rrhcn - Mathbox for Thierry Arnoux

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Theorem rrhcn 34181

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 24645	. . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
3		ngpxms 24584	. . . . 5 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ∞MetSp)
5		xmstps 24436	. . . 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 34180	. . 3 ⊢ (𝑅 ∈ TopSp → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
10	6, 9	syl 17	. 2 ⊢ (𝜑 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
11		rebase 21581	. . 3 ⊢ ℝ = (Base‘ℝ_fld)
12		rrhf.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
13		retopn 25364	. . . 4 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝ_fld)
14	7, 13	eqtri 2762	. . 3 ⊢ 𝐽 = (TopOpen‘ℝ_fld)
15		eqid 2739	. . 3 ⊢ (UnifSt‘ℝ_fld) = (UnifSt‘ℝ_fld)
16		df-refld 21580	. . . . . 6 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
17	16	oveq1i 7366	. . . . 5 ⊢ (ℝ_fld ↾_s ℚ) = ((ℂ_fld ↾_s ℝ) ↾_s ℚ)
18		reex 11120	. . . . . 6 ⊢ ℝ ∈ V
19		qssre 12900	. . . . . 6 ⊢ ℚ ⊆ ℝ
20		ressabs 17209	. . . . . 6 ⊢ ((ℝ ∈ V ∧ ℚ ⊆ ℝ) → ((ℂ_fld ↾_s ℝ) ↾_s ℚ) = (ℂ_fld ↾_s ℚ))
21	18, 19, 20	mp2an 698	. . . . 5 ⊢ ((ℂ_fld ↾_s ℝ) ↾_s ℚ) = (ℂ_fld ↾_s ℚ)
22	17, 21	eqtr2i 2763	. . . 4 ⊢ (ℂ_fld ↾_s ℚ) = (ℝ_fld ↾_s ℚ)
23	22	fveq2i 6830	. . 3 ⊢ (UnifSt‘(ℂ_fld ↾_s ℚ)) = (UnifSt‘(ℝ_fld ↾_s ℚ))
24		eqid 2739	. . 3 ⊢ (UnifSt‘𝑅) = (UnifSt‘𝑅)
25		recms 25365	. . . . 5 ⊢ ℝ_fld ∈ CMetSp
26		cmsms 25333	. . . . 5 ⊢ (ℝ_fld ∈ CMetSp → ℝ_fld ∈ MetSp)
27		mstps 24438	. . . . 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 25367	. . . 4 ⊢ ℝ_fld ∈ CUnifSp
31		cuspusp 24282	. . . 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 24434	. . . . 5 ⊢ (𝑅 ∈ ∞MetSp → 𝐾 = (MetOpen‘𝐷))
36	4, 35	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 = (MetOpen‘𝐷))
37	12, 34	xmsxmet 24439	. . . . 5 ⊢ (𝑅 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝐵))
38		eqid 2739	. . . . . 6 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
39	38	methaus 24503	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ Haus)
40	4, 37, 39	3syl 18	. . . 4 ⊢ (𝜑 → (MetOpen‘𝐷) ∈ Haus)
41	36, 40	eqeltrd 2839	. . 3 ⊢ (𝜑 → 𝐾 ∈ Haus)
42	19	a1i 11	. . 3 ⊢ (𝜑 → ℚ ⊆ ℝ)
43		eqid 2739	. . . . 5 ⊢ (ℂ_fld ↾_s ℚ) = (ℂ_fld ↾_s ℚ)
44		eqid 2739	. . . . 5 ⊢ (UnifSt‘(ℂ_fld ↾_s ℚ)) = (UnifSt‘(ℂ_fld ↾_s ℚ))
45	34	fveq2i 6830	. . . . 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 34176	. . . 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 2745	. . . . 5 ⊢ (𝜑 → (metUnif‘𝐷) = (UnifSt‘𝑅))
53	52	oveq2d 7372	. . . 4 ⊢ (𝜑 → ((UnifSt‘(ℂ_fld ↾_s ℚ)) Cn_u(metUnif‘𝐷)) = ((UnifSt‘(ℂ_fld ↾_s ℚ)) Cn_u(UnifSt‘𝑅)))
54	50, 53	eleqtrd 2841	. . 3 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ ((UnifSt‘(ℂ_fld ↾_s ℚ)) Cn_u(UnifSt‘𝑅)))
55	7	fveq2i 6830	. . . . . 6 ⊢ (cls‘𝐽) = (cls‘(topGen‘ran (,)))
56	55	fveq1i 6828	. . . . 5 ⊢ ((cls‘𝐽)‘ℚ) = ((cls‘(topGen‘ran (,)))‘ℚ)
57		qdensere 24752	. . . . 5 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ
58	56, 57	eqtri 2762	. . . 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 24286	. 2 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ (𝐽 Cn 𝐾))
61	10, 60	eqeltrd 2839	1 ⊢ (𝜑 → (ℝHom‘𝑅) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1547 ∈ wcel 2119 Vcvv 3431 ⊆ wss 3883 × cxp 5616 ran crn 5619 ↾ cres 5620 ‘cfv 6485 (class class class)co 7356 ℝcr 11028 0cc0 11029 ℚcq 12889 (,)cioo 13289 Basecbs 17170 ↾_s cress 17191 distcds 17220 TopOpenctopn 17375 topGenctg 17391 DivRingcdr 20701 ∞Metcxmet 21332 MetOpencmopn 21337 metUnifcmetu 21338 ℂ_fldccnfld 21347 ℤModczlm 21475 chrcchr 21476 ℝ_fldcrefld 21579 TopSpctps 22915 clsccl 23001 Cn ccn 23207 Hauscha 23291 CnExtccnext 24042 UnifStcuss 24236 UnifSpcusp 24237 Cn_ucucn 24257 CUnifSpccusp 24279 ∞MetSpcxms 24300 MetSpcms 24301 NrmGrpcngp 24560 NrmRingcnrg 24562 NrmModcnlm 24563 CMetSpccms 25317 ℚHomcqqh 34154 ℝHomcrrh 34177

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16213 df-gcd 16455 df-numer 16696 df-denom 16697 df-gz 16892 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-ghm 19179 df-cntz 19283 df-od 19494 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-rhm 20443 df-nzr 20485 df-subrng 20518 df-subrg 20542 df-drng 20703 df-abv 20781 df-lmod 20852 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-metu 21346 df-cnfld 21348 df-zring 21422 df-zrh 21478 df-zlm 21479 df-chr 21480 df-refld 21580 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-cn 23210 df-cnp 23211 df-haus 23298 df-reg 23299 df-cmp 23370 df-tx 23545 df-hmeo 23738 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-fcls 23924 df-cnext 24043 df-ust 24184 df-utop 24214 df-uss 24239 df-usp 24240 df-ucn 24258 df-cfilu 24269 df-cusp 24280 df-xms 24303 df-ms 24304 df-tms 24305 df-nm 24565 df-ngp 24566 df-nrg 24568 df-nlm 24569 df-cncf 24863 df-cfil 25240 df-cmet 25242 df-cms 25320 df-qqh 34155 df-rrh 34179

This theorem is referenced by: rrhf 34182 rrhcne 34197

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