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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 20419 | . . . 4 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
2 | 1 | fveq2i 6533 | . . 3 ⊢ (UnifSt‘ℝfld) = (UnifSt‘(ℂfld ↾s ℝ)) |
3 | reex 10463 | . . . 4 ⊢ ℝ ∈ V | |
4 | ressuss 22543 | . . . 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 2793 | . . . . 5 ⊢ (UnifSt‘ℂfld) = (UnifSt‘ℂfld) | |
7 | 6 | cnflduss 23630 | . . . 4 ⊢ (UnifSt‘ℂfld) = (metUnif‘(abs ∘ − )) |
8 | 7 | oveq1i 7017 | . . 3 ⊢ ((UnifSt‘ℂfld) ↾t (ℝ × ℝ)) = ((metUnif‘(abs ∘ − )) ↾t (ℝ × ℝ)) |
9 | 2, 5, 8 | 3eqtri 2821 | . 2 ⊢ (UnifSt‘ℝfld) = ((metUnif‘(abs ∘ − )) ↾t (ℝ × ℝ)) |
10 | 0re 10478 | . . . 4 ⊢ 0 ∈ ℝ | |
11 | 10 | ne0ii 4217 | . . 3 ⊢ ℝ ≠ ∅ |
12 | cnxmet 23052 | . . . 4 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
13 | xmetpsmet 22629 | . . . 4 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (abs ∘ − ) ∈ (PsMet‘ℂ)) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ (abs ∘ − ) ∈ (PsMet‘ℂ) |
15 | ax-resscn 10429 | . . 3 ⊢ ℝ ⊆ ℂ | |
16 | restmetu 22851 | . . 3 ⊢ ((ℝ ≠ ∅ ∧ (abs ∘ − ) ∈ (PsMet‘ℂ) ∧ ℝ ⊆ ℂ) → ((metUnif‘(abs ∘ − )) ↾t (ℝ × ℝ)) = (metUnif‘((abs ∘ − ) ↾ (ℝ × ℝ)))) | |
17 | 11, 14, 15, 16 | mp3an 1451 | . 2 ⊢ ((metUnif‘(abs ∘ − )) ↾t (ℝ × ℝ)) = (metUnif‘((abs ∘ − ) ↾ (ℝ × ℝ))) |
18 | reds 20430 | . . . 4 ⊢ (abs ∘ − ) = (dist‘ℝfld) | |
19 | 18 | reseq1i 5722 | . . 3 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((dist‘ℝfld) ↾ (ℝ × ℝ)) |
20 | 19 | fveq2i 6533 | . 2 ⊢ (metUnif‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (metUnif‘((dist‘ℝfld) ↾ (ℝ × ℝ))) |
21 | 9, 17, 20 | 3eqtri 2821 | 1 ⊢ (UnifSt‘ℝfld) = (metUnif‘((dist‘ℝfld) ↾ (ℝ × ℝ))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1520 ∈ wcel 2079 ≠ wne 2982 Vcvv 3432 ⊆ wss 3854 ∅c0 4206 × cxp 5433 ↾ cres 5437 ∘ ccom 5439 ‘cfv 6217 (class class class)co 7007 ℂcc 10370 ℝcr 10371 0cc0 10372 − cmin 10706 abscabs 14415 ↾s cress 16301 distcds 16391 ↾t crest 16511 PsMetcpsmet 20199 ∞Metcxmet 20200 metUnifcmetu 20206 ℂfldccnfld 20215 ℝfldcrefld 20418 UnifStcuss 22533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-oadd 7948 df-er 8130 df-map 8249 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-sup 8742 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-z 11819 df-dec 11937 df-uz 12083 df-rp 12229 df-xneg 12346 df-xadd 12347 df-xmul 12348 df-ico 12583 df-fz 12732 df-seq 13208 df-exp 13268 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-starv 16397 df-tset 16401 df-ple 16402 df-ds 16404 df-unif 16405 df-rest 16513 df-psmet 20207 df-xmet 20208 df-met 20209 df-fbas 20212 df-fg 20213 df-metu 20214 df-cnfld 20216 df-refld 20419 df-fil 22126 df-ust 22480 df-uss 22536 |
This theorem is referenced by: recusp 23656 rerrext 30823 |
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