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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrhcn | Structured version Visualization version GIF version |
Description: If the topology of 𝑅 is Hausdorff, and 𝑅 is a complete uniform space, then the canonical homomorphism from the real numbers to 𝑅 is continuous. (Contributed by Thierry Arnoux, 17-Jan-2018.) |
Ref | Expression |
---|---|
rrhf.d | ⊢ 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵)) |
rrhf.j | ⊢ 𝐽 = (topGen‘ran (,)) |
rrhf.b | ⊢ 𝐵 = (Base‘𝑅) |
rrhf.k | ⊢ 𝐾 = (TopOpen‘𝑅) |
rrhf.z | ⊢ 𝑍 = (ℤMod‘𝑅) |
rrhf.1 | ⊢ (𝜑 → 𝑅 ∈ DivRing) |
rrhf.2 | ⊢ (𝜑 → 𝑅 ∈ NrmRing) |
rrhf.3 | ⊢ (𝜑 → 𝑍 ∈ NrmMod) |
rrhf.4 | ⊢ (𝜑 → (chr‘𝑅) = 0) |
rrhf.5 | ⊢ (𝜑 → 𝑅 ∈ CUnifSp) |
rrhf.6 | ⊢ (𝜑 → (UnifSt‘𝑅) = (metUnif‘𝐷)) |
Ref | Expression |
---|---|
rrhcn | ⊢ (𝜑 → (ℝHom‘𝑅) ∈ (𝐽 Cn 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrhf.2 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ NrmRing) | |
2 | nrgngp 24667 | . . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp) | |
3 | ngpxms 24598 | . . . . 5 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp) | |
4 | 1, 2, 3 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ ∞MetSp) |
5 | xmstps 24447 | . . . 4 ⊢ (𝑅 ∈ ∞MetSp → 𝑅 ∈ TopSp) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ TopSp) |
7 | rrhf.j | . . . 4 ⊢ 𝐽 = (topGen‘ran (,)) | |
8 | rrhf.k | . . . 4 ⊢ 𝐾 = (TopOpen‘𝑅) | |
9 | 7, 8 | rrhval 33824 | . . 3 ⊢ (𝑅 ∈ TopSp → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅))) |
10 | 6, 9 | syl 17 | . 2 ⊢ (𝜑 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅))) |
11 | rebase 21598 | . . 3 ⊢ ℝ = (Base‘ℝfld) | |
12 | rrhf.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
13 | retopn 25395 | . . . 4 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝfld) | |
14 | 7, 13 | eqtri 2754 | . . 3 ⊢ 𝐽 = (TopOpen‘ℝfld) |
15 | eqid 2726 | . . 3 ⊢ (UnifSt‘ℝfld) = (UnifSt‘ℝfld) | |
16 | df-refld 21597 | . . . . . 6 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
17 | 16 | oveq1i 7426 | . . . . 5 ⊢ (ℝfld ↾s ℚ) = ((ℂfld ↾s ℝ) ↾s ℚ) |
18 | reex 11240 | . . . . . 6 ⊢ ℝ ∈ V | |
19 | qssre 12989 | . . . . . 6 ⊢ ℚ ⊆ ℝ | |
20 | ressabs 17258 | . . . . . 6 ⊢ ((ℝ ∈ V ∧ ℚ ⊆ ℝ) → ((ℂfld ↾s ℝ) ↾s ℚ) = (ℂfld ↾s ℚ)) | |
21 | 18, 19, 20 | mp2an 690 | . . . . 5 ⊢ ((ℂfld ↾s ℝ) ↾s ℚ) = (ℂfld ↾s ℚ) |
22 | 17, 21 | eqtr2i 2755 | . . . 4 ⊢ (ℂfld ↾s ℚ) = (ℝfld ↾s ℚ) |
23 | 22 | fveq2i 6896 | . . 3 ⊢ (UnifSt‘(ℂfld ↾s ℚ)) = (UnifSt‘(ℝfld ↾s ℚ)) |
24 | eqid 2726 | . . 3 ⊢ (UnifSt‘𝑅) = (UnifSt‘𝑅) | |
25 | recms 25396 | . . . . 5 ⊢ ℝfld ∈ CMetSp | |
26 | cmsms 25364 | . . . . 5 ⊢ (ℝfld ∈ CMetSp → ℝfld ∈ MetSp) | |
27 | mstps 24449 | . . . . 5 ⊢ (ℝfld ∈ MetSp → ℝfld ∈ TopSp) | |
28 | 25, 26, 27 | mp2b 10 | . . . 4 ⊢ ℝfld ∈ TopSp |
29 | 28 | a1i 11 | . . 3 ⊢ (𝜑 → ℝfld ∈ TopSp) |
30 | recusp 25398 | . . . 4 ⊢ ℝfld ∈ CUnifSp | |
31 | cuspusp 24293 | . . . 4 ⊢ (ℝfld ∈ CUnifSp → ℝfld ∈ UnifSp) | |
32 | 30, 31 | mp1i 13 | . . 3 ⊢ (𝜑 → ℝfld ∈ UnifSp) |
33 | rrhf.5 | . . 3 ⊢ (𝜑 → 𝑅 ∈ CUnifSp) | |
34 | rrhf.d | . . . . . 6 ⊢ 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵)) | |
35 | 8, 12, 34 | xmstopn 24445 | . . . . 5 ⊢ (𝑅 ∈ ∞MetSp → 𝐾 = (MetOpen‘𝐷)) |
36 | 4, 35 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 = (MetOpen‘𝐷)) |
37 | 12, 34 | xmsxmet 24450 | . . . . 5 ⊢ (𝑅 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝐵)) |
38 | eqid 2726 | . . . . . 6 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
39 | 38 | methaus 24517 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ Haus) |
40 | 4, 37, 39 | 3syl 18 | . . . 4 ⊢ (𝜑 → (MetOpen‘𝐷) ∈ Haus) |
41 | 36, 40 | eqeltrd 2826 | . . 3 ⊢ (𝜑 → 𝐾 ∈ Haus) |
42 | 19 | a1i 11 | . . 3 ⊢ (𝜑 → ℚ ⊆ ℝ) |
43 | eqid 2726 | . . . . 5 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ) | |
44 | eqid 2726 | . . . . 5 ⊢ (UnifSt‘(ℂfld ↾s ℚ)) = (UnifSt‘(ℂfld ↾s ℚ)) | |
45 | 34 | fveq2i 6896 | . . . . 5 ⊢ (metUnif‘𝐷) = (metUnif‘((dist‘𝑅) ↾ (𝐵 × 𝐵))) |
46 | rrhf.z | . . . . 5 ⊢ 𝑍 = (ℤMod‘𝑅) | |
47 | rrhf.1 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ DivRing) | |
48 | rrhf.3 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ NrmMod) | |
49 | rrhf.4 | . . . . 5 ⊢ (𝜑 → (chr‘𝑅) = 0) | |
50 | 12, 43, 44, 45, 46, 1, 47, 48, 49 | qqhucn 33820 | . . . 4 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ ((UnifSt‘(ℂfld ↾s ℚ)) Cnu(metUnif‘𝐷))) |
51 | rrhf.6 | . . . . . 6 ⊢ (𝜑 → (UnifSt‘𝑅) = (metUnif‘𝐷)) | |
52 | 51 | eqcomd 2732 | . . . . 5 ⊢ (𝜑 → (metUnif‘𝐷) = (UnifSt‘𝑅)) |
53 | 52 | oveq2d 7432 | . . . 4 ⊢ (𝜑 → ((UnifSt‘(ℂfld ↾s ℚ)) Cnu(metUnif‘𝐷)) = ((UnifSt‘(ℂfld ↾s ℚ)) Cnu(UnifSt‘𝑅))) |
54 | 50, 53 | eleqtrd 2828 | . . 3 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ ((UnifSt‘(ℂfld ↾s ℚ)) Cnu(UnifSt‘𝑅))) |
55 | 7 | fveq2i 6896 | . . . . . 6 ⊢ (cls‘𝐽) = (cls‘(topGen‘ran (,))) |
56 | 55 | fveq1i 6894 | . . . . 5 ⊢ ((cls‘𝐽)‘ℚ) = ((cls‘(topGen‘ran (,)))‘ℚ) |
57 | qdensere 24774 | . . . . 5 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ | |
58 | 56, 57 | eqtri 2754 | . . . 4 ⊢ ((cls‘𝐽)‘ℚ) = ℝ |
59 | 58 | a1i 11 | . . 3 ⊢ (𝜑 → ((cls‘𝐽)‘ℚ) = ℝ) |
60 | 11, 12, 14, 8, 15, 23, 24, 29, 32, 6, 33, 41, 42, 54, 59 | ucnextcn 24297 | . 2 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ (𝐽 Cn 𝐾)) |
61 | 10, 60 | eqeltrd 2826 | 1 ⊢ (𝜑 → (ℝHom‘𝑅) ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ⊆ wss 3946 × cxp 5672 ran crn 5675 ↾ cres 5676 ‘cfv 6546 (class class class)co 7416 ℝcr 11148 0cc0 11149 ℚcq 12978 (,)cioo 13372 Basecbs 17208 ↾s cress 17237 distcds 17270 TopOpenctopn 17431 topGenctg 17447 DivRingcdr 20703 ∞Metcxmet 21324 MetOpencmopn 21329 metUnifcmetu 21330 ℂfldccnfld 21339 ℤModczlm 21486 chrcchr 21487 ℝfldcrefld 21596 TopSpctps 22922 clsccl 23010 Cn ccn 23216 Hauscha 23300 CnExtccnext 24051 UnifStcuss 24246 UnifSpcusp 24247 Cnucucn 24268 CUnifSpccusp 24290 ∞MetSpcxms 24311 MetSpcms 24312 NrmGrpcngp 24574 NrmRingcnrg 24576 NrmModcnlm 24577 CMetSpccms 25348 ℚHomcqqh 33800 ℝHomcrrh 33821 |
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 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228 ax-mulf 11229 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-tpos 8233 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-pm 8850 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-fi 9447 df-sup 9478 df-inf 9479 df-oi 9546 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-ioo 13376 df-ico 13378 df-icc 13379 df-fz 13533 df-fzo 13676 df-fl 13806 df-mod 13884 df-seq 14016 df-exp 14076 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-dvds 16252 df-gcd 16490 df-numer 16732 df-denom 16733 df-gz 16927 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-rest 17432 df-topn 17433 df-0g 17451 df-gsum 17452 df-topgen 17453 df-pt 17454 df-prds 17457 df-xrs 17512 df-qtop 17517 df-imas 17518 df-xps 17520 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-submnd 18769 df-grp 18926 df-minusg 18927 df-sbg 18928 df-mulg 19058 df-subg 19113 df-ghm 19203 df-cntz 19307 df-od 19522 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-ring 20214 df-cring 20215 df-oppr 20312 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-rhm 20450 df-nzr 20491 df-subrng 20524 df-subrg 20549 df-drng 20705 df-abv 20784 df-lmod 20834 df-psmet 21331 df-xmet 21332 df-met 21333 df-bl 21334 df-mopn 21335 df-fbas 21336 df-fg 21337 df-metu 21338 df-cnfld 21340 df-zring 21433 df-zrh 21489 df-zlm 21490 df-chr 21491 df-refld 21597 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22937 df-cld 23011 df-ntr 23012 df-cls 23013 df-nei 23090 df-cn 23219 df-cnp 23220 df-haus 23307 df-reg 23308 df-cmp 23379 df-tx 23554 df-hmeo 23747 df-fil 23838 df-fm 23930 df-flim 23931 df-flf 23932 df-fcls 23933 df-cnext 24052 df-ust 24193 df-utop 24224 df-uss 24249 df-usp 24250 df-ucn 24269 df-cfilu 24280 df-cusp 24291 df-xms 24314 df-ms 24315 df-tms 24316 df-nm 24579 df-ngp 24580 df-nrg 24582 df-nlm 24583 df-cncf 24886 df-cfil 25271 df-cmet 25273 df-cms 25351 df-qqh 33801 df-rrh 33823 |
This theorem is referenced by: rrhf 33826 rrhcne 33841 |
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