![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 22874 | . . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp) | |
3 | ngpxms 22813 | . . . . 5 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp) | |
4 | 1, 2, 3 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ ∞MetSp) |
5 | xmstps 22666 | . . . 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 30638 | . . 3 ⊢ (𝑅 ∈ TopSp → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅))) |
10 | 6, 9 | syl 17 | . 2 ⊢ (𝜑 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅))) |
11 | rebase 20349 | . . 3 ⊢ ℝ = (Base‘ℝfld) | |
12 | rrhf.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
13 | retopn 23585 | . . . 4 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝfld) | |
14 | 7, 13 | eqtri 2801 | . . 3 ⊢ 𝐽 = (TopOpen‘ℝfld) |
15 | eqid 2777 | . . 3 ⊢ (UnifSt‘ℝfld) = (UnifSt‘ℝfld) | |
16 | df-refld 20348 | . . . . . 6 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
17 | 16 | oveq1i 6932 | . . . . 5 ⊢ (ℝfld ↾s ℚ) = ((ℂfld ↾s ℝ) ↾s ℚ) |
18 | reex 10363 | . . . . . 6 ⊢ ℝ ∈ V | |
19 | qssre 12106 | . . . . . 6 ⊢ ℚ ⊆ ℝ | |
20 | ressabs 16336 | . . . . . 6 ⊢ ((ℝ ∈ V ∧ ℚ ⊆ ℝ) → ((ℂfld ↾s ℝ) ↾s ℚ) = (ℂfld ↾s ℚ)) | |
21 | 18, 19, 20 | mp2an 682 | . . . . 5 ⊢ ((ℂfld ↾s ℝ) ↾s ℚ) = (ℂfld ↾s ℚ) |
22 | 17, 21 | eqtr2i 2802 | . . . 4 ⊢ (ℂfld ↾s ℚ) = (ℝfld ↾s ℚ) |
23 | 22 | fveq2i 6449 | . . 3 ⊢ (UnifSt‘(ℂfld ↾s ℚ)) = (UnifSt‘(ℝfld ↾s ℚ)) |
24 | eqid 2777 | . . 3 ⊢ (UnifSt‘𝑅) = (UnifSt‘𝑅) | |
25 | recms 23586 | . . . . 5 ⊢ ℝfld ∈ CMetSp | |
26 | cmsms 23554 | . . . . 5 ⊢ (ℝfld ∈ CMetSp → ℝfld ∈ MetSp) | |
27 | mstps 22668 | . . . . 5 ⊢ (ℝfld ∈ MetSp → ℝfld ∈ TopSp) | |
28 | 25, 26, 27 | mp2b 10 | . . . 4 ⊢ ℝfld ∈ TopSp |
29 | 28 | a1i 11 | . . 3 ⊢ (𝜑 → ℝfld ∈ TopSp) |
30 | recusp 23588 | . . . 4 ⊢ ℝfld ∈ CUnifSp | |
31 | cuspusp 22512 | . . . 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 22664 | . . . . 5 ⊢ (𝑅 ∈ ∞MetSp → 𝐾 = (MetOpen‘𝐷)) |
36 | 4, 35 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 = (MetOpen‘𝐷)) |
37 | 12, 34 | xmsxmet 22669 | . . . . 5 ⊢ (𝑅 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝐵)) |
38 | eqid 2777 | . . . . . 6 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
39 | 38 | methaus 22733 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ Haus) |
40 | 4, 37, 39 | 3syl 18 | . . . 4 ⊢ (𝜑 → (MetOpen‘𝐷) ∈ Haus) |
41 | 36, 40 | eqeltrd 2858 | . . 3 ⊢ (𝜑 → 𝐾 ∈ Haus) |
42 | 19 | a1i 11 | . . 3 ⊢ (𝜑 → ℚ ⊆ ℝ) |
43 | eqid 2777 | . . . . 5 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ) | |
44 | eqid 2777 | . . . . 5 ⊢ (UnifSt‘(ℂfld ↾s ℚ)) = (UnifSt‘(ℂfld ↾s ℚ)) | |
45 | 34 | fveq2i 6449 | . . . . 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 30634 | . . . 4 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ ((UnifSt‘(ℂfld ↾s ℚ)) Cnu(metUnif‘𝐷))) |
51 | rrhf.6 | . . . . . 6 ⊢ (𝜑 → (UnifSt‘𝑅) = (metUnif‘𝐷)) | |
52 | 51 | eqcomd 2783 | . . . . 5 ⊢ (𝜑 → (metUnif‘𝐷) = (UnifSt‘𝑅)) |
53 | 52 | oveq2d 6938 | . . . 4 ⊢ (𝜑 → ((UnifSt‘(ℂfld ↾s ℚ)) Cnu(metUnif‘𝐷)) = ((UnifSt‘(ℂfld ↾s ℚ)) Cnu(UnifSt‘𝑅))) |
54 | 50, 53 | eleqtrd 2860 | . . 3 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ ((UnifSt‘(ℂfld ↾s ℚ)) Cnu(UnifSt‘𝑅))) |
55 | 7 | fveq2i 6449 | . . . . . 6 ⊢ (cls‘𝐽) = (cls‘(topGen‘ran (,))) |
56 | 55 | fveq1i 6447 | . . . . 5 ⊢ ((cls‘𝐽)‘ℚ) = ((cls‘(topGen‘ran (,)))‘ℚ) |
57 | qdensere 22981 | . . . . 5 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ | |
58 | 56, 57 | eqtri 2801 | . . . 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 22516 | . 2 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ (𝐽 Cn 𝐾)) |
61 | 10, 60 | eqeltrd 2858 | 1 ⊢ (𝜑 → (ℝHom‘𝑅) ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 Vcvv 3397 ⊆ wss 3791 × cxp 5353 ran crn 5356 ↾ cres 5357 ‘cfv 6135 (class class class)co 6922 ℝcr 10271 0cc0 10272 ℚcq 12095 (,)cioo 12487 Basecbs 16255 ↾s cress 16256 distcds 16347 TopOpenctopn 16468 topGenctg 16484 DivRingcdr 19139 ∞Metcxmet 20127 MetOpencmopn 20132 metUnifcmetu 20133 ℂfldccnfld 20142 ℤModczlm 20245 chrcchr 20246 ℝfldcrefld 20347 TopSpctps 21144 clsccl 21230 Cn ccn 21436 Hauscha 21520 CnExtccnext 22271 UnifStcuss 22465 UnifSpcusp 22466 Cnucucn 22487 CUnifSpccusp 22509 ∞MetSpcxms 22530 MetSpcms 22531 NrmGrpcngp 22790 NrmRingcnrg 22792 NrmModcnlm 22793 CMetSpccms 23538 ℚHomcqqh 30614 ℝHomcrrh 30635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-dvds 15388 df-gcd 15623 df-numer 15847 df-denom 15848 df-gz 16038 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mulg 17928 df-subg 17975 df-ghm 18042 df-cntz 18133 df-od 18332 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-rnghom 19104 df-drng 19141 df-subrg 19170 df-abv 19209 df-lmod 19257 df-nzr 19655 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-metu 20141 df-cnfld 20143 df-zring 20215 df-zrh 20248 df-zlm 20249 df-chr 20250 df-refld 20348 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-cn 21439 df-cnp 21440 df-haus 21527 df-reg 21528 df-cmp 21599 df-tx 21774 df-hmeo 21967 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-fcls 22153 df-cnext 22272 df-ust 22412 df-utop 22443 df-uss 22468 df-usp 22469 df-ucn 22488 df-cfilu 22499 df-cusp 22510 df-xms 22533 df-ms 22534 df-tms 22535 df-nm 22795 df-ngp 22796 df-nrg 22798 df-nlm 22799 df-cncf 23089 df-cfil 23461 df-cmet 23463 df-cms 23541 df-qqh 30615 df-rrh 30637 |
This theorem is referenced by: rrhf 30640 rrhcne 30655 |
Copyright terms: Public domain | W3C validator |