 6, 7 | mp2an 690 | . . . . 5 ⊢ (metUnif‘(abs ∘ − )) ∈ (UnifOn‘ℂ) |
9 | ustuni 24217 | . . . . 5 ⊢ ((metUnif‘(abs ∘ − )) ∈ (UnifOn‘ℂ) → ∪ (metUnif‘(abs ∘ − )) = (ℂ × ℂ)) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ ∪ (metUnif‘(abs ∘ − )) = (ℂ × ℂ) |
11 | 10 | eqcomi 2735 | . . 3 ⊢ (ℂ × ℂ) = ∪ (metUnif‘(abs ∘ − )) |
12 | cnfldbas 21341 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
13 | cnfldunif 21350 | . . . 4 ⊢ (metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld) | |
14 | 12, 13 | ussid 24251 | . . 3 ⊢ ((ℂ × ℂ) = ∪ (metUnif‘(abs ∘ − )) → (metUnif‘(abs ∘ − )) = (UnifSt‘ℂfld)) |
15 | 11, 14 | ax-mp 5 | . 2 ⊢ (metUnif‘(abs ∘ − )) = (UnifSt‘ℂfld) |
16 | 1, 15 | eqtr4i 2757 | 1 ⊢ 𝑈 = (metUnif‘(abs ∘ − )) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∅c0 4323 ∪ cuni 4906 × cxp 5671 ∘ ccom 5677 ‘cfv 6544 ℂcc 11145 0cc0 11147 − cmin 11483 abscabs 15232 PsMetcpsmet 21321 ∞Metcxmet 21322 metUnifcmetu 21328 ℂfldccnfld 21337 UnifOncust 24190 UnifStcuss 24244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-pre-sup 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9476 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-nn 12257 df-2 12319 df-3 12320 df-4 12321 df-5 12322 df-6 12323 df-7 12324 df-8 12325 df-9 12326 df-n0 12517 df-z 12603 df-dec 12722 df-uz 12867 df-rp 13021 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ico 13376 df-fz 13531 df-seq 14014 df-exp 14074 df-cj 15097 df-re 15098 df-im 15099 df-sqrt 15233 df-abs 15234 df-struct 17142 df-slot 17177 df-ndx 17189 df-base 17207 df-plusg 17272 df-mulr 17273 df-starv 17274 df-tset 17278 df-ple 17279 df-ds 17281 df-unif 17282 df-rest 17430 df-psmet 21329 df-xmet 21330 df-met 21331 df-fbas 21334 df-fg 21335 df-metu 21336 df-cnfld 21338 df-fil 23836 df-ust 24191 df-uss 24247 |
This theorem is referenced by: cnfldcusp 25371 reust 25395 qqhucn 33818 cnrrext 33836 |
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