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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 10633 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
3 | 2 | ne0ii 4303 | . . . . . 6 ⊢ ℂ ≠ ∅ |
4 | cnxmet 23381 | . . . . . . 7 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
5 | xmetpsmet 22958 | . . . . . . 7 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (abs ∘ − ) ∈ (PsMet‘ℂ)) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ (abs ∘ − ) ∈ (PsMet‘ℂ) |
7 | metuust 23170 | . . . . . 6 ⊢ ((ℂ ≠ ∅ ∧ (abs ∘ − ) ∈ (PsMet‘ℂ)) → (metUnif‘(abs ∘ − )) ∈ (UnifOn‘ℂ)) | |
8 | 3, 6, 7 | mp2an 690 | . . . . 5 ⊢ (metUnif‘(abs ∘ − )) ∈ (UnifOn‘ℂ) |
9 | ustuni 22835 | . . . . 5 ⊢ ((metUnif‘(abs ∘ − )) ∈ (UnifOn‘ℂ) → ∪ (metUnif‘(abs ∘ − )) = (ℂ × ℂ)) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ ∪ (metUnif‘(abs ∘ − )) = (ℂ × ℂ) |
11 | 10 | eqcomi 2830 | . . 3 ⊢ (ℂ × ℂ) = ∪ (metUnif‘(abs ∘ − )) |
12 | cnfldbas 20549 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
13 | cnfldunif 20556 | . . . 4 ⊢ (metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld) | |
14 | 12, 13 | ussid 22869 | . . 3 ⊢ ((ℂ × ℂ) = ∪ (metUnif‘(abs ∘ − )) → (metUnif‘(abs ∘ − )) = (UnifSt‘ℂfld)) |
15 | 11, 14 | ax-mp 5 | . 2 ⊢ (metUnif‘(abs ∘ − )) = (UnifSt‘ℂfld) |
16 | 1, 15 | eqtr4i 2847 | 1 ⊢ 𝑈 = (metUnif‘(abs ∘ − )) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 ∪ cuni 4838 × cxp 5553 ∘ ccom 5559 ‘cfv 6355 ℂcc 10535 0cc0 10537 − cmin 10870 abscabs 14593 PsMetcpsmet 20529 ∞Metcxmet 20530 metUnifcmetu 20536 ℂfldccnfld 20545 UnifOncust 22808 UnifStcuss 22862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ico 12745 df-fz 12894 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-rest 16696 df-psmet 20537 df-xmet 20538 df-met 20539 df-fbas 20542 df-fg 20543 df-metu 20544 df-cnfld 20546 df-fil 22454 df-ust 22809 df-uss 22865 |
This theorem is referenced by: cnfldcusp 23960 reust 23984 qqhucn 31233 cnrrext 31251 |
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