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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 11147 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
3 | 2 | ne0ii 4297 | . . . . . 6 ⊢ ℂ ≠ ∅ |
4 | cnxmet 24136 | . . . . . . 7 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
5 | xmetpsmet 23701 | . . . . . . 7 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (abs ∘ − ) ∈ (PsMet‘ℂ)) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ (abs ∘ − ) ∈ (PsMet‘ℂ) |
7 | metuust 23916 | . . . . . 6 ⊢ ((ℂ ≠ ∅ ∧ (abs ∘ − ) ∈ (PsMet‘ℂ)) → (metUnif‘(abs ∘ − )) ∈ (UnifOn‘ℂ)) | |
8 | 3, 6, 7 | mp2an 690 | . . . . 5 ⊢ (metUnif‘(abs ∘ − )) ∈ (UnifOn‘ℂ) |
9 | ustuni 23578 | . . . . 5 ⊢ ((metUnif‘(abs ∘ − )) ∈ (UnifOn‘ℂ) → ∪ (metUnif‘(abs ∘ − )) = (ℂ × ℂ)) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ ∪ (metUnif‘(abs ∘ − )) = (ℂ × ℂ) |
11 | 10 | eqcomi 2745 | . . 3 ⊢ (ℂ × ℂ) = ∪ (metUnif‘(abs ∘ − )) |
12 | cnfldbas 20800 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
13 | cnfldunif 20807 | . . . 4 ⊢ (metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld) | |
14 | 12, 13 | ussid 23612 | . . 3 ⊢ ((ℂ × ℂ) = ∪ (metUnif‘(abs ∘ − )) → (metUnif‘(abs ∘ − )) = (UnifSt‘ℂfld)) |
15 | 11, 14 | ax-mp 5 | . 2 ⊢ (metUnif‘(abs ∘ − )) = (UnifSt‘ℂfld) |
16 | 1, 15 | eqtr4i 2767 | 1 ⊢ 𝑈 = (metUnif‘(abs ∘ − )) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∅c0 4282 ∪ cuni 4865 × cxp 5631 ∘ ccom 5637 ‘cfv 6496 ℂcc 11049 0cc0 11051 − cmin 11385 abscabs 15119 PsMetcpsmet 20780 ∞Metcxmet 20781 metUnifcmetu 20787 ℂfldccnfld 20796 UnifOncust 23551 UnifStcuss 23605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ico 13270 df-fz 13425 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-struct 17019 df-slot 17054 df-ndx 17066 df-base 17084 df-plusg 17146 df-mulr 17147 df-starv 17148 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-rest 17304 df-psmet 20788 df-xmet 20789 df-met 20790 df-fbas 20793 df-fg 20794 df-metu 20795 df-cnfld 20797 df-fil 23197 df-ust 23552 df-uss 23608 |
This theorem is referenced by: cnfldcusp 24721 reust 24745 qqhucn 32573 cnrrext 32591 |
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