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Mirrors > Home > MPE Home > Th. List > metdcn2 | Structured version Visualization version GIF version |
Description: The metric function of a metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 4-Sep-2015.) |
Ref | Expression |
---|---|
xmetdcn2.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
metdcn2.2 | ⊢ 𝐾 = (topGen‘ran (,)) |
Ref | Expression |
---|---|
metdcn2 | ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metxmet 22637 | . . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
2 | xmetdcn2.1 | . . . . 5 ⊢ 𝐽 = (MetOpen‘𝐷) | |
3 | eqid 2772 | . . . . 5 ⊢ (ordTop‘ ≤ ) = (ordTop‘ ≤ ) | |
4 | 2, 3 | xmetdcn 23139 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ((𝐽 ×t 𝐽) Cn (ordTop‘ ≤ ))) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ ((𝐽 ×t 𝐽) Cn (ordTop‘ ≤ ))) |
6 | letopon 21507 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) | |
7 | metf 22633 | . . . . 5 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) | |
8 | 7 | frnd 6345 | . . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → ran 𝐷 ⊆ ℝ) |
9 | ressxr 10476 | . . . . 5 ⊢ ℝ ⊆ ℝ* | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → ℝ ⊆ ℝ*) |
11 | cnrest2 21588 | . . . 4 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) ∧ ran 𝐷 ⊆ ℝ ∧ ℝ ⊆ ℝ*) → (𝐷 ∈ ((𝐽 ×t 𝐽) Cn (ordTop‘ ≤ )) ↔ 𝐷 ∈ ((𝐽 ×t 𝐽) Cn ((ordTop‘ ≤ ) ↾t ℝ)))) | |
12 | 6, 8, 10, 11 | mp3an2i 1445 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 ∈ ((𝐽 ×t 𝐽) Cn (ordTop‘ ≤ )) ↔ 𝐷 ∈ ((𝐽 ×t 𝐽) Cn ((ordTop‘ ≤ ) ↾t ℝ)))) |
13 | 5, 12 | mpbid 224 | . 2 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ ((𝐽 ×t 𝐽) Cn ((ordTop‘ ≤ ) ↾t ℝ))) |
14 | metdcn2.2 | . . . 4 ⊢ 𝐾 = (topGen‘ran (,)) | |
15 | eqid 2772 | . . . . 5 ⊢ ((ordTop‘ ≤ ) ↾t ℝ) = ((ordTop‘ ≤ ) ↾t ℝ) | |
16 | 15 | xrtgioo 23107 | . . . 4 ⊢ (topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾t ℝ) |
17 | 14, 16 | eqtri 2796 | . . 3 ⊢ 𝐾 = ((ordTop‘ ≤ ) ↾t ℝ) |
18 | 17 | oveq2i 6981 | . 2 ⊢ ((𝐽 ×t 𝐽) Cn 𝐾) = ((𝐽 ×t 𝐽) Cn ((ordTop‘ ≤ ) ↾t ℝ)) |
19 | 13, 18 | syl6eleqr 2871 | 1 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1507 ∈ wcel 2048 ⊆ wss 3825 × cxp 5398 ran crn 5401 ‘cfv 6182 (class class class)co 6970 ℝcr 10326 ℝ*cxr 10465 ≤ cle 10467 (,)cioo 12547 ↾t crest 16540 topGenctg 16557 ordTopcordt 16618 ∞Metcxmet 20222 Metcmet 20223 MetOpencmopn 20227 TopOnctopon 21212 Cn ccn 21526 ×t ctx 21862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7494 df-2nd 7495 df-supp 7627 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-er 8081 df-ec 8083 df-map 8200 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-fsupp 8621 df-fi 8662 df-sup 8693 df-inf 8694 df-oi 8761 df-card 9154 df-cda 9380 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-q 12156 df-rp 12198 df-xneg 12317 df-xadd 12318 df-xmul 12319 df-ioo 12551 df-ioc 12552 df-ico 12553 df-icc 12554 df-fz 12702 df-fzo 12843 df-seq 13178 df-exp 13238 df-hash 13499 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-sca 16427 df-vsca 16428 df-ip 16429 df-tset 16430 df-ple 16431 df-ds 16433 df-hom 16435 df-cco 16436 df-rest 16542 df-topn 16543 df-0g 16561 df-gsum 16562 df-topgen 16563 df-pt 16564 df-prds 16567 df-ordt 16620 df-xrs 16621 df-qtop 16626 df-imas 16627 df-xps 16629 df-mre 16705 df-mrc 16706 df-acs 16708 df-ps 17658 df-tsr 17659 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-submnd 17794 df-mulg 18002 df-cntz 18208 df-cmn 18658 df-psmet 20229 df-xmet 20230 df-met 20231 df-bl 20232 df-mopn 20233 df-top 21196 df-topon 21213 df-topsp 21235 df-bases 21248 df-cn 21529 df-cnp 21530 df-tx 21864 df-hmeo 22057 df-xms 22623 df-tms 22625 |
This theorem is referenced by: metdcn 23141 msdcn 23142 |
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