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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 21467 | . . . 4 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
2 | 1 | fveq2i 6894 | . . 3 ⊢ (UnifSt‘ℝfld) = (UnifSt‘(ℂfld ↾s ℝ)) |
3 | reex 11207 | . . . 4 ⊢ ℝ ∈ V | |
4 | ressuss 24086 | . . . 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 2731 | . . . . 5 ⊢ (UnifSt‘ℂfld) = (UnifSt‘ℂfld) | |
7 | 6 | cnflduss 25203 | . . . 4 ⊢ (UnifSt‘ℂfld) = (metUnif‘(abs ∘ − )) |
8 | 7 | oveq1i 7422 | . . 3 ⊢ ((UnifSt‘ℂfld) ↾t (ℝ × ℝ)) = ((metUnif‘(abs ∘ − )) ↾t (ℝ × ℝ)) |
9 | 2, 5, 8 | 3eqtri 2763 | . 2 ⊢ (UnifSt‘ℝfld) = ((metUnif‘(abs ∘ − )) ↾t (ℝ × ℝ)) |
10 | 0re 11223 | . . . 4 ⊢ 0 ∈ ℝ | |
11 | 10 | ne0ii 4337 | . . 3 ⊢ ℝ ≠ ∅ |
12 | cnxmet 24608 | . . . 4 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
13 | xmetpsmet 24173 | . . . 4 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (abs ∘ − ) ∈ (PsMet‘ℂ)) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ (abs ∘ − ) ∈ (PsMet‘ℂ) |
15 | ax-resscn 11173 | . . 3 ⊢ ℝ ⊆ ℂ | |
16 | restmetu 24398 | . . 3 ⊢ ((ℝ ≠ ∅ ∧ (abs ∘ − ) ∈ (PsMet‘ℂ) ∧ ℝ ⊆ ℂ) → ((metUnif‘(abs ∘ − )) ↾t (ℝ × ℝ)) = (metUnif‘((abs ∘ − ) ↾ (ℝ × ℝ)))) | |
17 | 11, 14, 15, 16 | mp3an 1460 | . 2 ⊢ ((metUnif‘(abs ∘ − )) ↾t (ℝ × ℝ)) = (metUnif‘((abs ∘ − ) ↾ (ℝ × ℝ))) |
18 | reds 21478 | . . . 4 ⊢ (abs ∘ − ) = (dist‘ℝfld) | |
19 | 18 | reseq1i 5977 | . . 3 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((dist‘ℝfld) ↾ (ℝ × ℝ)) |
20 | 19 | fveq2i 6894 | . 2 ⊢ (metUnif‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (metUnif‘((dist‘ℝfld) ↾ (ℝ × ℝ))) |
21 | 9, 17, 20 | 3eqtri 2763 | 1 ⊢ (UnifSt‘ℝfld) = (metUnif‘((dist‘ℝfld) ↾ (ℝ × ℝ))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 ≠ wne 2939 Vcvv 3473 ⊆ wss 3948 ∅c0 4322 × cxp 5674 ↾ cres 5678 ∘ ccom 5680 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 ℝcr 11115 0cc0 11116 − cmin 11451 abscabs 15188 ↾s cress 17180 distcds 17213 ↾t crest 17373 PsMetcpsmet 21216 ∞Metcxmet 21217 metUnifcmetu 21223 ℂfldccnfld 21232 ℝfldcrefld 21466 UnifStcuss 24077 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ico 13337 df-fz 13492 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-rest 17375 df-psmet 21224 df-xmet 21225 df-met 21226 df-fbas 21229 df-fg 21230 df-metu 21231 df-cnfld 21233 df-refld 21467 df-fil 23669 df-ust 24024 df-uss 24080 |
This theorem is referenced by: recusp 25229 rerrext 33452 |
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