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Mirrors > Home > MPE Home > Th. List > cnxmet | Structured version Visualization version GIF version |
Description: The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
cnxmet | ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmet 23073 | . 2 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) | |
2 | metxmet 22637 | . 2 ⊢ ((abs ∘ − ) ∈ (Met‘ℂ) → (abs ∘ − ) ∈ (∞Met‘ℂ)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2048 ∘ ccom 5404 ‘cfv 6182 ℂcc 10325 − cmin 10662 abscabs 14444 ∞Metcxmet 20222 Metcmet 20223 |
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-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-iun 4788 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-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-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-sup 8693 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-n0 11701 df-z 11787 df-uz 12052 df-rp 12198 df-xadd 12318 df-seq 13178 df-exp 13238 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-xmet 20230 df-met 20231 |
This theorem is referenced by: cnbl0 23075 cnfldms 23077 cnfldtopn 23083 cnfldhaus 23086 blcvx 23099 tgioo2 23104 recld2 23115 zdis 23117 reperflem 23119 addcnlem 23165 divcn 23169 iitopon 23180 dfii3 23184 cncfmet 23209 cncfcn 23210 cnheibor 23252 cnllycmp 23253 ipcn 23542 lmclim 23599 cnflduss 23652 reust 23677 ellimc3 24170 dvlipcn 24284 dvlip2 24285 dv11cn 24291 lhop1lem 24303 ftc1lem6 24331 ulmdvlem1 24681 ulmdvlem3 24683 psercn 24707 pserdvlem2 24709 pserdv 24710 abelthlem2 24713 abelthlem3 24714 abelthlem5 24716 abelthlem7 24719 abelth 24722 dvlog2lem 24926 dvlog2 24927 efopnlem2 24931 efopn 24932 logtayl 24934 logtayl2 24936 cxpcn3 25020 rlimcnp 25235 xrlimcnp 25238 efrlim 25239 lgamucov 25307 lgamcvg2 25324 ftalem3 25344 smcnlem 28241 hhcnf 29453 tpr2rico 30756 qqhucn 30834 blsconn 32036 cnllysconn 32037 ftc1cnnc 34355 cntotbnd 34464 reheibor 34507 binomcxplemdvbinom 40045 binomcxplemnotnn0 40048 iooabslt 41151 limcrecl 41287 islpcn 41297 stirlinglem5 41740 |
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