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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 11148 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
3 | 2 | ne0ii 4298 | . . . . . 6 ⊢ ℂ ≠ ∅ |
4 | cnxmet 24139 | . . . . . . 7 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
5 | xmetpsmet 23704 | . . . . . . 7 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (abs ∘ − ) ∈ (PsMet‘ℂ)) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ (abs ∘ − ) ∈ (PsMet‘ℂ) |
7 | metuust 23919 | . . . . . 6 ⊢ ((ℂ ≠ ∅ ∧ (abs ∘ − ) ∈ (PsMet‘ℂ)) → (metUnif‘(abs ∘ − )) ∈ (UnifOn‘ℂ)) | |
8 | 3, 6, 7 | mp2an 691 | . . . . 5 ⊢ (metUnif‘(abs ∘ − )) ∈ (UnifOn‘ℂ) |
9 | ustuni 23581 | . . . . 5 ⊢ ((metUnif‘(abs ∘ − )) ∈ (UnifOn‘ℂ) → ∪ (metUnif‘(abs ∘ − )) = (ℂ × ℂ)) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ ∪ (metUnif‘(abs ∘ − )) = (ℂ × ℂ) |
11 | 10 | eqcomi 2746 | . . 3 ⊢ (ℂ × ℂ) = ∪ (metUnif‘(abs ∘ − )) |
12 | cnfldbas 20803 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
13 | cnfldunif 20810 | . . . 4 ⊢ (metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld) | |
14 | 12, 13 | ussid 23615 | . . 3 ⊢ ((ℂ × ℂ) = ∪ (metUnif‘(abs ∘ − )) → (metUnif‘(abs ∘ − )) = (UnifSt‘ℂfld)) |
15 | 11, 14 | ax-mp 5 | . 2 ⊢ (metUnif‘(abs ∘ − )) = (UnifSt‘ℂfld) |
16 | 1, 15 | eqtr4i 2768 | 1 ⊢ 𝑈 = (metUnif‘(abs ∘ − )) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∅c0 4283 ∪ cuni 4866 × cxp 5632 ∘ ccom 5638 ‘cfv 6497 ℂcc 11050 0cc0 11052 − cmin 11386 abscabs 15120 PsMetcpsmet 20783 ∞Metcxmet 20784 metUnifcmetu 20790 ℂfldccnfld 20799 UnifOncust 23554 UnifStcuss 23608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9379 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-dec 12620 df-uz 12765 df-rp 12917 df-xneg 13034 df-xadd 13035 df-xmul 13036 df-ico 13271 df-fz 13426 df-seq 13908 df-exp 13969 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-struct 17020 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-mulr 17148 df-starv 17149 df-tset 17153 df-ple 17154 df-ds 17156 df-unif 17157 df-rest 17305 df-psmet 20791 df-xmet 20792 df-met 20793 df-fbas 20796 df-fg 20797 df-metu 20798 df-cnfld 20800 df-fil 23200 df-ust 23555 df-uss 23611 |
This theorem is referenced by: cnfldcusp 24724 reust 24748 qqhucn 32576 cnrrext 32594 |
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