rrhcn - Mathbox for Thierry Arnoux

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Theorem rrhcn 31943

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 23824	. . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
3		ngpxms 23755	. . . . 5 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ∞MetSp)
5		xmstps 23604	. . . 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 31942	. . 3 ⊢ (𝑅 ∈ TopSp → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
10	6, 9	syl 17	. 2 ⊢ (𝜑 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
11		rebase 20809	. . 3 ⊢ ℝ = (Base‘ℝ_fld)
12		rrhf.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
13		retopn 24541	. . . 4 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝ_fld)
14	7, 13	eqtri 2768	. . 3 ⊢ 𝐽 = (TopOpen‘ℝ_fld)
15		eqid 2740	. . 3 ⊢ (UnifSt‘ℝ_fld) = (UnifSt‘ℝ_fld)
16		df-refld 20808	. . . . . 6 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
17	16	oveq1i 7281	. . . . 5 ⊢ (ℝ_fld ↾_s ℚ) = ((ℂ_fld ↾_s ℝ) ↾_s ℚ)
18		reex 10963	. . . . . 6 ⊢ ℝ ∈ V
19		qssre 12698	. . . . . 6 ⊢ ℚ ⊆ ℝ
20		ressabs 16957	. . . . . 6 ⊢ ((ℝ ∈ V ∧ ℚ ⊆ ℝ) → ((ℂ_fld ↾_s ℝ) ↾_s ℚ) = (ℂ_fld ↾_s ℚ))
21	18, 19, 20	mp2an 689	. . . . 5 ⊢ ((ℂ_fld ↾_s ℝ) ↾_s ℚ) = (ℂ_fld ↾_s ℚ)
22	17, 21	eqtr2i 2769	. . . 4 ⊢ (ℂ_fld ↾_s ℚ) = (ℝ_fld ↾_s ℚ)
23	22	fveq2i 6774	. . 3 ⊢ (UnifSt‘(ℂ_fld ↾_s ℚ)) = (UnifSt‘(ℝ_fld ↾_s ℚ))
24		eqid 2740	. . 3 ⊢ (UnifSt‘𝑅) = (UnifSt‘𝑅)
25		recms 24542	. . . . 5 ⊢ ℝ_fld ∈ CMetSp
26		cmsms 24510	. . . . 5 ⊢ (ℝ_fld ∈ CMetSp → ℝ_fld ∈ MetSp)
27		mstps 23606	. . . . 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 24544	. . . 4 ⊢ ℝ_fld ∈ CUnifSp
31		cuspusp 23450	. . . 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 23602	. . . . 5 ⊢ (𝑅 ∈ ∞MetSp → 𝐾 = (MetOpen‘𝐷))
36	4, 35	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 = (MetOpen‘𝐷))
37	12, 34	xmsxmet 23607	. . . . 5 ⊢ (𝑅 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝐵))
38		eqid 2740	. . . . . 6 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
39	38	methaus 23674	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ Haus)
40	4, 37, 39	3syl 18	. . . 4 ⊢ (𝜑 → (MetOpen‘𝐷) ∈ Haus)
41	36, 40	eqeltrd 2841	. . 3 ⊢ (𝜑 → 𝐾 ∈ Haus)
42	19	a1i 11	. . 3 ⊢ (𝜑 → ℚ ⊆ ℝ)
43		eqid 2740	. . . . 5 ⊢ (ℂ_fld ↾_s ℚ) = (ℂ_fld ↾_s ℚ)
44		eqid 2740	. . . . 5 ⊢ (UnifSt‘(ℂ_fld ↾_s ℚ)) = (UnifSt‘(ℂ_fld ↾_s ℚ))
45	34	fveq2i 6774	. . . . 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 31938	. . . 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 2746	. . . . 5 ⊢ (𝜑 → (metUnif‘𝐷) = (UnifSt‘𝑅))
53	52	oveq2d 7287	. . . 4 ⊢ (𝜑 → ((UnifSt‘(ℂ_fld ↾_s ℚ)) Cn_u(metUnif‘𝐷)) = ((UnifSt‘(ℂ_fld ↾_s ℚ)) Cn_u(UnifSt‘𝑅)))
54	50, 53	eleqtrd 2843	. . 3 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ ((UnifSt‘(ℂ_fld ↾_s ℚ)) Cn_u(UnifSt‘𝑅)))
55	7	fveq2i 6774	. . . . . 6 ⊢ (cls‘𝐽) = (cls‘(topGen‘ran (,)))
56	55	fveq1i 6772	. . . . 5 ⊢ ((cls‘𝐽)‘ℚ) = ((cls‘(topGen‘ran (,)))‘ℚ)
57		qdensere 23931	. . . . 5 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ
58	56, 57	eqtri 2768	. . . 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 23454	. 2 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ (𝐽 Cn 𝐾))
61	10, 60	eqeltrd 2841	1 ⊢ (𝜑 → (ℝHom‘𝑅) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ⊆ wss 3892 × cxp 5588 ran crn 5591 ↾ cres 5592 ‘cfv 6432 (class class class)co 7271 ℝcr 10871 0cc0 10872 ℚcq 12687 (,)cioo 13078 Basecbs 16910 ↾_s cress 16939 distcds 16969 TopOpenctopn 17130 topGenctg 17146 DivRingcdr 19989 ∞Metcxmet 20580 MetOpencmopn 20585 metUnifcmetu 20586 ℂ_fldccnfld 20595 ℤModczlm 20700 chrcchr 20701 ℝ_fldcrefld 20807 TopSpctps 22079 clsccl 22167 Cn ccn 22373 Hauscha 22457 CnExtccnext 23208 UnifStcuss 23403 UnifSpcusp 23404 Cn_ucucn 23425 CUnifSpccusp 23447 ∞MetSpcxms 23468 MetSpcms 23469 NrmGrpcngp 23731 NrmRingcnrg 23733 NrmModcnlm 23734 CMetSpccms 24494 ℚHomcqqh 31918 ℝHomcrrh 31939

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 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-tpos 8033 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-er 8481 df-map 8600 df-pm 8601 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-fi 9148 df-sup 9179 df-inf 9180 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-ioo 13082 df-ico 13084 df-icc 13085 df-fz 13239 df-fzo 13382 df-fl 13510 df-mod 13588 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-dvds 15962 df-gcd 16200 df-numer 16437 df-denom 16438 df-gz 16629 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-0g 17150 df-gsum 17151 df-topgen 17152 df-pt 17153 df-prds 17156 df-xrs 17211 df-qtop 17216 df-imas 17217 df-xps 17219 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-mulg 18699 df-subg 18750 df-ghm 18830 df-cntz 18921 df-od 19134 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-cring 19784 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-dvr 19923 df-rnghom 19957 df-drng 19991 df-subrg 20020 df-abv 20075 df-lmod 20123 df-nzr 20527 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-fbas 20592 df-fg 20593 df-metu 20594 df-cnfld 20596 df-zring 20669 df-zrh 20703 df-zlm 20704 df-chr 20705 df-refld 20808 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cld 22168 df-ntr 22169 df-cls 22170 df-nei 22247 df-cn 22376 df-cnp 22377 df-haus 22464 df-reg 22465 df-cmp 22536 df-tx 22711 df-hmeo 22904 df-fil 22995 df-fm 23087 df-flim 23088 df-flf 23089 df-fcls 23090 df-cnext 23209 df-ust 23350 df-utop 23381 df-uss 23406 df-usp 23407 df-ucn 23426 df-cfilu 23437 df-cusp 23448 df-xms 23471 df-ms 23472 df-tms 23473 df-nm 23736 df-ngp 23737 df-nrg 23739 df-nlm 23740 df-cncf 24039 df-cfil 24417 df-cmet 24419 df-cms 24497 df-qqh 31919 df-rrh 31941

This theorem is referenced by: rrhf 31944 rrhcne 31959

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