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Mirrors > Home > MPE Home > Th. List > cnmetdval | Structured version Visualization version GIF version |
Description: Value of the distance function of the metric space of complex numbers. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
cnmetdval.1 | ⊢ 𝐷 = (abs ∘ − ) |
Ref | Expression |
---|---|
cnmetdval | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subf 10446 | . . 3 ⊢ − :(ℂ × ℂ)⟶ℂ | |
2 | opelxpi 5293 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 〈𝐴, 𝐵〉 ∈ (ℂ × ℂ)) | |
3 | fvco3 6425 | . . 3 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ 〈𝐴, 𝐵〉 ∈ (ℂ × ℂ)) → ((abs ∘ − )‘〈𝐴, 𝐵〉) = (abs‘( − ‘〈𝐴, 𝐵〉))) | |
4 | 1, 2, 3 | sylancr 698 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs ∘ − )‘〈𝐴, 𝐵〉) = (abs‘( − ‘〈𝐴, 𝐵〉))) |
5 | df-ov 6804 | . . 3 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
6 | cnmetdval.1 | . . . 4 ⊢ 𝐷 = (abs ∘ − ) | |
7 | 6 | fveq1i 6341 | . . 3 ⊢ (𝐷‘〈𝐴, 𝐵〉) = ((abs ∘ − )‘〈𝐴, 𝐵〉) |
8 | 5, 7 | eqtri 2770 | . 2 ⊢ (𝐴𝐷𝐵) = ((abs ∘ − )‘〈𝐴, 𝐵〉) |
9 | df-ov 6804 | . . 3 ⊢ (𝐴 − 𝐵) = ( − ‘〈𝐴, 𝐵〉) | |
10 | 9 | fveq2i 6343 | . 2 ⊢ (abs‘(𝐴 − 𝐵)) = (abs‘( − ‘〈𝐴, 𝐵〉)) |
11 | 4, 8, 10 | 3eqtr4g 2807 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1620 ∈ wcel 2127 〈cop 4315 × cxp 5252 ∘ ccom 5258 ⟶wf 6033 ‘cfv 6037 (class class class)co 6801 ℂcc 10097 − cmin 10429 abscabs 14144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-op 4316 df-uni 4577 df-iun 4662 df-br 4793 df-opab 4853 df-mpt 4870 df-id 5162 df-po 5175 df-so 5176 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-1st 7321 df-2nd 7322 df-er 7899 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10239 df-mnf 10240 df-ltxr 10242 df-sub 10431 |
This theorem is referenced by: cnmet 22747 cnbl0 22749 cnblcld 22750 cnfldnm 22754 remetdval 22764 blcvx 22773 recld2 22789 zdis 22791 reperflem 22793 addcnlem 22839 divcn 22843 cncfmet 22883 cnheibor 22926 cnllycmp 22927 ipcn 23216 lmclim 23272 cncmet 23290 ovolfsval 23410 ellimc3 23813 lhop1lem 23946 ftc1lem6 23974 ulmdvlem1 24324 psercn 24350 pserdvlem2 24352 abelthlem2 24356 abelthlem3 24357 abelthlem5 24359 abelthlem7 24362 abelth 24365 dvlog2lem 24568 efopn 24574 logtayl 24576 logtayl2 24578 cxpcn3 24659 rlimcnp 24862 xrlimcnp 24865 efrlim 24866 lgamucov 24934 lgamcvg2 24951 ftalem3 24971 smcnlem 27832 hhcnf 29044 tpr2rico 30238 qqhcn 30315 qqhucn 30316 ftc1cnnc 33766 cntotbnd 33877 iccbnd 33921 cnmetcoval 39862 iooabslt 40193 limcrecl 40333 islpcn 40343 stirlinglem5 40767 ovolval2lem 41332 ovolval3 41336 |
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