cnxmet - Metamath Proof Explorer

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Theorem cnxmet 24717

Description: The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)

**Assertion**
Ref	Expression
cnxmet	⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)

**Proof of Theorem cnxmet**
Step	Hyp	Ref	Expression
1		cnmet 24716	. 2 ⊢ (abs ∘ − ) ∈ (Met‘ℂ)
2		metxmet 24268	. 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 2098 ∘ ccom 5686 ‘cfv 6553 ℂcc 11146 − cmin 11484 abscabs 15223 ∞Metcxmet 21278 Metcmet 21279

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-sup 9475 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-n0 12513 df-z 12599 df-uz 12863 df-rp 13017 df-xadd 13135 df-seq 14009 df-exp 14069 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-xmet 21286 df-met 21287

This theorem is referenced by: cnbl0 24718 cnfldms 24720 cnfldtopn 24726 cnfldhaus 24729 blcvx 24742 tgioo2 24747 recld2 24758 zdis 24760 reperflem 24762 addcnlem 24808 divcnOLD 24812 divcn 24814 iitopon 24827 dfii3 24831 cncfmet 24857 cncfcn 24858 cnheibor 24909 cnllycmp 24910 ipcn 25202 lmclim 25259 cnflduss 25312 reust 25337 ellimc3 25836 dvlipcn 25955 dvlip2 25956 dv11cn 25962 lhop1lem 25974 ftc1lem6 26004 ulmdvlem1 26364 ulmdvlem3 26366 psercn 26391 pserdvlem2 26393 pserdv 26394 abelthlem2 26397 abelthlem3 26398 abelthlem5 26400 abelthlem7 26403 abelth 26406 dvlog2lem 26614 dvlog2 26615 efopnlem2 26619 efopn 26620 logtayl 26622 logtayl2 26624 cxpcn3 26711 rlimcnp 26925 xrlimcnp 26928 efrlim 26929 efrlimOLD 26930 lgamucov 26998 lgamcvg2 27015 ftalem3 27035 smcnlem 30535 hhcnf 31743 tpr2rico 33554 qqhucn 33634 blsconn 34895 cnllysconn 34896 ftc1cnnc 37206 cntotbnd 37310 reheibor 37353 binomcxplemdvbinom 43839 binomcxplemnotnn0 43842 iooabslt 44931 limcrecl 45064 islpcn 45074 stirlinglem5 45513

	
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