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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rrhf | Structured version Visualization version GIF version | ||
| Description: If the topology of 𝑅 is Hausdorff, Cauchy sequences have at most one limit, i.e. the canonical homomorphism of ℝ into 𝑅 is a function. (Contributed by Thierry Arnoux, 2-Nov-2017.) | 
| 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)) | 
| Ref | Expression | 
|---|---|
| rrhf | ⊢ (𝜑 → (ℝHom‘𝑅):ℝ⟶𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rrhf.d | . . . 4 ⊢ 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵)) | |
| 2 | eqid 2736 | . . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
| 3 | rrhf.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
| 4 | rrhf.k | . . . 4 ⊢ 𝐾 = (TopOpen‘𝑅) | |
| 5 | rrhf.z | . . . 4 ⊢ 𝑍 = (ℤMod‘𝑅) | |
| 6 | rrhf.1 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ DivRing) | |
| 7 | rrhf.2 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ NrmRing) | |
| 8 | rrhf.3 | . . . 4 ⊢ (𝜑 → 𝑍 ∈ NrmMod) | |
| 9 | rrhf.4 | . . . 4 ⊢ (𝜑 → (chr‘𝑅) = 0) | |
| 10 | rrhf.5 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ CUnifSp) | |
| 11 | rrhf.6 | . . . 4 ⊢ (𝜑 → (UnifSt‘𝑅) = (metUnif‘𝐷)) | |
| 12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | rrhcn 33999 | . . 3 ⊢ (𝜑 → (ℝHom‘𝑅) ∈ ((topGen‘ran (,)) Cn 𝐾)) | 
| 13 | uniretop 24784 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
| 14 | eqid 2736 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
| 15 | 13, 14 | cnf 23255 | . . 3 ⊢ ((ℝHom‘𝑅) ∈ ((topGen‘ran (,)) Cn 𝐾) → (ℝHom‘𝑅):ℝ⟶∪ 𝐾) | 
| 16 | 12, 15 | syl 17 | . 2 ⊢ (𝜑 → (ℝHom‘𝑅):ℝ⟶∪ 𝐾) | 
| 17 | nrgngp 24684 | . . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp) | |
| 18 | ngpxms 24615 | . . . . 5 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp) | |
| 19 | 7, 17, 18 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ ∞MetSp) | 
| 20 | xmstps 24464 | . . . 4 ⊢ (𝑅 ∈ ∞MetSp → 𝑅 ∈ TopSp) | |
| 21 | 3, 4 | tpsuni 22943 | . . . 4 ⊢ (𝑅 ∈ TopSp → 𝐵 = ∪ 𝐾) | 
| 22 | 19, 20, 21 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐵 = ∪ 𝐾) | 
| 23 | 22 | feq3d 6722 | . 2 ⊢ (𝜑 → ((ℝHom‘𝑅):ℝ⟶𝐵 ↔ (ℝHom‘𝑅):ℝ⟶∪ 𝐾)) | 
| 24 | 16, 23 | mpbird 257 | 1 ⊢ (𝜑 → (ℝHom‘𝑅):ℝ⟶𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∪ cuni 4906 × cxp 5682 ran crn 5685 ↾ cres 5686 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ℝcr 11155 0cc0 11156 (,)cioo 13388 Basecbs 17248 distcds 17307 TopOpenctopn 17467 topGenctg 17483 DivRingcdr 20730 metUnifcmetu 21356 ℤModczlm 21512 chrcchr 21513 TopSpctps 22939 Cn ccn 23233 UnifStcuss 24263 CUnifSpccusp 24307 ∞MetSpcxms 24328 NrmGrpcngp 24591 NrmRingcnrg 24593 NrmModcnlm 24594 ℝHomcrrh 33995 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-dvds 16292 df-gcd 16533 df-numer 16773 df-denom 16774 df-gz 16969 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-ghm 19232 df-cntz 19336 df-od 19547 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-dvr 20402 df-rhm 20473 df-nzr 20514 df-subrng 20547 df-subrg 20571 df-drng 20732 df-abv 20811 df-lmod 20861 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-metu 21364 df-cnfld 21366 df-zring 21459 df-zrh 21515 df-zlm 21516 df-chr 21517 df-refld 21624 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-cn 23236 df-cnp 23237 df-haus 23324 df-reg 23325 df-cmp 23396 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-fcls 23950 df-cnext 24069 df-ust 24210 df-utop 24241 df-uss 24266 df-usp 24267 df-ucn 24286 df-cfilu 24297 df-cusp 24308 df-xms 24331 df-ms 24332 df-tms 24333 df-nm 24596 df-ngp 24597 df-nrg 24599 df-nlm 24600 df-cncf 24905 df-cfil 25290 df-cmet 25292 df-cms 25370 df-qqh 33973 df-rrh 33997 | 
| This theorem is referenced by: rrhfe 34014 sitgclg 34345 | 
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