rrhcn - Mathbox for Thierry Arnoux

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Theorem rrhcn 30639

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.)

**Hypotheses**
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))

**Assertion**
Ref	Expression
rrhcn	⊢ (𝜑 → (ℝHom‘𝑅) ∈ (𝐽 Cn 𝐾))

**Proof of Theorem rrhcn**
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‘ℝ_fld)
12		rrhf.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
13		retopn 23585	. . . 4 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝ_fld)
14	7, 13	eqtri 2801	. . 3 ⊢ 𝐽 = (TopOpen‘ℝ_fld)
15		eqid 2777	. . 3 ⊢ (UnifSt‘ℝ_fld) = (UnifSt‘ℝ_fld)
16		df-refld 20348	. . . . . 6 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
17	16	oveq1i 6932	. . . . 5 ⊢ (ℝ_fld ↾_s ℚ) = ((ℂ_fld ↾_s ℝ) ↾_s ℚ)
18		reex 10363	. . . . . 6 ⊢ ℝ ∈ V
19		qssre 12106	. . . . . 6 ⊢ ℚ ⊆ ℝ
20		ressabs 16336	. . . . . 6 ⊢ ((ℝ ∈ V ∧ ℚ ⊆ ℝ) → ((ℂ_fld ↾_s ℝ) ↾_s ℚ) = (ℂ_fld ↾_s ℚ))
21	18, 19, 20	mp2an 682	. . . . 5 ⊢ ((ℂ_fld ↾_s ℝ) ↾_s ℚ) = (ℂ_fld ↾_s ℚ)
22	17, 21	eqtr2i 2802	. . . 4 ⊢ (ℂ_fld ↾_s ℚ) = (ℝ_fld ↾_s ℚ)
23	22	fveq2i 6449	. . 3 ⊢ (UnifSt‘(ℂ_fld ↾_s ℚ)) = (UnifSt‘(ℝ_fld ↾_s ℚ))
24		eqid 2777	. . 3 ⊢ (UnifSt‘𝑅) = (UnifSt‘𝑅)
25		recms 23586	. . . . 5 ⊢ ℝ_fld ∈ CMetSp
26		cmsms 23554	. . . . 5 ⊢ (ℝ_fld ∈ CMetSp → ℝ_fld ∈ MetSp)
27		mstps 22668	. . . . 5 ⊢ (ℝ_fld ∈ MetSp → ℝ_fld ∈ TopSp)
28	25, 26, 27	mp2b 10	. . . 4 ⊢ ℝ_fld ∈ TopSp
29	28	a1i 11	. . 3 ⊢ (𝜑 → ℝ_fld ∈ TopSp)
30		recusp 23588	. . . 4 ⊢ ℝ_fld ∈ CUnifSp
31		cuspusp 22512	. . . 4 ⊢ (ℝ_fld ∈ CUnifSp → ℝ_fld ∈ UnifSp)
32	30, 31	mp1i 13	. . 3 ⊢ (𝜑 → ℝ_fld ∈ 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 ⊢ (ℂ_fld ↾_s ℚ) = (ℂ_fld ↾_s ℚ)
44		eqid 2777	. . . . 5 ⊢ (UnifSt‘(ℂ_fld ↾_s ℚ)) = (UnifSt‘(ℂ_fld ↾_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‘(ℂ_fld ↾_s ℚ)) Cn_u(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‘(ℂ_fld ↾_s ℚ)) Cn_u(metUnif‘𝐷)) = ((UnifSt‘(ℂ_fld ↾_s ℚ)) Cn_u(UnifSt‘𝑅)))
54	50, 53	eleqtrd 2860	. . 3 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ ((UnifSt‘(ℂ_fld ↾_s ℚ)) Cn_u(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 ℂ_fldccnfld 20142 ℤModczlm 20245 chrcchr 20246 ℝ_fldcrefld 20347 TopSpctps 21144 clsccl 21230 Cn ccn 21436 Hauscha 21520 CnExtccnext 22271 UnifStcuss 22465 UnifSpcusp 22466 Cn_ucucn 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

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