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Mirrors > Home > MPE Home > Th. List > reust | Structured version Visualization version GIF version |
Description: The Uniform structure of the real numbers. (Contributed by Thierry Arnoux, 14-Feb-2018.) |
Ref | Expression |
---|---|
reust | ⊢ (UnifSt‘ℝfld) = (metUnif‘((dist‘ℝfld) ↾ (ℝ × ℝ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-refld 21487 | . . . 4 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
2 | 1 | fveq2i 6885 | . . 3 ⊢ (UnifSt‘ℝfld) = (UnifSt‘(ℂfld ↾s ℝ)) |
3 | reex 11198 | . . . 4 ⊢ ℝ ∈ V | |
4 | ressuss 24111 | . . . 4 ⊢ (ℝ ∈ V → (UnifSt‘(ℂfld ↾s ℝ)) = ((UnifSt‘ℂfld) ↾t (ℝ × ℝ))) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (UnifSt‘(ℂfld ↾s ℝ)) = ((UnifSt‘ℂfld) ↾t (ℝ × ℝ)) |
6 | eqid 2724 | . . . . 5 ⊢ (UnifSt‘ℂfld) = (UnifSt‘ℂfld) | |
7 | 6 | cnflduss 25228 | . . . 4 ⊢ (UnifSt‘ℂfld) = (metUnif‘(abs ∘ − )) |
8 | 7 | oveq1i 7412 | . . 3 ⊢ ((UnifSt‘ℂfld) ↾t (ℝ × ℝ)) = ((metUnif‘(abs ∘ − )) ↾t (ℝ × ℝ)) |
9 | 2, 5, 8 | 3eqtri 2756 | . 2 ⊢ (UnifSt‘ℝfld) = ((metUnif‘(abs ∘ − )) ↾t (ℝ × ℝ)) |
10 | 0re 11215 | . . . 4 ⊢ 0 ∈ ℝ | |
11 | 10 | ne0ii 4330 | . . 3 ⊢ ℝ ≠ ∅ |
12 | cnxmet 24633 | . . . 4 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
13 | xmetpsmet 24198 | . . . 4 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (abs ∘ − ) ∈ (PsMet‘ℂ)) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ (abs ∘ − ) ∈ (PsMet‘ℂ) |
15 | ax-resscn 11164 | . . 3 ⊢ ℝ ⊆ ℂ | |
16 | restmetu 24423 | . . 3 ⊢ ((ℝ ≠ ∅ ∧ (abs ∘ − ) ∈ (PsMet‘ℂ) ∧ ℝ ⊆ ℂ) → ((metUnif‘(abs ∘ − )) ↾t (ℝ × ℝ)) = (metUnif‘((abs ∘ − ) ↾ (ℝ × ℝ)))) | |
17 | 11, 14, 15, 16 | mp3an 1457 | . 2 ⊢ ((metUnif‘(abs ∘ − )) ↾t (ℝ × ℝ)) = (metUnif‘((abs ∘ − ) ↾ (ℝ × ℝ))) |
18 | reds 21498 | . . . 4 ⊢ (abs ∘ − ) = (dist‘ℝfld) | |
19 | 18 | reseq1i 5968 | . . 3 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((dist‘ℝfld) ↾ (ℝ × ℝ)) |
20 | 19 | fveq2i 6885 | . 2 ⊢ (metUnif‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (metUnif‘((dist‘ℝfld) ↾ (ℝ × ℝ))) |
21 | 9, 17, 20 | 3eqtri 2756 | 1 ⊢ (UnifSt‘ℝfld) = (metUnif‘((dist‘ℝfld) ↾ (ℝ × ℝ))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ≠ wne 2932 Vcvv 3466 ⊆ wss 3941 ∅c0 4315 × cxp 5665 ↾ cres 5669 ∘ ccom 5671 ‘cfv 6534 (class class class)co 7402 ℂcc 11105 ℝcr 11106 0cc0 11107 − cmin 11443 abscabs 15183 ↾s cress 17178 distcds 17211 ↾t crest 17371 PsMetcpsmet 21218 ∞Metcxmet 21219 metUnifcmetu 21225 ℂfldccnfld 21234 ℝfldcrefld 21486 UnifStcuss 24102 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-ico 13331 df-fz 13486 df-seq 13968 df-exp 14029 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-rest 17373 df-psmet 21226 df-xmet 21227 df-met 21228 df-fbas 21231 df-fg 21232 df-metu 21233 df-cnfld 21235 df-refld 21487 df-fil 23694 df-ust 24049 df-uss 24105 |
This theorem is referenced by: recusp 25254 rerrext 33508 |
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