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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 22385 | . 2 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) | |
2 | metxmet 21949 | . 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 1977 ∘ ccom 5042 ‘cfv 5804 ℂcc 9813 − cmin 10145 abscabs 13822 ∞Metcxmt 19552 Metcme 19553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-xadd 11823 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-xmet 19560 df-met 19561 |
This theorem is referenced by: cnbl0 22387 cnfldms 22389 cnfldtopn 22395 cnfldhaus 22398 blcvx 22409 tgioo2 22414 recld2 22425 zdis 22427 reperflem 22429 addcnlem 22475 divcn 22479 iitopon 22490 dfii3 22494 cncfmet 22519 cncfcn 22520 cnheibor 22562 cnllycmp 22563 ipcn 22853 lmclim 22909 cnflduss 22960 reust 22977 ellimc3 23449 dvlipcn 23561 dvlip2 23562 dv11cn 23568 lhop1lem 23580 ftc1lem6 23608 ulmdvlem1 23958 ulmdvlem3 23960 psercn 23984 pserdvlem2 23986 pserdv 23987 abelthlem2 23990 abelthlem3 23991 abelthlem5 23993 abelthlem7 23996 abelth 23999 dvlog2lem 24198 dvlog2 24199 efopnlem2 24203 efopn 24204 logtayl 24206 logtayl2 24208 cxpcn3 24289 rlimcnp 24492 xrlimcnp 24495 efrlim 24496 lgamucov 24564 lgamcvg2 24581 ftalem3 24601 smcnlem 26936 hhcnf 28148 tpr2rico 29286 qqhucn 29364 blscon 30480 cnllyscon 30481 ftc1cnnc 32654 cntotbnd 32765 reheibor 32808 binomcxplemdvbinom 37574 binomcxplemnotnn0 37577 iooabslt 38568 limcrecl 38696 islpcn 38706 stirlinglem5 38971 |
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