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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 21536 | . . . 4 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
2 | 1 | fveq2i 6900 | . . 3 ⊢ (UnifSt‘ℝfld) = (UnifSt‘(ℂfld ↾s ℝ)) |
3 | reex 11229 | . . . 4 ⊢ ℝ ∈ V | |
4 | ressuss 24166 | . . . 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 2728 | . . . . 5 ⊢ (UnifSt‘ℂfld) = (UnifSt‘ℂfld) | |
7 | 6 | cnflduss 25283 | . . . 4 ⊢ (UnifSt‘ℂfld) = (metUnif‘(abs ∘ − )) |
8 | 7 | oveq1i 7430 | . . 3 ⊢ ((UnifSt‘ℂfld) ↾t (ℝ × ℝ)) = ((metUnif‘(abs ∘ − )) ↾t (ℝ × ℝ)) |
9 | 2, 5, 8 | 3eqtri 2760 | . 2 ⊢ (UnifSt‘ℝfld) = ((metUnif‘(abs ∘ − )) ↾t (ℝ × ℝ)) |
10 | 0re 11246 | . . . 4 ⊢ 0 ∈ ℝ | |
11 | 10 | ne0ii 4338 | . . 3 ⊢ ℝ ≠ ∅ |
12 | cnxmet 24688 | . . . 4 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
13 | xmetpsmet 24253 | . . . 4 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (abs ∘ − ) ∈ (PsMet‘ℂ)) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ (abs ∘ − ) ∈ (PsMet‘ℂ) |
15 | ax-resscn 11195 | . . 3 ⊢ ℝ ⊆ ℂ | |
16 | restmetu 24478 | . . 3 ⊢ ((ℝ ≠ ∅ ∧ (abs ∘ − ) ∈ (PsMet‘ℂ) ∧ ℝ ⊆ ℂ) → ((metUnif‘(abs ∘ − )) ↾t (ℝ × ℝ)) = (metUnif‘((abs ∘ − ) ↾ (ℝ × ℝ)))) | |
17 | 11, 14, 15, 16 | mp3an 1458 | . 2 ⊢ ((metUnif‘(abs ∘ − )) ↾t (ℝ × ℝ)) = (metUnif‘((abs ∘ − ) ↾ (ℝ × ℝ))) |
18 | reds 21547 | . . . 4 ⊢ (abs ∘ − ) = (dist‘ℝfld) | |
19 | 18 | reseq1i 5981 | . . 3 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((dist‘ℝfld) ↾ (ℝ × ℝ)) |
20 | 19 | fveq2i 6900 | . 2 ⊢ (metUnif‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (metUnif‘((dist‘ℝfld) ↾ (ℝ × ℝ))) |
21 | 9, 17, 20 | 3eqtri 2760 | 1 ⊢ (UnifSt‘ℝfld) = (metUnif‘((dist‘ℝfld) ↾ (ℝ × ℝ))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ≠ wne 2937 Vcvv 3471 ⊆ wss 3947 ∅c0 4323 × cxp 5676 ↾ cres 5680 ∘ ccom 5682 ‘cfv 6548 (class class class)co 7420 ℂcc 11136 ℝcr 11137 0cc0 11138 − cmin 11474 abscabs 15213 ↾s cress 17208 distcds 17241 ↾t crest 17401 PsMetcpsmet 21262 ∞Metcxmet 21263 metUnifcmetu 21269 ℂfldccnfld 21278 ℝfldcrefld 21535 UnifStcuss 24157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ico 13362 df-fz 13517 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-rest 17403 df-psmet 21270 df-xmet 21271 df-met 21272 df-fbas 21275 df-fg 21276 df-metu 21277 df-cnfld 21279 df-refld 21536 df-fil 23749 df-ust 24104 df-uss 24160 |
This theorem is referenced by: recusp 25309 rerrext 33610 |
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