rrhcn - Mathbox for Thierry Arnoux

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Theorem rrhcn 33987

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 24550	. . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
3		ngpxms 24489	. . . . 5 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ∞MetSp)
5		xmstps 24341	. . . 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 33986	. . 3 ⊢ (𝑅 ∈ TopSp → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
10	6, 9	syl 17	. 2 ⊢ (𝜑 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
11		rebase 21515	. . 3 ⊢ ℝ = (Base‘ℝ_fld)
12		rrhf.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
13		retopn 25279	. . . 4 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝ_fld)
14	7, 13	eqtri 2752	. . 3 ⊢ 𝐽 = (TopOpen‘ℝ_fld)
15		eqid 2729	. . 3 ⊢ (UnifSt‘ℝ_fld) = (UnifSt‘ℝ_fld)
16		df-refld 21514	. . . . . 6 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
17	16	oveq1i 7397	. . . . 5 ⊢ (ℝ_fld ↾_s ℚ) = ((ℂ_fld ↾_s ℝ) ↾_s ℚ)
18		reex 11159	. . . . . 6 ⊢ ℝ ∈ V
19		qssre 12918	. . . . . 6 ⊢ ℚ ⊆ ℝ
20		ressabs 17218	. . . . . 6 ⊢ ((ℝ ∈ V ∧ ℚ ⊆ ℝ) → ((ℂ_fld ↾_s ℝ) ↾_s ℚ) = (ℂ_fld ↾_s ℚ))
21	18, 19, 20	mp2an 692	. . . . 5 ⊢ ((ℂ_fld ↾_s ℝ) ↾_s ℚ) = (ℂ_fld ↾_s ℚ)
22	17, 21	eqtr2i 2753	. . . 4 ⊢ (ℂ_fld ↾_s ℚ) = (ℝ_fld ↾_s ℚ)
23	22	fveq2i 6861	. . 3 ⊢ (UnifSt‘(ℂ_fld ↾_s ℚ)) = (UnifSt‘(ℝ_fld ↾_s ℚ))
24		eqid 2729	. . 3 ⊢ (UnifSt‘𝑅) = (UnifSt‘𝑅)
25		recms 25280	. . . . 5 ⊢ ℝ_fld ∈ CMetSp
26		cmsms 25248	. . . . 5 ⊢ (ℝ_fld ∈ CMetSp → ℝ_fld ∈ MetSp)
27		mstps 24343	. . . . 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 25282	. . . 4 ⊢ ℝ_fld ∈ CUnifSp
31		cuspusp 24187	. . . 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 24339	. . . . 5 ⊢ (𝑅 ∈ ∞MetSp → 𝐾 = (MetOpen‘𝐷))
36	4, 35	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 = (MetOpen‘𝐷))
37	12, 34	xmsxmet 24344	. . . . 5 ⊢ (𝑅 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝐵))
38		eqid 2729	. . . . . 6 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
39	38	methaus 24408	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ Haus)
40	4, 37, 39	3syl 18	. . . 4 ⊢ (𝜑 → (MetOpen‘𝐷) ∈ Haus)
41	36, 40	eqeltrd 2828	. . 3 ⊢ (𝜑 → 𝐾 ∈ Haus)
42	19	a1i 11	. . 3 ⊢ (𝜑 → ℚ ⊆ ℝ)
43		eqid 2729	. . . . 5 ⊢ (ℂ_fld ↾_s ℚ) = (ℂ_fld ↾_s ℚ)
44		eqid 2729	. . . . 5 ⊢ (UnifSt‘(ℂ_fld ↾_s ℚ)) = (UnifSt‘(ℂ_fld ↾_s ℚ))
45	34	fveq2i 6861	. . . . 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 33982	. . . 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 2735	. . . . 5 ⊢ (𝜑 → (metUnif‘𝐷) = (UnifSt‘𝑅))
53	52	oveq2d 7403	. . . 4 ⊢ (𝜑 → ((UnifSt‘(ℂ_fld ↾_s ℚ)) Cn_u(metUnif‘𝐷)) = ((UnifSt‘(ℂ_fld ↾_s ℚ)) Cn_u(UnifSt‘𝑅)))
54	50, 53	eleqtrd 2830	. . 3 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ ((UnifSt‘(ℂ_fld ↾_s ℚ)) Cn_u(UnifSt‘𝑅)))
55	7	fveq2i 6861	. . . . . 6 ⊢ (cls‘𝐽) = (cls‘(topGen‘ran (,)))
56	55	fveq1i 6859	. . . . 5 ⊢ ((cls‘𝐽)‘ℚ) = ((cls‘(topGen‘ran (,)))‘ℚ)
57		qdensere 24657	. . . . 5 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ
58	56, 57	eqtri 2752	. . . 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 24191	. 2 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ (𝐽 Cn 𝐾))
61	10, 60	eqeltrd 2828	1 ⊢ (𝜑 → (ℝHom‘𝑅) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 Vcvv 3447 ⊆ wss 3914 × cxp 5636 ran crn 5639 ↾ cres 5640 ‘cfv 6511 (class class class)co 7387 ℝcr 11067 0cc0 11068 ℚcq 12907 (,)cioo 13306 Basecbs 17179 ↾_s cress 17200 distcds 17229 TopOpenctopn 17384 topGenctg 17400 DivRingcdr 20638 ∞Metcxmet 21249 MetOpencmopn 21254 metUnifcmetu 21255 ℂ_fldccnfld 21264 ℤModczlm 21410 chrcchr 21411 ℝ_fldcrefld 21513 TopSpctps 22819 clsccl 22905 Cn ccn 23111 Hauscha 23195 CnExtccnext 23946 UnifStcuss 24141 UnifSpcusp 24142 Cn_ucucn 24162 CUnifSpccusp 24184 ∞MetSpcxms 24205 MetSpcms 24206 NrmGrpcngp 24465 NrmRingcnrg 24467 NrmModcnlm 24468 CMetSpccms 25232 ℚHomcqqh 33960 ℝHomcrrh 33983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-dvds 16223 df-gcd 16465 df-numer 16705 df-denom 16706 df-gz 16901 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-ghm 19145 df-cntz 19249 df-od 19458 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-rhm 20381 df-nzr 20422 df-subrng 20455 df-subrg 20479 df-drng 20640 df-abv 20718 df-lmod 20768 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-metu 21263 df-cnfld 21265 df-zring 21357 df-zrh 21413 df-zlm 21414 df-chr 21415 df-refld 21514 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-cn 23114 df-cnp 23115 df-haus 23202 df-reg 23203 df-cmp 23274 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-fcls 23828 df-cnext 23947 df-ust 24088 df-utop 24119 df-uss 24144 df-usp 24145 df-ucn 24163 df-cfilu 24174 df-cusp 24185 df-xms 24208 df-ms 24209 df-tms 24210 df-nm 24470 df-ngp 24471 df-nrg 24473 df-nlm 24474 df-cncf 24771 df-cfil 25155 df-cmet 25157 df-cms 25235 df-qqh 33961 df-rrh 33985

This theorem is referenced by: rrhf 33988 rrhcne 34003

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