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 23265 | . . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp) | |
3 | ngpxms 23204 | . . . . 5 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp) | |
4 | 1, 2, 3 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ ∞MetSp) |
5 | xmstps 23057 | . . . 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 31232 | . . 3 ⊢ (𝑅 ∈ TopSp → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅))) |
10 | 6, 9 | syl 17 | . 2 ⊢ (𝜑 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅))) |
11 | rebase 20744 | . . 3 ⊢ ℝ = (Base‘ℝfld) | |
12 | rrhf.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
13 | retopn 23976 | . . . 4 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝfld) | |
14 | 7, 13 | eqtri 2844 | . . 3 ⊢ 𝐽 = (TopOpen‘ℝfld) |
15 | eqid 2821 | . . 3 ⊢ (UnifSt‘ℝfld) = (UnifSt‘ℝfld) | |
16 | df-refld 20743 | . . . . . 6 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
17 | 16 | oveq1i 7160 | . . . . 5 ⊢ (ℝfld ↾s ℚ) = ((ℂfld ↾s ℝ) ↾s ℚ) |
18 | reex 10622 | . . . . . 6 ⊢ ℝ ∈ V | |
19 | qssre 12352 | . . . . . 6 ⊢ ℚ ⊆ ℝ | |
20 | ressabs 16557 | . . . . . 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 2845 | . . . 4 ⊢ (ℂfld ↾s ℚ) = (ℝfld ↾s ℚ) |
23 | 22 | fveq2i 6668 | . . 3 ⊢ (UnifSt‘(ℂfld ↾s ℚ)) = (UnifSt‘(ℝfld ↾s ℚ)) |
24 | eqid 2821 | . . 3 ⊢ (UnifSt‘𝑅) = (UnifSt‘𝑅) | |
25 | recms 23977 | . . . . 5 ⊢ ℝfld ∈ CMetSp | |
26 | cmsms 23945 | . . . . 5 ⊢ (ℝfld ∈ CMetSp → ℝfld ∈ MetSp) | |
27 | mstps 23059 | . . . . 5 ⊢ (ℝfld ∈ MetSp → ℝfld ∈ TopSp) | |
28 | 25, 26, 27 | mp2b 10 | . . . 4 ⊢ ℝfld ∈ TopSp |
29 | 28 | a1i 11 | . . 3 ⊢ (𝜑 → ℝfld ∈ TopSp) |
30 | recusp 23979 | . . . 4 ⊢ ℝfld ∈ CUnifSp | |
31 | cuspusp 22903 | . . . 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 23055 | . . . . 5 ⊢ (𝑅 ∈ ∞MetSp → 𝐾 = (MetOpen‘𝐷)) |
36 | 4, 35 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 = (MetOpen‘𝐷)) |
37 | 12, 34 | xmsxmet 23060 | . . . . 5 ⊢ (𝑅 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝐵)) |
38 | eqid 2821 | . . . . . 6 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
39 | 38 | methaus 23124 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ Haus) |
40 | 4, 37, 39 | 3syl 18 | . . . 4 ⊢ (𝜑 → (MetOpen‘𝐷) ∈ Haus) |
41 | 36, 40 | eqeltrd 2913 | . . 3 ⊢ (𝜑 → 𝐾 ∈ Haus) |
42 | 19 | a1i 11 | . . 3 ⊢ (𝜑 → ℚ ⊆ ℝ) |
43 | eqid 2821 | . . . . 5 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ) | |
44 | eqid 2821 | . . . . 5 ⊢ (UnifSt‘(ℂfld ↾s ℚ)) = (UnifSt‘(ℂfld ↾s ℚ)) | |
45 | 34 | fveq2i 6668 | . . . . 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 31228 | . . . 4 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ ((UnifSt‘(ℂfld ↾s ℚ)) Cnu(metUnif‘𝐷))) |
51 | rrhf.6 | . . . . . 6 ⊢ (𝜑 → (UnifSt‘𝑅) = (metUnif‘𝐷)) | |
52 | 51 | eqcomd 2827 | . . . . 5 ⊢ (𝜑 → (metUnif‘𝐷) = (UnifSt‘𝑅)) |
53 | 52 | oveq2d 7166 | . . . 4 ⊢ (𝜑 → ((UnifSt‘(ℂfld ↾s ℚ)) Cnu(metUnif‘𝐷)) = ((UnifSt‘(ℂfld ↾s ℚ)) Cnu(UnifSt‘𝑅))) |
54 | 50, 53 | eleqtrd 2915 | . . 3 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ ((UnifSt‘(ℂfld ↾s ℚ)) Cnu(UnifSt‘𝑅))) |
55 | 7 | fveq2i 6668 | . . . . . 6 ⊢ (cls‘𝐽) = (cls‘(topGen‘ran (,))) |
56 | 55 | fveq1i 6666 | . . . . 5 ⊢ ((cls‘𝐽)‘ℚ) = ((cls‘(topGen‘ran (,)))‘ℚ) |
57 | qdensere 23372 | . . . . 5 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ | |
58 | 56, 57 | eqtri 2844 | . . . 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 22907 | . 2 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ (𝐽 Cn 𝐾)) |
61 | 10, 60 | eqeltrd 2913 | 1 ⊢ (𝜑 → (ℝHom‘𝑅) ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ⊆ wss 3936 × cxp 5548 ran crn 5551 ↾ cres 5552 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 0cc0 10531 ℚcq 12342 (,)cioo 12732 Basecbs 16477 ↾s cress 16478 distcds 16568 TopOpenctopn 16689 topGenctg 16705 DivRingcdr 19496 ∞Metcxmet 20524 MetOpencmopn 20529 metUnifcmetu 20530 ℂfldccnfld 20539 ℤModczlm 20642 chrcchr 20643 ℝfldcrefld 20742 TopSpctps 21534 clsccl 21620 Cn ccn 21826 Hauscha 21910 CnExtccnext 22661 UnifStcuss 22856 UnifSpcusp 22857 Cnucucn 22878 CUnifSpccusp 22900 ∞MetSpcxms 22921 MetSpcms 22922 NrmGrpcngp 23181 NrmRingcnrg 23183 NrmModcnlm 23184 CMetSpccms 23929 ℚHomcqqh 31208 ℝHomcrrh 31229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-dvds 15602 df-gcd 15838 df-numer 16069 df-denom 16070 df-gz 16260 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-cntz 18441 df-od 18650 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-rnghom 19461 df-drng 19498 df-subrg 19527 df-abv 19582 df-lmod 19630 df-nzr 20025 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-metu 20538 df-cnfld 20540 df-zring 20612 df-zrh 20645 df-zlm 20646 df-chr 20647 df-refld 20743 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-cn 21829 df-cnp 21830 df-haus 21917 df-reg 21918 df-cmp 21989 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-fcls 22543 df-cnext 22662 df-ust 22803 df-utop 22834 df-uss 22859 df-usp 22860 df-ucn 22879 df-cfilu 22890 df-cusp 22901 df-xms 22924 df-ms 22925 df-tms 22926 df-nm 23186 df-ngp 23187 df-nrg 23189 df-nlm 23190 df-cncf 23480 df-cfil 23852 df-cmet 23854 df-cms 23932 df-qqh 31209 df-rrh 31231 |
This theorem is referenced by: rrhf 31234 rrhcne 31249 |
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