rrhcn - Mathbox for Thierry Arnoux

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Theorem rrhcn 34679

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 24990	. . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
3		ngpxms 24929	. . . . 5 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
4	1, 2, 3	3syl 19	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ∞MetSp)
5		xmstps 24781	. . . 4 ⊢ (𝑅 ∈ ∞MetSp → 𝑅 ∈ TopSp)
6	4, 5	syl 18	. . 3 ⊢ (𝜑 → 𝑅 ∈ TopSp)
7		rrhf.j	. . . 4 ⊢ 𝐽 = (topGen‘ran (,))
8		rrhf.k	. . . 4 ⊢ 𝐾 = (TopOpen‘𝑅)
9	7, 8	rrhval 34678	. . 3 ⊢ (𝑅 ∈ TopSp → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
10	6, 9	syl 18	. 2 ⊢ (𝜑 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
11		rebase 21921	. . 3 ⊢ ℝ = (Base‘ℝ_fld)
12		rrhf.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
13		retopn 25709	. . . 4 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝ_fld)
14	7, 13	eqtri 2783	. . 3 ⊢ 𝐽 = (TopOpen‘ℝ_fld)
15		eqid 2760	. . 3 ⊢ (UnifSt‘ℝ_fld) = (UnifSt‘ℝ_fld)
16		df-refld 21920	. . . . . 6 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
17	16	oveq1i 7434	. . . . 5 ⊢ (ℝ_fld ↾_s ℚ) = ((ℂ_fld ↾_s ℝ) ↾_s ℚ)
18		reex 11298	. . . . . 6 ⊢ ℝ ∈ V
19		qssre 13093	. . . . . 6 ⊢ ℚ ⊆ ℝ
20		ressabs 17435	. . . . . 6 ⊢ ((ℝ ∈ V ∧ ℚ ⊆ ℝ) → ((ℂ_fld ↾_s ℝ) ↾_s ℚ) = (ℂ_fld ↾_s ℚ))
21	18, 19, 20	mp2an 705	. . . . 5 ⊢ ((ℂ_fld ↾_s ℝ) ↾_s ℚ) = (ℂ_fld ↾_s ℚ)
22	17, 21	eqtr2i 2784	. . . 4 ⊢ (ℂ_fld ↾_s ℚ) = (ℝ_fld ↾_s ℚ)
23	22	fveq2i 6893	. . 3 ⊢ (UnifSt‘(ℂ_fld ↾_s ℚ)) = (UnifSt‘(ℝ_fld ↾_s ℚ))
24		eqid 2760	. . 3 ⊢ (UnifSt‘𝑅) = (UnifSt‘𝑅)
25		recms 25710	. . . . 5 ⊢ ℝ_fld ∈ CMetSp
26		cmsms 25678	. . . . 5 ⊢ (ℝ_fld ∈ CMetSp → ℝ_fld ∈ MetSp)
27		mstps 24783	. . . . 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 25712	. . . 4 ⊢ ℝ_fld ∈ CUnifSp
31		cuspusp 24627	. . . 4 ⊢ (ℝ_fld ∈ CUnifSp → ℝ_fld ∈ UnifSp)
32	30, 31	mp1i 14	. . 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 24779	. . . . 5 ⊢ (𝑅 ∈ ∞MetSp → 𝐾 = (MetOpen‘𝐷))
36	4, 35	syl 18	. . . 4 ⊢ (𝜑 → 𝐾 = (MetOpen‘𝐷))
37	12, 34	xmsxmet 24784	. . . . 5 ⊢ (𝑅 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝐵))
38		eqid 2760	. . . . . 6 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
39	38	methaus 24848	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ Haus)
40	4, 37, 39	3syl 19	. . . 4 ⊢ (𝜑 → (MetOpen‘𝐷) ∈ Haus)
41	36, 40	eqeltrd 2860	. . 3 ⊢ (𝜑 → 𝐾 ∈ Haus)
42	19	a1i 11	. . 3 ⊢ (𝜑 → ℚ ⊆ ℝ)
43		eqid 2760	. . . . 5 ⊢ (ℂ_fld ↾_s ℚ) = (ℂ_fld ↾_s ℚ)
44		eqid 2760	. . . . 5 ⊢ (UnifSt‘(ℂ_fld ↾_s ℚ)) = (UnifSt‘(ℂ_fld ↾_s ℚ))
45	34	fveq2i 6893	. . . . 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 34674	. . . 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 2766	. . . . 5 ⊢ (𝜑 → (metUnif‘𝐷) = (UnifSt‘𝑅))
53	52	oveq2d 7440	. . . 4 ⊢ (𝜑 → ((UnifSt‘(ℂ_fld ↾_s ℚ)) Cn_u(metUnif‘𝐷)) = ((UnifSt‘(ℂ_fld ↾_s ℚ)) Cn_u(UnifSt‘𝑅)))
54	50, 53	eleqtrd 2862	. . 3 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ ((UnifSt‘(ℂ_fld ↾_s ℚ)) Cn_u(UnifSt‘𝑅)))
55	7	fveq2i 6893	. . . . . 6 ⊢ (cls‘𝐽) = (cls‘(topGen‘ran (,)))
56	55	fveq1i 6891	. . . . 5 ⊢ ((cls‘𝐽)‘ℚ) = ((cls‘(topGen‘ran (,)))‘ℚ)
57		qdensere 25097	. . . . 5 ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ
58	56, 57	eqtri 2783	. . . 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 24631	. 2 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)) ∈ (𝐽 Cn 𝐾))
61	10, 60	eqeltrd 2860	1 ⊢ (𝜑 → (ℝHom‘𝑅) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∈ wcel 2144 Vcvv 3450 ⊆ wss 3896 × cxp 5649 ran crn 5652 ↾ cres 5653 ‘cfv 6543 (class class class)co 7424 ℝcr 11206 0cc0 11207 ℚcq 13082 (,)cioo 13483 Basecbs 17396 ↾_s cress 17417 distcds 17446 TopOpenctopn 17601 topGenctg 17617 DivRingcdr 20989 ∞Metcxmet 21672 MetOpencmopn 21677 metUnifcmetu 21678 ℂ_fldccnfld 21687 ℤModczlm 21815 chrcchr 21816 ℝ_fldcrefld 21919 TopSpctps 23259 clsccl 23345 Cn ccn 23551 Hauscha 23635 CnExtccnext 24387 UnifStcuss 24581 UnifSpcusp 24582 Cn_ucucn 24602 CUnifSpccusp 24624 ∞MetSpcxms 24645 MetSpcms 24646 NrmGrpcngp 24905 NrmRingcnrg 24907 NrmModcnlm 24908 CMetSpccms 25662 ℚHomcqqh 34652 ℝHomcrrh 34675

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285 ax-addf 11286 ax-mulf 11287

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-om 7882 df-1st 8005 df-2nd 8006 df-supp 8185 df-tpos 8251 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-2o 8485 df-er 8725 df-map 8857 df-pm 8858 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9362 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9511 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-q 13083 df-rp 13128 df-xneg 13248 df-xadd 13249 df-xmul 13250 df-ioo 13487 df-ico 13489 df-icc 13490 df-fz 13647 df-fzo 13796 df-fl 13939 df-mod 14017 df-seq 14152 df-exp 14212 df-hash 14482 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-dvds 16432 df-gcd 16674 df-numer 16920 df-denom 16921 df-gz 17117 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-starv 17452 df-sca 17453 df-vsca 17454 df-ip 17455 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-hom 17461 df-cco 17462 df-rest 17602 df-topn 17603 df-0g 17621 df-gsum 17622 df-topgen 17623 df-pt 17624 df-prds 17627 df-xrs 17683 df-qtop 17688 df-imas 17689 df-xps 17691 df-mre 17765 df-mrc 17766 df-acs 17768 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-mhm 18987 df-submnd 18988 df-grp 19156 df-minusg 19157 df-sbg 19158 df-mulg 19287 df-subg 19342 df-ghm 19437 df-cntz 19540 df-od 19751 df-cmn 20005 df-abl 20006 df-mgp 20370 df-rng 20384 df-ur 20417 df-ring 20470 df-cring 20471 df-oppr 20576 df-dvdsr 20596 df-unit 20597 df-invr 20627 df-dvr 20640 df-rhm 20711 df-nzr 20772 df-subrng 20807 df-subrg 20831 df-drng 20991 df-abv 21075 df-lmod 21146 df-psmet 21679 df-xmet 21680 df-met 21681 df-bl 21682 df-mopn 21683 df-fbas 21684 df-fg 21685 df-metu 21686 df-cnfld 21688 df-zring 21762 df-zrh 21818 df-zlm 21819 df-chr 21820 df-refld 21920 df-top 23221 df-topon 23238 df-topsp 23260 df-bases 23273 df-cld 23346 df-ntr 23347 df-cls 23348 df-nei 23425 df-cn 23554 df-cnp 23555 df-haus 23642 df-reg 23643 df-cmp 23714 df-tx 23890 df-hmeo 24083 df-fil 24174 df-fm 24266 df-flim 24267 df-flf 24268 df-fcls 24269 df-cnext 24388 df-ust 24529 df-utop 24559 df-uss 24584 df-usp 24585 df-ucn 24603 df-cfilu 24614 df-cusp 24625 df-xms 24648 df-ms 24649 df-tms 24650 df-nm 24910 df-ngp 24911 df-nrg 24913 df-nlm 24914 df-cncf 25208 df-cfil 25585 df-cmet 25587 df-cms 25665 df-qqh 34653 df-rrh 34677

This theorem is used by: rrhf 34680 rrhcne 34695

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