![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11207 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
3 | 2 | ne0ii 4332 | . . . . . 6 ⊢ ℂ ≠ ∅ |
4 | cnxmet 24639 | . . . . . . 7 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
5 | xmetpsmet 24204 | . . . . . . 7 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (abs ∘ − ) ∈ (PsMet‘ℂ)) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ (abs ∘ − ) ∈ (PsMet‘ℂ) |
7 | metuust 24419 | . . . . . 6 ⊢ ((ℂ ≠ ∅ ∧ (abs ∘ − ) ∈ (PsMet‘ℂ)) → (metUnif‘(abs ∘ − )) ∈ (UnifOn‘ℂ)) | |
8 | 3, 6, 7 | mp2an 689 | . . . . 5 ⊢ (metUnif‘(abs ∘ − )) ∈ (UnifOn‘ℂ) |
9 | ustuni 24081 | . . . . 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 21239 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
13 | cnfldunif 21248 | . . . 4 ⊢ (metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld) | |
14 | 12, 13 | ussid 24115 | . . 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 1533 ∈ wcel 2098 ≠ wne 2934 ∅c0 4317 ∪ cuni 4902 × cxp 5667 ∘ ccom 5673 ‘cfv 6536 ℂcc 11107 0cc0 11109 − cmin 11445 abscabs 15184 PsMetcpsmet 21219 ∞Metcxmet 21220 metUnifcmetu 21226 ℂfldccnfld 21235 UnifOncust 24054 UnifStcuss 24108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ico 13333 df-fz 13488 df-seq 13970 df-exp 14030 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-mulr 17217 df-starv 17218 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-rest 17374 df-psmet 21227 df-xmet 21228 df-met 21229 df-fbas 21232 df-fg 21233 df-metu 21234 df-cnfld 21236 df-fil 23700 df-ust 24055 df-uss 24111 |
This theorem is referenced by: cnfldcusp 25235 reust 25259 qqhucn 33501 cnrrext 33519 |
Copyright terms: Public domain | W3C validator |